1.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
2.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
3.
Subtraction
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Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign, for example, in the picture on the right, there are 5 −2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. It is anticommutative, meaning that changing the order changes the sign of the answer and it is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number, subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers, general binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks, subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the system, starting with single digits. Subtraction is written using the minus sign − between the terms, that is, in infix notation, the result is expressed with an equals sign. This is most common in accounting, formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. All of this terminology derives from Latin, subtraction is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub from under and trahere to pull, thus to subtract is to draw from below, take away. Using the gerundive suffix -nd results in subtrahend, thing to be subtracted, likewise from minuere to reduce or diminish, one gets minuend, thing to be diminished. Imagine a line segment of length b with the left end labeled a, starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition, a + b = c, from c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction, c − b = a, now, a line segment labeled with the numbers 1,2, and 3. From position 3, it takes no steps to the left to stay at 3 and it takes 2 steps to the left to get to position 1, so 3 −2 =1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3, to represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number, from 3, it takes 3 steps to the left to get to 0, so 3 −3 =0. But 3 −4 is still invalid since it leaves the line
4.
Measured
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
5.
Logarithmic scale
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A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the strength, sound loudness, light intensity. It is based on orders of magnitude, rather than a linear scale. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch, the top left graph is linear in the X and Y axis, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the left graph. The top right graph uses a scale for just the X axis. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales, the geometric mean of two numbers is midway between the numbers. Before the advent of graphics, logarithmic graph paper was a commonly used scientific tool. If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot, if only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot. Bit Byte Decade John Napier Level Logarithm Logarithmic mean Preferred number Dehaene, Stanislas, Izard, Véronique, Spelke, Elizabeth, Pica, distinct intuitions of the number scale in Western and Amazonian indigene cultures. American Association for the Advancement of Science, why using logarithmic scale to display share prices
6.
Decade (log scale)
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One decade is a factor of 10 difference between two numbers measured on a logarithmic scale. Along with the octave, it is a unit used to describe frequency bands or frequency ratios. It is especially useful when referring to frequencies and when describing frequency response of electronic systems, the factor-of-ten in a decade can be in either direction, so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, log 10 =4 decades How many decades is it from 3.2 GHz to 4.7 MHz. log 10 = −2.83 decades How many decades is one octave. 101 /30 =1.079775 - or each step is 7. 9775% larger than the last. For example, an amplifier will usually have a frequency band ranging from 20 Hz to 20 kHz. Typically the graph for such a representation would begin at 1 Hz and go up to perhaps 100 kHz, to include the full audio band in a standard-sized graph paper. Whereas in the distance on a linear scale, with 10 as the major step-size. Electronic frequency responses are often described in terms of per decade, the example Bode plot shows a slope of -20 dB/decade in the stopband, which means that for every factor-of-ten increase in frequency, the gain decreases by 20 dB
7.
Orders of magnitude (numbers)
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This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Mathematics – Writing, Approximately 10−183,800 is a rough first estimate of the probability that a monkey, however, taking punctuation, capitalization, and spacing into account, the actual probability is far lower, around 10−360,783. Computing, The number 1×10−6176 is equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value, Computing, The number 6. 5×10−4966 is approximately equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE floating-point value. Computing, The number 3. 6×10−4951 is approximately equal to the smallest positive non-zero value that can be represented by a 80-bit x86 double-extended IEEE floating-point value. Computing, The number 1×10−398 is equal to the smallest positive non-zero value that can be represented by a double-precision IEEE decimal floating-point value, Computing, The number 4. 9×10−324 is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value. Computing, The number 1×10−101 is equal to the smallest positive non-zero value that can be represented by a single-precision IEEE decimal floating-point value, Mathematics, The probability in a game of bridge of all four players getting a complete suit is approximately 4. 47×10−28. ISO, yocto- ISO, zepto- Mathematics, The probability of matching 20 numbers for 20 in a game of keno is approximately 2.83 × 10−19. ISO, atto- Mathematics, The probability of rolling snake eyes 10 times in a row on a pair of dice is about 2. 74×10−16. ISO, micro- Mathematics – Poker, The odds of being dealt a flush in poker are 649,739 to 1 against. Mathematics – Poker, The odds of being dealt a flush in poker are 72,192 to 1 against. Mathematics – Poker, The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4 × 10−4. ISO, milli- Mathematics – Poker, The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10−3. Mathematics – Poker, The odds of being dealt a flush in poker are 507.8 to 1 against, Mathematics – Poker, The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10−3. Physics, α =0.007297352570, the fine-structure constant, ISO, deci- Mathematics – Poker, The odds of being dealt only one pair in poker are about 5 to 2 against, for a probability of 0.42. Demography, The population of Monowi, a village in Nebraska. Mathematics, √2 ≈1.414213562373095489, the ratio of the diagonal of a square to its side length. Mathematics, φ ≈1.618033988749895848, the golden ratio Mathematics, the number system understood by most computers, human scale, There are 10 digits on a pair of human hands, and 10 toes on a pair of human feet. Mathematics, The number system used in life, the decimal system, has 10 digits,0,1,2,3,4,5,6,7,8,9
8.
Geometric mean
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In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i. e. for a set of numbers x1, x2. As another example, the mean of the three numbers 4,1, and 1/32 is the cube root of their product, which is 1/2. A geometric mean is used when comparing different items—finding a single figure of merit for these items—when each item has multiple properties that have different numeric ranges. So, a 20% change in environmental sustainability from 4 to 4.8 has the effect on the geometric mean as a 20% change in financial viability from 60 to 72. The geometric mean can be understood in terms of geometry, the geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. The geometric mean applies only to numbers of the same sign, the geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. The above figure uses capital pi notation to show a series of multiplications. For example, in a set of four numbers, the product of 1 ×2 ×3 ×4 is 24, note that the exponent 1 / n on the left side is equivalent to the taking nth root. For example,241 /4 =244, the geometric mean of a data set is less than the data sets arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the mean, a mixture of the two which always lies in between. The geometric mean can also be expressed as the exponential of the mean of logarithms. This is sometimes called the log-average and this is less likely to occur with the sum of the logarithms for each number. Instead, the mean is simply 1 n, where n is the number of steps from the initial to final state. If the values are a 0, …, a n and this is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources. In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of time of computer programs. However, by presenting appropriately normalized values and using the arithmetic mean, however, this reasoning has been questioned
9.
Arithmetic mean
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In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, the term arithmetic mean is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the mean is used frequently in fields such as economics, sociology, and history. For example, per capita income is the average income of a nations population. While the arithmetic mean is used to report central tendencies, it is not a robust statistic. In a more obscure usage, any sequence of values that form a sequence between two numbers x and y can be called arithmetic means between x and y. The arithmetic mean is the most commonly used and readily understood measure of central tendency, in statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the values of each. For example, let us consider the monthly salary of 10 employees of a firm,2500,2700,2400,2300,2550,2650,2750,2450,2600,2400. The arithmetic mean is 2500 +2700 +2400 +2300 +2550 +2650 +2750 +2450 +2600 +240010 =2530, If the data set is a statistical population, then the mean of that population is called the population mean. If the data set is a sample, we call the statistic resulting from this calculation a sample mean. The arithmetic mean of a variable is denoted by a bar, for example as in x ¯. The arithmetic mean has several properties that make it useful, especially as a measure of central tendency and these include, If numbers x 1, …, x n have mean x ¯, then + ⋯ + =0. The mean is the single number for which the residuals sum to zero. If the arithmetic mean of a population of numbers is desired, the arithmetic mean may be contrasted with the median. The median is defined such that half the values are larger than, and half are smaller than, If elements in the sample data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample 1,2,3,4, the average is 2.5, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as 1,2,4,8,16, in this case, the arithmetic average is 6.2 and the median is 4
10.
Scientific notation
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Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
11.
Long and short scales
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Thus, billion means a million millions, trillion means a million billions, and so on. Short scale Every new term greater than million is one thousand times larger than the previous term, thus, billion means a thousand millions, trillion means a thousand billions, and so on. For whole numbers less than a million the two scales are identical. From a thousand million up the two scales diverge, using the words for different numbers, this can cause misunderstanding. Countries where the scale is currently used include most countries in continental Europe and most French-speaking, Spanish-speaking. The short scale is now used in most English-speaking and Arabic-speaking countries, in Brazil, in former Soviet Union, number names are rendered in the language of the country, but are similar everywhere due to shared etymology. Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, for example the Indian numbering system. After several decades of increasing informal British usage of the scale, in 1974 the government of the UK adopted it. With very few exceptions, the British usage and American usage are now identical, the first recorded use of the terms short scale and long scale was by the French mathematician Geneviève Guitel in 1975. At and above a million the same names are used to refer to numbers differing by a factor of an integer power of 1,000. Each scale has a justification to explain the use of each such differing numerical name. The short-scale logic is based on powers of one thousand, whereas the long-scale logic is based on powers of one million, in both scales, the prefix bi- refers to 2 and tri- refers to 3, etc. However only in the scale do the prefixes beyond one million indicate the actual power or exponent. In the short scale, the prefixes refer to one less than the exponent, the word, million, derives from the Old French, milion, from the earlier Old Italian, milione, an intensification of the Latin word, mille, a thousand. That is, a million is a big thousand, much as a great gross is a dozen gross or 12×144 =1728, the word, milliard, or its translation, is found in many European languages and is used in those languages for 109. However, it is unknown in American English, which uses billion, and not used in British English, which preferred to use thousand million before the current usage of billion. The financial term, yard, which derives from milliard, is used on financial markets, as, unlike the term, billion, it is internationally unambiguous and phonetically distinct from million. Likewise, many long scale use the word billiard for one thousand long scale billions
12.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
13.
Mathematical operators and symbols in Unicode
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The Unicode Standard encodes almost all standard characters used in mathematics. Unicode Technical Report #25 provides comprehensive information about the repertoire, their properties. Mathematical operators and symbols are in multiple Unicode blocks, some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a property of Math. The Mathematical Operators block contains characters for mathematical, logical, the Mathematical Alphanumeric Symbols block contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The holes in the ranges are filled by previously defined characters in the Letter like Symbols block shown below. The Letterlike Symbols block includes variables, most alphabetic math symbols are in the Mathematical Alphanumeric Symbols block shown above. The math subset of this block is U+2102, U+2107, U+210A–U+2113, U+2115, U+2118–U+2119, U+2124, U+2128–U+2129, U+212C, U+212F, U+2133, U+2135, U+213C–U+2149, the Miscellaneous Mathematical Symbols-A block contains characters for mathematical, logical, and database notation. The Miscellaneous Mathematical Symbols-B block contains miscellaneous mathematical symbols, including brackets, angles, the Miscellaneous Technical block includes braces and operators. The math subset of this block is U+2308–U+230B, U+2320-U+2321, U+237C, U+239B-U+23B5, 23B7, U+23D0, the Geometric Shapes block contains geometric shape symbols. The math subset of this block is U+25A0–25A1, U+25AE–25B7, U+25BC–25C1, U+25C6–25C7, U+25CA–25CB, U+25CF–25D3, U+25E2, U+25E4, U+25E7–25EC, the Miscellaneous Symbols and Arrows block contains arrows and geometric shapes with various fills. The math subset of this block is U+2B30–2B44 and U+2B47–2B4C, the Arrows block contains line, curve, and semicircle arrows and arrow-like operators. The Supplemental Arrows-A block contains arrows and arrow-like operators, the Supplemental Arrows-B block contains arrows and arrow-like operators. The Combining Diacritical Marks for Symbols block contains arrows, dots, enclosures, the math subset of this block is U+20D0–U+20DC, U+20E1, U+20E5–U+20E6, and U+20EB–U+20EF. The Arabic Mathematical Alphabetic Symbols block contains characters used in Arabic mathematical expressions, Mathematical characters also appear in other blocks. Below is a list of characters as of Unicode version 9. Images of glyphs in section 6.3.3 of the Mathematical Markup Language W3C Recommendation
14.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
15.
Mu (letter)
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Mu is the 12th letter of the Greek alphabet. In the system of Greek numerals it has a value of 40, Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water, to become
16.
Decibel
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The decibel is a logarithmic unit used to express the ratio of two values of a physical quantity. One of these values is often a reference value, in which case the decibel is used to express the level of the other value relative to this reference. When used in way, the decibel symbol is often qualified with a suffix that indicates the reference quantity that has been used or some other property of the quantity being measured. For example, dBm indicates a power of one milliwatt. There are two different scales used when expressing a ratio in decibels depending on the nature of the quantities, when expressing power quantities, the number of decibels is ten times the logarithm to base 10 of the ratio of two power quantities. That is, a change in power by a factor of 10 corresponds to a 10 dB change in level, when expressing field quantities, a change in amplitude by a factor of 10 corresponds to a 20 dB change in level. The difference in scales relates to the square law of fields in three-dimensional linear space. The decibel scales differ so that comparisons can be made between related power and field quantities when they are expressed in decibels. The definition of the decibel is based on the measurement of power in telephony of the early 20th century in the Bell System in the United States. One decibel is one tenth of one bel, named in honor of Alexander Graham Bell, however, today, the decibel is used for a wide variety of measurements in science and engineering, most prominently in acoustics, electronics, and control theory. In electronics, the gains of amplifiers, attenuation of signals, the decibel originates from methods used to quantify signal loss in telegraph and telephone circuits. The unit for loss was originally Miles of Standard Cable, the standard telephone cable implied was a cable having uniformly distributed resistance of 88 ohms per loop mile and uniformly distributed shunt capacitance of 0.054 microfarad per mile. 1 TU was defined such that the number of TUs was ten times the logarithm of the ratio of measured power to a reference power level. The definition was conveniently chosen such that 1 TU approximated 1 MSC, in 1928, the Bell system renamed the TU into the decibel, being one tenth of a newly defined unit for the base-10 logarithm of the power ratio. It was named the bel, in honor of the telecommunications pioneer Alexander Graham Bell, the bel is seldom used, as the decibel was the proposed working unit. However, the decibel is recognized by international bodies such as the International Electrotechnical Commission. The term field quantity is deprecated by ISO 80000-1, which favors root-power, in spite of their widespread use, suffixes are not recognized by the IEC or ISO. The ISO Standard 80000-3,2006 defines the following quantities, the decibel is one-tenth of a bel,1 dB =0.1 B
17.
Metre
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator
18.
Hectare
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The hectare is an SI accepted metric system unit of area equal to 100 ares and primarily used in the measurement of land as a metric replacement for the imperial acre. An acre is about 0.405 hectare and one hectare contains about 2.47 acres, in 1795, when the metric system was introduced, the are was defined as 100 square metres and the hectare was thus 100 ares or 1⁄100 km2. When the metric system was further rationalised in 1960, resulting in the International System of Units, the are was not included as a recognised unit. The hectare, however, remains as a non-SI unit accepted for use with the SI units, the metric system of measurement was first given a legal basis in 1795 by the French Revolutionary government. At the first meeting of the CGPM in 1889 when a new standard metre, manufactured by Johnson Matthey & Co of London was adopted, in 1960, when the metric system was updated as the International System of Units, the are did not receive international recognition. The units that were catalogued replicated the recommendations of the CGPM, many farmers, especially older ones, still use the acre for everyday calculations, and convert to hectares only for official paperwork. Farm fields can have long histories which are resistant to change, with names such as the six acre field stretching back hundreds of years. The names centiare, deciare, decare and hectare are derived by adding the standard metric prefixes to the base unit of area. The centiare is a synonym for one square metre, the deciare is ten square metres. The are is a unit of area, equal to 100 square metres and it was defined by older forms of the metric system, but is now outside of the modern International System of Units. It is commonly used to measure real estate, in particular in Indonesia, India, and in French-, Portuguese-, Slovakian-, Serbian-, Czech-, Polish-, Dutch-, in Russia and other former Soviet Union states, the are is called sotka. It is used to describe the size of suburban dacha or allotment garden plots or small city parks where the hectare would be too large, the decare is derived from deka, the prefix for 10 and are, and is equal to 10 ares or 1000 square metres. It is used in Norway and in the former Ottoman areas of the Middle East, the hectare, although not strictly a unit of SI, is the only named unit of area that is accepted for use within the SI. The United Kingdom, United States, Burma, and to some extent Canada instead use the acre, others, such as South Africa, published conversion factors which were to be used particularly when preparing consolidation diagrams by compilation. In many countries, metrication redefined or clarified existing measures in terms of metric units, non-SI units accepted for use with the International System of Units
19.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
20.
Second
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The second is the base unit of time in the International System of Units. It is qualitatively defined as the division of the hour by sixty. SI definition of second is the duration of 9192631770 periods of the corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. Seconds may be measured using a mechanical, electrical or an atomic clock, SI prefixes are combined with the word second to denote subdivisions of the second, e. g. the millisecond, the microsecond, and the nanosecond. Though SI prefixes may also be used to form multiples of the such as kilosecond. The second is also the unit of time in other systems of measurement, the centimetre–gram–second, metre–kilogram–second, metre–tonne–second. Absolute zero implies no movement, and therefore zero external radiation effects, the second thus defined is consistent with the ephemeris second, which was based on astronomical measurements. The realization of the second is described briefly in a special publication from the National Institute of Standards and Technology. 1 international second is equal to, 1⁄60 minute 1⁄3,600 hour 1⁄86,400 day 1⁄31,557,600 Julian year 1⁄, more generally, = 1⁄, the Hellenistic astronomers Hipparchus and Ptolemy subdivided the day into sixty parts. They also used an hour, simple fractions of an hour. No sexagesimal unit of the day was used as an independent unit of time. The modern second is subdivided using decimals - although the third remains in some languages. The earliest clocks to display seconds appeared during the last half of the 16th century, the second became accurately measurable with the development of mechanical clocks keeping mean time, as opposed to the apparent time displayed by sundials. The earliest spring-driven timepiece with a hand which marked seconds is an unsigned clock depicting Orpheus in the Fremersdorf collection. During the 3rd quarter of the 16th century, Taqi al-Din built a clock with marks every 1/5 minute, in 1579, Jost Bürgi built a clock for William of Hesse that marked seconds. In 1581, Tycho Brahe redesigned clocks that displayed minutes at his observatory so they also displayed seconds, however, they were not yet accurate enough for seconds. In 1587, Tycho complained that his four clocks disagreed by plus or minus four seconds, in 1670, London clockmaker William Clement added this seconds pendulum to the original pendulum clock of Christiaan Huygens. From 1670 to 1680, Clement made many improvements to his clock and this clock used an anchor escapement mechanism with a seconds pendulum to display seconds in a small subdial
21.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
22.
Gray (unit)
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The gray is a derived unit of ionizing radiation dose in the International System of Units. It is defined as the absorption of one joule of energy per kilogram of matter. It is used as a measure of absorbed dose, specific energy and it is a physical quantity, and does not take into account any biological context. Unlike the pre-1971 non-SI roentgen unit of exposure, the gray when used for absorbed dose is defined independently of any target material. However, when measuring kerma the reference target material must be defined explicitly, usually as dry air at standard temperature and pressure. The corresponding cgs unit, the rad, remains common in the United States, though strongly discouraged in the guide for U. S. National Institute of Standards. The gray was named after British physicist Louis Harold Gray, a pioneer in the measurement of X-ray and radium radiation and it was adopted as part of the International System of Units in 1975. One gray is the absorption of one joule of energy, in the form of ionizing radiation, measuring and controlling the value of absorbed dose is vital to ensuring correct operation of these processes. Kerma is a measure of the energy of ionisation due to irradiation. Importantly, kerma dose is different from absorbed dose, depending on the energies involved. The measurement of absorbed dose is a problem, and so many different dosimeters are available for these measurements. These dosimeters cover measurements that can be done in 1-D, 2-D and 3-D, in radiation therapy, the amount of radiation applied varies depending on the type and stage of cancer being treated. For curative cases, the dose for a solid epithelial tumor ranges from 60 to 80 Gy. Preventive doses are typically around 45–60 Gy in 1. 8–2 Gy fractions, the absorbed dose also plays an important role in radiation protection, as it is the starting point for calculating the effect of low levels of radiation. In radiation protection dose assessment, the health risk is defined as the probability of cancer induction. This probability is related to the equivalent dose in sieverts, which has the dimensions as the gray. It is related to the gray by weighting factors described in the articles on equivalent dose, to avoid any risk of confusion between the absorbed dose and the equivalent dose, the absorbed dose is stated in grays and the equivalent dose is stated in sieverts. The accompanying diagrams show how absorbed dose is first obtained by computational techniques, radiation poisoning - The gray is conventionally used to express the severity of what are known as tissue effects from doses received in acute exposure to high levels of ionizing radiation
23.
Pascal (unit)
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The pascal is the SI derived unit of pressure used to quantify internal pressure, stress, Youngs modulus and ultimate tensile strength. It is defined as one newton per square meter and it is named after the French polymath Blaise Pascal. Common multiple units of the pascal are the hectopascal which is equal to one millibar, the unit of measurement called standard atmosphere is defined as 101,325 Pa and approximates to the average pressure at sea-level at the latitude 45° N. Meteorological reports typically state atmospheric pressure in hectopascals, the unit is named after Blaise Pascal, noted for his contributions to hydrodynamics and hydrostatics, and experiments with a barometer. The name pascal was adopted for the SI unit newton per square metre by the 14th General Conference on Weights, one pascal is the pressure exerted by a force of magnitude one newton perpendicularly upon an area of one square metre. The unit of measurement called atmosphere or standard atmosphere is 101325 Pa and this value is often used as a reference pressure and specified as such in some national and international standards, such as ISO2787, ISO2533 and ISO5024. In contrast, IUPAC recommends the use of 100 kPa as a standard pressure when reporting the properties of substances, geophysicists use the gigapascal in measuring or calculating tectonic stresses and pressures within the Earth. Medical elastography measures tissue stiffness non-invasively with ultrasound or magnetic resonance imaging, in materials science and engineering, the pascal measures the stiffness, tensile strength and compressive strength of materials. In engineering use, because the pascal represents a small quantity. The pascal is also equivalent to the SI unit of energy density and this applies not only to the thermodynamics of pressurised gases, but also to the energy density of electric, magnetic, and gravitational fields. In measurements of sound pressure, or loudness of sound, one pascal is equal to 94 decibels SPL, the quietest sound a human can hear, known as the threshold of hearing, is 0 dB SPL, or 20 µPa. The airtightness of buildings is measured at 50 Pa, the units of atmospheric pressure commonly used in meteorology were formerly the bar, which was close to the average air pressure on Earth, and the millibar. Since the introduction of SI units, meteorologists generally measure pressures in hectopascals unit, exceptions include Canada and Portugal, which use kilopascals. In many other fields of science, the SI is preferred, many countries also use the millibar or hectopascal to give aviation altimeter settings. In practically all fields, the kilopascal is used instead. Centimetre of water Metric prefix Orders of magnitude Pascals law
24.
Bar (unit)
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The bar is a metric unit of pressure, but is not approved as part of the International System of Units. It is defined as equal to 100000 Pa, which is slightly less than the current average atmospheric pressure on Earth at sea level. The bar and the millibar were introduced by the Norwegian meteorologist Vilhelm Bjerknes, use of the bar is deprecated by some professional bodies in some fields. The International Astronomical Union also lists it under Non-SI units and symbols whose continued use is deprecated, as of 2004, the bar is legally recognized in countries of the European Union. Units derived from the bar include the megabar, kilobar, decibar, centibar, the notation bar, though deprecated by various bodies, represents gauge pressure, i. e. pressure in bars above ambient or atmospheric pressure. The bar is defined using the SI derived unit, pascal,1 bar ≡100000 Pa. Thus,1 bar is equal to,100 kPa 1×105 N/m21000000 Ba, notes,1 millibar =1 one-thousandth bar, or 1×10−3 bar 1 millibar =1 hectopascal. The word bar has its origin in the Greek word βάρος, the units official symbol is bar, the earlier symbol b is now deprecated and conflicts with the use of b denoting the unit barn, but it is still encountered, especially as mb to denote the millibar. Between 1793 and 1795, the bar was used for a unit of weight in an early version of the metric system. Atmospheric air pressure is given in millibars where standard sea level pressure is defined as 1013 mbar,101.3,1.013 bar. Despite the millibar not being an SI unit, meteorologists and weather reporters worldwide have long measured air pressure in millibars as the values are convenient, for example, the weather office of Environment Canada uses kilopascals and hectopascals on their weather maps. In contrast, Americans are familiar with the use of the millibar in US reports of hurricanes, in fresh water, there is an approximate numerical equivalence between the change in pressure in decibars and the change in depth from the water surface in metres. Specifically, an increase of 1 decibar occurs for every 1.019716 m increase in depth, in sea water with respect to the gravity variation, the latitude and the geopotential anomaly the pressure can be converted into meters depth according to an empirical formula. As a result, decibars are commonly used in oceanography, many engineers worldwide use the bar as a unit of pressure because, in much of their work, using pascals would involve using very large numbers. In the automotive field, turbocharger boost is often described in bars outside the USA), unicode has characters for mb and bar, but they exist only for compatibility with legacy Asian encodings and are not intended to be used in new documents. The kilobar, equivalent to 100 MPa, is used in geological systems. Bar and bara are sometimes used to indicate absolute pressures and bar and this usage is deprecated and fuller descriptions such as gauge pressure of 2 bar or 2 bar gauge are recommended.0 Unported License but not under the GFDL. Non-SI units accepted for use with the SI
25.
Mole (unit)
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The mole is the unit of measurement in the International System of Units for amount of substance. This number is expressed by the Avogadro constant, which has a value of 6. 022140857×1023 mol−1, the mole is one of the base units of the SI, and has the unit symbol mol. The mole is used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. For example, the chemical equation 2 H2 + O2 →2 H2O implies that 2 moles of dihydrogen and 1 mole of dioxygen react to form 2 moles of water. The mole may also be used to express the number of atoms, ions, the concentration of a solution is commonly expressed by its molarity, defined as the number of moles of the dissolved substance per litre of solution. For example, the relative molecular mass of natural water is about 18.015, therefore. The term gram-molecule was formerly used for essentially the same concept, the term gram-atom has been used for a related but distinct concept, namely a quantity of a substance that contains Avogadros number of atoms, whether isolated or combined in molecules. Thus, for example,1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2, in honor of the unit, some chemists celebrate October 23, which is a reference to the 1023 scale of the Avogadro constant, as Mole Day. Some also do the same for February 6 and June 2, thus, by definition, one mole of pure 12C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the number of molecules as X moles of any other substance. The mass per mole of a substance is called its molar mass, the number of elementary entities in a sample of a substance is technically called its amount. Therefore, the mole is a convenient unit for that physical quantity, one can determine the chemical amount of a known substance, in moles, by dividing the samples mass by the substances molar mass. Other methods include the use of the volume or the measurement of electric charge. The mass of one mole of a substance depends not only on its molecular formula, since the definition of the gram is not mathematically tied to that of the atomic mass unit, the number NA of molecules in a mole must be determined experimentally. The value adopted by CODATA in 2010 is NA =6. 02214129×1023 ±0. 00000027×1023, in 2011 the measurement was refined to 6. 02214078×1023 ±0. 00000018×1023. The number of moles of a sample is the sample mass divided by the mass of the material. The history of the mole is intertwined with that of mass, atomic mass unit, Avogadros number. The first table of atomic mass was published by John Dalton in 1805
26.
Litre
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The litre or liter is an SI accepted metric system unit of volume equal to 1 cubic decimetre,1,000 cubic centimetres or 1/1,000 cubic metre. A cubic decimetre occupies a volume of 10×10×10 centimetres and is equal to one-thousandth of a cubic metre. The original French metric system used the litre as a base unit. The word litre is derived from an older French unit, the litron, whose name came from Greek — where it was a unit of weight, not volume — via Latin, and which equalled approximately 0.831 litres. The litre was also used in subsequent versions of the metric system and is accepted for use with the SI. The spelling used by the International Bureau of Weights and Measures is litre, the less common spelling of liter is more predominantly used in American English. One litre of water has a mass of almost exactly one kilogram. Subsequent redefinitions of the metre and kilogram mean that this relationship is no longer exact, a litre is defined as a special name for a cubic decimetre or 10 centimetres ×10 centimetres ×10 centimetres. Hence 1 L ≡0.001 m3 ≡1000 cm3, from 1901 to 1964, the litre was defined as the volume of one kilogram of pure water at maximum density and standard pressure. The kilogram was in turn specified as the mass of a platinum/iridium cylinder held at Sèvres in France and was intended to be of the mass as the 1 litre of water referred to above. It was subsequently discovered that the cylinder was around 28 parts per million too large and thus, during this time, additionally, the mass-volume relationship of water depends on temperature, pressure, purity and isotopic uniformity. In 1964, the definition relating the litre to mass was abandoned in favour of the current one, although the litre is not an official SI unit, it is accepted by the CGPM for use with the SI. CGPM defines the litre and its acceptable symbols, a litre is equal in volume to the millistere, an obsolete non-SI metric unit customarily used for dry measure. The litre is often used in some calculated measurements, such as density. One litre of water has a mass of almost exactly one kilogram when measured at its maximal density, similarly,1 millilitre of water has a mass of about 1 g,1,000 litres of water has a mass of about 1,000 kg. It is now known that density of water depends on the isotopic ratios of the oxygen and hydrogen atoms in a particular sample. The litre, though not an official SI unit, may be used with SI prefixes, the most commonly used derived unit is the millilitre, defined as one-thousandth of a litre, and also often referred to by the SI derived unit name cubic centimetre. It is a commonly used measure, especially in medicine and cooking, Other units may be found in the table below, where the more often used terms are in bold
27.
PH
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In chemistry, pH is a numeric scale used to specify the acidity or basicity of an aqueous solution. It is approximately the negative of the base 10 logarithm of the concentration, measured in units of moles per liter. More precisely it is the negative of the logarithm to base 10 of the activity of the hydrogen ion, solutions with a pH less than 7 are acidic and solutions with a pH greater than 7 are basic. Pure water is neutral, at pH7, being neither an acid nor a base, contrary to popular belief, the pH value can be less than 0 or greater than 14 for very strong acids and bases respectively. The pH scale is traceable to a set of standard solutions whose pH is established by international agreement, the pH of aqueous solutions can be measured with a glass electrode and a pH meter, or an indicator. In the first papers, the notation had the H as a subscript to the p, as so. The exact meaning of the p in pH is disputed, but according to the Carlsberg Foundation and it has also been suggested that the p stands for the German Potenz, others refer to French puissance. Another suggestion is that the p stands for the Latin terms pondus hydrogenii, potentia hydrogenii and it is also suggested that Sørensen used the letters p and q simply to label the test solution and the reference solution. Currently in chemistry, the p stands for decimal cologarithm of, PH is defined as the decimal logarithm of the reciprocal of the hydrogen ion activity, aH+, in a solution. P H = − log 10 = log 10 For example and this definition was adopted because ion-selective electrodes, which are used to measure pH, respond to activity. For H+ number of electrons transferred is one and it follows that electrode potential is proportional to pH when pH is defined in terms of activity. The reference electrode may be a silver chloride electrode or a calomel electrode, the hydrogen-ion selective electrode is a standard hydrogen electrode. Reference electrode | concentrated solution of KCl || test solution | H2 | Pt Firstly, the cell is filled with a solution of hydrogen ion activity. Then the emf, EX, of the cell containing the solution of unknown pH is measured. PH = pH + E S − E X z The difference between the two measured emf values is proportional to pH and this method of calibration avoids the need to know the standard electrode potential. The proportionality constant, 1/z is ideally equal to 12.303 R T / F the Nernstian slope, to apply this process in practice, a glass electrode is used rather than the cumbersome hydrogen electrode. A combined glass electrode has a reference electrode. It is calibrated against buffer solutions of hydrogen ion activity
28.
Astronomical unit
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The astronomical unit is a unit of length, roughly the distance from Earth to the Sun. However, that varies as Earth orbits the Sun, from a maximum to a minimum. Originally conceived as the average of Earths aphelion and perihelion, it is now defined as exactly 149597870700 metres, the astronomical unit is used primarily as a convenient yardstick for measuring distances within the Solar System or around other stars. However, it is also a component in the definition of another unit of astronomical length. A variety of symbols and abbreviations have been in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union used the symbol A for the astronomical unit, in 2006, the International Bureau of Weights and Measures recommended ua as the symbol for the unit. In 2012, the IAU, noting that various symbols are presently in use for the astronomical unit, in the 2014 revision of the SI Brochure, the BIPM used the unit symbol au. In ISO 80000-3, the symbol of the unit is ua. Earths orbit around the Sun is an ellipse, the semi-major axis of this ellipse is defined to be half of the straight line segment that joins the aphelion and perihelion. The centre of the sun lies on this line segment. In addition, it mapped out exactly the largest straight-line distance that Earth traverses over the course of a year, knowing Earths shift and a stars shift enabled the stars distance to be calculated. But all measurements are subject to some degree of error or uncertainty, improvements in precision have always been a key to improving astronomical understanding. Improving measurements were continually checked and cross-checked by means of our understanding of the laws of celestial mechanics, the expected positions and distances of objects at an established time are calculated from these laws, and assembled into a collection of data called an ephemeris. NASAs Jet Propulsion Laboratory provides one of several ephemeris computation services, in 1976, in order to establish a yet more precise measure for the astronomical unit, the IAU formally adopted a new definition. Equivalently, by definition, one AU is the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass. As with all measurements, these rely on measuring the time taken for photons to be reflected from an object. However, for precision the calculations require adjustment for such as the motions of the probe. In addition, the measurement of the time itself must be translated to a scale that accounts for relativistic time dilation
29.
Becquerel
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The becquerel is the SI derived unit of radioactivity. One becquerel is defined as the activity of a quantity of material in which one nucleus decays per second. The becquerel is therefore equivalent to a second, s−1. The becquerel is named after Henri Becquerel, who shared a Nobel Prize in Physics with Pierre, as with every International System of Units unit named for a person, the first letter of its symbol is uppercase. 1 Bq =1 s−1 A special name was introduced for the second to represent radioactivity to avoid potentially dangerous mistakes with prefixes. For example,1 µs−1 could be taken to mean 106 disintegrations per second, other names considered were hertz, a special name already in use for the reciprocal second, and fourier. The hertz is now used for periodic phenomena. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radioactivity event per second, the gray and the becquerel were introduced in 1975. Between 1953 and 1975, absorbed dose was often measured in rads, decay activity was measured in curies before 1946 and often in rutherfords between 1946 and 1975. Like any SI unit, Bq can be prefixed, commonly used multiples are kBq, MBq, GBq, TBq, for practical applications,1 Bq is a small unit, therefore, the prefixes are common. For example, the roughly 0.0169 g of potassium-40 present in a human body produces approximately 4,400 disintegrations per second or 4.4 kBq of activity. The global inventory of carbon-14 is estimated to be 8. 5×1018 Bq, the nuclear explosion in Hiroshima is estimated to have produced 8×1024 Bq. The becquerel succeeded the curie, an older, non-SI unit of radioactivity based on the activity of 1 gram of radium-226, the curie is defined as 3. 7·1010 s−1, or 37 GBq. The following table shows radiation quantities in SI and non-SI units
30.
Sievert
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The sievert, named after Rolf Maximilian Sievert, is a derived unit of ionizing radiation dose in the International System of Units. It is a measure of the effect of low levels of ionizing radiation on the human body. These are under review, and changes are advised in the formal Reports of those bodies. The sievert is used for radiation dose quantities such as equivalent dose, effective dose and it is used to represent both the risk of the effect of external radiation from sources outside the body and the effect of internal irradiation due to inhaled or ingested radioactive substances. Conventionally, the sievert is not used for high rates of radiation that produce deterministic effects. Such effects are compared to the physical quantity absorbed dose measured by the unit gray, one sievert carries with it a 5. 5% chance of eventually developing cancer based on the linear no-threshold model. The rem is an older, non-SI unit of measurement, in summary, The gray - quantity D1 Gy =1 joule/kilogram - a physical quantity. 1 Gy is the deposit of a joule of energy in a kg of matter or tissue. The sievert - quantity H1 Sv =1 joule/kilogram - a biological effect, the sievert represents the equivalent biological effect of the deposit of a joule of radiation energy in a kilogram of human tissue. The equivalence to absorbed dose is denoted by Q, the ICRP definition of the sievert is, The sievert is the special name for the SI unit of equivalent dose, effective dose, and operational dose quantities. The unit is joule per kilogram, the sievert is used for a number of dose quantities which are described in this article and are part of the international radiological protection system devised and defined by the ICRP and ICRU. The ICRU/ICRP dose quantities have specific purposes and meanings, but some use common words in a different order, there can be confusion between, for instance, equivalent dose and dose equivalent. Only the operational dose quantities which still use Q for calculation retain the phrase dose equivalent, however, there are joint ICRU/ICRP proposals to simplify this system by changes to the operational dose definitions to harmonise with those of protection quantities. In the USA there are differently named dose quantities which are not part of the ICRP nomenclature, the sievert is used to represent the biological effects of different forms of external ionizing radiation on various types of human tissue. Some quantities cannot be measured, but they must be related to actual instrumentation. The resultant complexity has required the creation of a number of different dose quantities within a coherent system developed by the ICRU working with the ICRP, the external dose quantities and their relationships are shown in the accompanying diagram. These are directly measurable physical quantities in which no allowance has been made for biological effects and these quantities cannot be practically measured but are a calculated value of dose of organs of the human body, which is arrived at by using anthropomorphic phantoms. These are 3D computational models of the body which take into account a number of complex effects such as body self-shielding
31.
Steradian
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The steradian or square radian is the SI unit of solid angle. It is used in geometry, and is analogous to the radian which quantifies planar angles. The name is derived from the Greek stereos for solid and the Latin radius for ray and it is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol sr is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian, the steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit. A steradian can be defined as the angle subtended at the center of a unit sphere by a unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian, because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π steradians. By the same argument, the solid angle that can be subtended at any point is 4π sr. Since A = r2, it corresponds to the area of a cap. Therefore one steradian corresponds to the angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by, θ = arccos = arccos = arccos ≈0.572 rad. This angle corresponds to the plane angle of 2θ ≈1.144 rad or 65. 54°. A steradian is also equal to the area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere. The solid angle of a cone whose cross-section subtends the angle 2θ is, Ω =2 π s r. In two dimensions, an angle is related to the length of the arc that it spans, θ = l r r a d where l is arc length, r is the radius of the circle. For example, a measurement of the width of an object would be given in radians. At the same time its visible area over ones visible field would be given in steradians. Just as the area of a circle is related to its diameter or radius. One-dimensional circular measure has units of radians or degrees, while two-dimensional spherical measure is expressed in steradians, in higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases and that is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or point set expressed in spherical coordinates
32.
Candela
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The candela is the SI base unit of luminous intensity, that is, luminous power per unit solid angle emitted by a point light source in a particular direction. A common candle emits light with an intensity of roughly one candela. If emission in some directions is blocked by an opaque barrier, the word candela means candle in Latin. Like most other SI base units, the candela has an operational definition—it is defined by a description of a process that will produce one candela of luminous intensity. The definition describes how to produce a source that emits one candela. Such a source could then be used to calibrate instruments designed to measure luminous intensity, the candela is sometimes still called by the old name candle, such as in foot-candle and the modern definition of candlepower. The frequency chosen is in the spectrum near green, corresponding to a wavelength of about 555 nanometres. The human eye is most sensitive to frequency, when adapted for bright conditions. At other frequencies, more radiant intensity is required to achieve the same luminous intensity, if more than one wavelength is present, one must sum or integrate over the spectrum of wavelengths present to get the total luminous intensity. A common candle emits light with roughly 1 cd luminous intensity. A25 W compact fluorescent light bulb puts out around 1700 lumens, if light is radiated equally in all directions. Focused into a 20° beam, the light bulb would have an intensity of around 18,000 cd. The luminous intensity of light-emitting diodes is measured in millicandelas, or thousandths of a candela, indicator LEDs are typically in the 50 mcd range, ultra-bright LEDs can reach 15,000 mcd, or higher. Prior to 1948, various standards for luminous intensity were in use in a number of countries and these were typically based on the brightness of the flame from a standard candle of defined composition, or the brightness of an incandescent filament of specific design. One of the best-known of these was the English standard of candlepower, one candlepower was the light produced by a pure spermaceti candle weighing one sixth of a pound and burning at a rate of 120 grains per hour. Germany, Austria and Scandinavia used the Hefnerkerze, a based on the output of a Hefner lamp. It became clear that a unit was needed. Jules Violle had proposed a standard based on the emitted by 1 cm2 of platinum at its melting point
33.
Lux
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The lux is the SI unit of illuminance and luminous emittance, measuring luminous flux per unit area. It is equal to one lumen per square metre, in photometry, this is used as a measure of the intensity, as perceived by the human eye, of light that hits or passes through a surface. In English, lux is used as both the singular and plural form, illuminance is a measure of how much luminous flux is spread over a given area. One can think of flux as a measure of the total amount of visible light present. A given amount of light will illuminate a surface more dimly if it is spread over a larger area, however, the same 1000 lumens, spread out over ten square metres, produces a dimmer illuminance of only 100 lux. Achieving an illuminance of 500 lux might be possible in a kitchen with a single fluorescent light fixture with an output of 12000 lumens. To light a factory floor with dozens of times the area of the kitchen would require dozens of such fixtures, thus, lighting a larger area to the same level of lux requires a greater number of lumens. As with other SI units, SI prefixes can be used, for instance, a star of apparent magnitude 0 provides 2.08 microlux at the earths surface. A barely perceptible magnitude 6 star provides 8 nanolux, the unobscured sun provides an illumination of up to 100 kilolux on the Earths surface, the exact value depending on time of year and atmospheric conditions. This direct normal illuminance is related to the solar illuminance constant Esc, the illumination provided on a surface by a point source equals the number of lux just described times the cosine of the angle between a ray coming from the source and a normal to the surface. The number of lux falling on the surface equals this cosine times a number that characterizes the source from the point of view in question and it differs from the luminance, which does depend on the angular distribution of the emission. A perfectly white surface with one lux falling on it will emit one lux, like all photometric units, the lux has a corresponding radiometric unit. The weighting factor is known as the luminosity function, the lux is one lumen per square metre, and the corresponding radiometric unit, which measures irradiance, is the watt per square metre. The peak of the luminosity function is at 555 nm, the eyes image-forming visual system is sensitive to light of this wavelength than any other. Other wavelengths of visible light produce fewer lux per watt-per-meter-squared, the luminosity function falls to zero for wavelengths outside the visible spectrum. For a light source with mixed wavelengths, the number of lumens per watt can be calculated by means of the luminosity function and this means that white light sources produce far fewer lumens per watt than the theoretical maximum of 683.002 lm/W. The ratio between the number of lumens per watt and the theoretical maximum is expressed as a percentage known as the luminous efficiency. For example, an incandescent light bulb has a luminous efficiency of only about 2%
34.
Lumen (unit)
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The lumen is the SI derived unit of luminous flux, a measure of the total quantity of visible light emitted by a source. Lumens are related to lux in that one lux is one lumen per square meter, the lumen is defined in relation to the candela as 1 lm =1 cd ⋅ sr. A full sphere has an angle of 4π steradians, so a light source that uniformly radiates one candela in all directions has a total luminous flux of 1 cd × 4π sr = 4π cd⋅sr ≈12.57 lumens. If a light source emits one candela of luminous intensity uniformly across a solid angle of one steradian, alternatively, an isotropic one-candela light-source emits a total luminous flux of exactly 4π lumens. If the source were partly covered by an ideal absorbing hemisphere, the luminous intensity would still be one candela in those directions that are not obscured. The lumen can be thought of casually as a measure of the amount of visible light in some defined beam or angle. The number of candelas or lumens from a source also depends on its spectrum, the difference between the units lumen and lux is that the lux takes into account the area over which the luminous flux is spread. A flux of 1000 lumens, concentrated into an area of one square metre, the same 1000 lumens, spread out over ten square metres, produces a dimmer illuminance of only 100 lux. Mathematically,1 lx =1 lm/m2, a source radiating a power of one watt of light in the color for which the eye is most efficient has luminous flux of 683 lumens. So a lumen represents at least 1/683 watts of light power. Lamps used for lighting are commonly labelled with their output in lumens. A23 W spiral compact fluorescent lamp emits about 1, 400–1,600 lm, many compact fluorescent lamps and other alternative light sources are labelled as being equivalent to an incandescent bulb with a specific wattage. Below is a table that shows typical luminous flux for common incandescent bulbs, on September 1,2010, European Union legislation came into force mandating that lighting equipment must be labelled primarily in terms of luminous flux, instead of electric power. This change is a result of the EUs Eco-design Directive for Energy-using Products, for example, according to the European Union standard, an energy-efficient bulb that claims to be the equivalent of a 60 W tungsten bulb must have a minimum light output of 700–750 lm. The light output of projectors is typically measured in lumens, a standardized procedure for testing projectors has been established by the American National Standards Institute, which involves averaging together several measurements taken at different positions. For marketing purposes, the flux of projectors that have been tested according to this procedure may be quoted in ANSI lumens. ANSI lumen measurements are in more accurate than the other measurement techniques used in the projector industry. This allows projectors to be easily compared on the basis of their brightness specifications
35.
Weber (unit)
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In physics, the weber /ˈweɪbər/ is the SI unit of magnetic flux. A flux density of one Wb/m2 is one tesla, the weber is named after the German physicist Wilhelm Eduard Weber. The weber may be defined in terms of Faradays law, which relates a changing magnetic flux through a loop to the field around the loop. A change in flux of one weber per second will induce an electromotive force of one volt. Officially, Weber — The weber is the flux that, linking a circuit of one turn. This SI unit is named after Wilhelm Eduard Weber, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, in 1861, the British Association for the Advancement of Science established a committee under William Thomson to study electrical units. It was not until 1927 that TC1 dealt with the study of various outstanding problems concerning electrical and magnetic quantities, as disagreement continued, the IEC decided on an effort to remedy the situation. It instructed a task force to study the question in readiness for the next meeting, in 1935, TC1 recommended names for several electrical units, including the weber for the practical unit of magnetic flux. This system was given the designation of Giorgi system, also in 1936, TC1 passed responsibility for electric and magnetic magnitudes and units to the new TC24. This led eventually to the adoption of the Giorgi system, which unified electromagnetic units with the MKS dimensional system of units. In 1938, TC24 recommended as a link the permeability of free space with the value of µ0 = 4π×10−7 H/m. This group also recognized that any one of the units already in use. After consultation, the ampere was adopted as the unit of the Giorgi system in Paris in 1950
36.
TNT equivalent
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TNT equivalent is a convention for expressing energy, typically used to describe the energy released in an explosion. The ton of TNT is a unit of energy defined by convention to be 4.184 gigajoules. The convention intends to compare the destructiveness of an event with that of conventional explosives, the kiloton is a unit of energy equal to 4.184 terajoules. The megaton is a unit of equal to 4.184 petajoules. The kiloton and megaton of TNT have traditionally used to describe the energy output. The TNT equivalent appears in various nuclear weapon control treaties, and has used to characterize the energy released in such other highly destructive events as an asteroid impact. A gram of TNT releases 2673–6702 J upon explosion, the energy liberated by one gram of TNT was arbitrarily defined as a matter of convention to be 4184 J, which is exactly one kilocalorie. The measured, pure heat output of a gram of TNT is only 2724 J, alternative TNT equivalency can be calculated as a function of when in the detonation the value is measured and which property is being compared. A kiloton of TNT can be visualized as a cube of TNT8.46 metres on a side, the RE factor is the relative mass of TNT to which an explosive is equivalent, the greater the RE, the more powerful the explosive. This enables engineers to determine the proper masses of different explosives when applying blasting formulas developed specifically for TNT. For example, if a timber-cutting formula calls for a charge of 1 kg of TNT, then based on octanitrocubanes RE factor of 2.38, using PETN, engineers would need 1. 0/1.66 kg to obtain the same effects as 1 kg of TNT. With ANFO or ammonium nitrate, they would require 1. 0/0.74 kg or 1. 0/0.42 kg, *, TBX or EBX, in a small, confined space, may have over twice the power of destruction. The total power of aluminized mixtures strictly depends on the condition of explosions, guide for the Use of the International System of Units. National Institute of Standards and Technology, nuclear Weapons FAQ Part 1.3 Rhodes, Richard. The Making of the Atomic Bomb, cooper, Paul W. Explosives Engineering, New York, Wiley-VCH, ISBN 0-471-18636-8 HQ Department of the Army, Field Manual 5-25, Explosives and Demolitions, Washington, D. C. Pentagon Publishing, pp. 83–84, ISBN 0-9759009-5-1 Explosives - Compositions, Alexandria, VA, thermobaric Explosives, Advanced Energetic Materials,2004. THE NATIONAL ACADEMIES PRESS, nap. edu, retrieved September 2004
37.
Horsepower
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Horsepower is a unit of measurement of power. There are many different standards and types of horsepower, two common definitions being used today are the mechanical horsepower, which is approximately 746 watts, and the metric horsepower, which is approximately 735.5 watts. The term was adopted in the late 18th century by Scottish engineer James Watt to compare the output of engines with the power of draft horses. It was later expanded to include the power of other types of piston engines, as well as turbines, electric motors. The definition of the unit varied among geographical regions, most countries now use the SI unit watt for measurement of power. With the implementation of the EU Directive 80/181/EEC on January 1,2010, units called horsepower have differing definitions, The mechanical horsepower, also known as imperial horsepower equals approximately 745.7 watts. It was defined originally as exactly 550 foot-pounds per second [745.7 N. m/s), the metric horsepower equals approximately 735.5 watts. It was defined originally as 75 kgf-m per second is approximately equivalent to 735.5 watts, the Pferdestärke PS is a name for a group of similar power measurements used in Germany around the end of the 19th century, all of about one metric horsepower in size. The boiler horsepower equals 9809.5 watts and it was used for rating steam boilers and is equivalent to 34.5 pounds of water evaporated per hour at 212 degrees Fahrenheit. One horsepower for rating electric motors is equal to 746 watts, one horsepower for rating Continental European electric motors is equal to 735 watts. Continental European electric motors used to have dual ratings, one British Royal Automobile Club horsepower can equal a range of values based on estimates of several engine dimensions. It is one of the tax horsepower systems adopted around Europe, the development of the steam engine provided a reason to compare the output of horses with that of the engines that could replace them. He had previously agreed to take royalties of one third of the savings in coal from the older Newcomen steam engines and this royalty scheme did not work with customers who did not have existing steam engines but used horses instead. Watt determined that a horse could turn a mill wheel 144 times in an hour, the wheel was 12 feet in radius, therefore, the horse travelled 2.4 × 2π ×12 feet in one minute. Watt judged that the horse could pull with a force of 180 pounds-force. So, P = W t = F d t =180 l b f ×2.4 ×2 π ×12 f t 1 m i n =32,572 f t ⋅ l b f m i n. Watt defined and calculated the horsepower as 32,572 ft·lbf/min, Watt determined that a pony could lift an average 220 lbf 100 ft per minute over a four-hour working shift. Watt then judged a horse was 50% more powerful than a pony, engineering in History recounts that John Smeaton initially estimated that a horse could produce 22,916 foot-pounds per minute
38.
Kilogram
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The kilogram or kilogramme is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype of the Kilogram. The avoirdupois pound, used in both the imperial and US customary systems, is defined as exactly 0.45359237 kg, making one kilogram approximately equal to 2.2046 avoirdupois pounds. Other traditional units of weight and mass around the world are also defined in terms of the kilogram, the gram, 1/1000 of a kilogram, was provisionally defined in 1795 as the mass of one cubic centimeter of water at the melting point of ice. The final kilogram, manufactured as a prototype in 1799 and from which the IPK was derived in 1875, had an equal to the mass of 1 dm3 of water at its maximum density. The kilogram is the only SI base unit with an SI prefix as part of its name and it is also the only SI unit that is still directly defined by an artifact rather than a fundamental physical property that can be reproduced in different laboratories. Three other base units and 17 derived units in the SI system are defined relative to the kilogram, only 8 other units do not require the kilogram in their definition, temperature, time and frequency, length, and angle. At its 2011 meeting, the CGPM agreed in principle that the kilogram should be redefined in terms of the Planck constant, the decision was originally deferred until 2014, in 2014 it was deferred again until the next meeting. There are currently several different proposals for the redefinition, these are described in the Proposed Future Definitions section below, the International Prototype Kilogram is rarely used or handled. In the decree of 1795, the term gramme thus replaced gravet, the French spelling was adopted in the United Kingdom when the word was used for the first time in English in 1797, with the spelling kilogram being adopted in the United States. In the United Kingdom both spellings are used, with kilogram having become by far the more common, UK law regulating the units to be used when trading by weight or measure does not prevent the use of either spelling. In the 19th century the French word kilo, a shortening of kilogramme, was imported into the English language where it has used to mean both kilogram and kilometer. In 1935 this was adopted by the IEC as the Giorgi system, now known as MKS system. In 1948 the CGPM commissioned the CIPM to make recommendations for a practical system of units of measurement. This led to the launch of SI in 1960 and the subsequent publication of the SI Brochure, the kilogram is a unit of mass, a property which corresponds to the common perception of how heavy an object is. Mass is a property, that is, it is related to the tendency of an object at rest to remain at rest, or if in motion to remain in motion at a constant velocity. Accordingly, for astronauts in microgravity, no effort is required to hold objects off the cabin floor, they are weightless. However, since objects in microgravity still retain their mass and inertia, the ratio of the force of gravity on the two objects, measured by the scale, is equal to the ratio of their masses. On April 7,1795, the gram was decreed in France to be the weight of a volume of pure water equal to the cube of the hundredth part of the metre
39.
Common logarithm
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In mathematics, the common logarithm is the logarithm with base 10. It is indicated by log10, or sometimes Log with a capital L, on calculators it is usually log, but mathematicians usually mean natural logarithm rather than common logarithm when they write log. To mitigate this ambiguity the ISO80000 specification recommends that log10 should be written lg, before the early 1970s, handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of logarithms were used in science, engineering. Use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions, because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well, see log table for the history of such tables. The fractional part is known as the mantissa, thus log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to 4 or 5 decimal places or more, of number in a range. Such a range would cover all possible values of the mantissa, the integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by, log 10 120 = log 10 =2 + log 10 1.2 ≈2 +0.07918. The last number —the fractional part or the mantissa of the logarithm of 120—can be found in the table shown. The location of the point in 120 tells us that the integer part of the common logarithm of 120. Numbers greater than 0 and less than 1 have negative logarithms, when reading a number in bar notation out loud, the symbol n ¯ is read as bar n, so that 2 ¯.07918 is read as bar 2 point 07918. The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten and this holds for any positive real number x because, log 10 = log 10 + log 10 = log 10 + i. Since i is always an integer the mantissa comes from log 10 which is constant for given x and this allows a table of logarithms to include only one entry for each mantissa. In the example of 5×10i,0.698970 will be listed once indexed by 5, or 0.5, common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician. In 1616 and 1617 Briggs visited John Napier, the inventor of what are now called natural logarithms at Edinburgh in order to suggest a change to Napiers logarithms. During these conferences the alteration proposed by Briggs was agreed upon, because base 10 logarithms were most useful for computations, engineers generally simply wrote log when they meant log10
40.
Human
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Modern humans are the only extant members of Hominina tribe, a branch of the tribe Hominini belonging to the family of great apes. Several of these hominins used fire, occupied much of Eurasia and they began to exhibit evidence of behavioral modernity around 50,000 years ago. In several waves of migration, anatomically modern humans ventured out of Africa, the spread of humans and their large and increasing population has had a profound impact on large areas of the environment and millions of native species worldwide. Humans are uniquely adept at utilizing systems of communication for self-expression and the exchange of ideas. Humans create complex structures composed of many cooperating and competing groups, from families. Social interactions between humans have established a wide variety of values, social norms, and rituals. These human societies subsequently expanded in size, establishing various forms of government, religion, today the global human population is estimated by the United Nations to be near 7.5 billion. In common usage, the word generally refers to the only extant species of the genus Homo—anatomically and behaviorally modern Homo sapiens. In scientific terms, the meanings of hominid and hominin have changed during the recent decades with advances in the discovery, there is also a distinction between anatomically modern humans and Archaic Homo sapiens, the earliest fossil members of the species. The English adjective human is a Middle English loanword from Old French humain, ultimately from Latin hūmānus, the words use as a noun dates to the 16th century. The native English term man can refer to the species generally, the species binomial Homo sapiens was coined by Carl Linnaeus in his 18th century work Systema Naturae. The generic name Homo is a learned 18th century derivation from Latin homō man, the species-name sapiens means wise or sapient. Note that the Latin word homo refers to humans of either gender, the genus Homo evolved and diverged from other hominins in Africa, after the human clade split from the chimpanzee lineage of the hominids branch of the primates. The closest living relatives of humans are chimpanzees and gorillas, with the sequencing of both the human and chimpanzee genome, current estimates of similarity between human and chimpanzee DNA sequences range between 95% and 99%. The gibbons and orangutans were the first groups to split from the leading to the humans. The splitting date between human and chimpanzee lineages is placed around 4–8 million years ago during the late Miocene epoch, during this split, chromosome 2 was formed from two other chromosomes, leaving humans with only 23 pairs of chromosomes, compared to 24 for the other apes. There is little evidence for the divergence of the gorilla, chimpanzee. Each of these species has been argued to be an ancestor of later hominins
41.
Population
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A population is the number of all the organisms of the same group or species, which live in a particular geographical area, and have the capability of interbreeding. In sociology, population refers to a collection of humans, Demography is a social science which entails the statistical study of human populations. This article refers mainly to human population, in population genetics a sexual population is a set of organisms in which any pair of members can breed together. This means that they can regularly exchange gametes to produce normally-fertile offspring and this also implies that all members belong to the same species. If the gamodeme is very large, and all gene alleles are uniformly distributed by the gametes within it, however, there may be low frequencies of exchange with these neighbours. This may be viewed as the breaking up of a sexual population into smaller overlapping sexual populations. The overall rise in homozygosity is quantified by the inbreeding coefficient, note that all homozygotes are increased in frequency – both the deleterious and the desirable. The mean phenotype of the collection is lower than that of the panmictic original – which is known as inbreeding depression. It is most important to note, however, that some lines will be superior to the panmictic original, while some will be about the same. The probabilities of each can be estimated from those binomial equations, in plant and animal breeding, procedures have been developed which deliberately utilise the effects of dispersion. It can be shown that dispersion-assisted selection leads to the greatest genetic advance and this is so for both allogamous and autogamous gamodemes. In ecology, the population of a species in a certain area can be estimated using the Lincoln Index. As of todays date, the population is estimated by the United States Census Bureau to be 7.496 billion. The US Census Bureau estimates the 7 billion number was surpassed on 12 March 2012, according to papers published by the United States Census Bureau, the world population hit 6.5 billion on 24 February 2006. The United Nations Population Fund designated 12 October 1999 as the day on which world population reached 6 billion. This was about 12 years after world population reached 5 billion in 1987, the population of countries such as Nigeria, is not even known to the nearest million, so there is a considerable margin of error in such estimates. Researcher Carl Haub calculated that a total of over 100 billion people have probably been born in the last 2000 years, Population growth increased significantly as the Industrial Revolution gathered pace from 1700 onwards. In 2007 the United Nations Population Division projected that the population will likely surpass 10 billion in 2055
42.
Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
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1,000,000,000
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1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn, in scientific notation, it is written as 1 ×109. The SI prefix giga indicates 1,000,000,000 times the base unit, one billion years may be called eon in astronomy and geology. Previously in British English, the word billion referred exclusively to a million millions, however, this is no longer as common as earlier, and the word has been used to mean one thousand million for some time. The alternative term one thousand million is used in the U. K. or countries such as Spain that uses one thousand million as one million million constitutes a billion. The worded figure, as opposed to the figure is used to differentiate between one thousand million or one billion. The term milliard can also be used to refer to 1,000,000,000, whereas milliard is seldom used in English, in the South Asian numbering system, it is known as 100 crore or 1 Arab. 1000000007 – smallest prime number with 10 digits,1023456789 – smallest pandigital number in base 10. 1026753849 – smallest pandigital square that includes 0,1073741824 –2301073807359 – 14th Kynea number. 1162261467 –3191220703125 –513 1232922769- 35113^2 Centered hexagonal number,1234567890 – pandigital number with the digits in order. 1882341361 – The least prime whose reversal is both square and triangular,1977326743 –7112147483647 – 8th Mersenne prime and the largest signed 32-bit integer. 2147483648 –2312176782336 –6122214502422 – 6th primary pseudoperfect number,2357947691 –1192971215073 – 11th Fibonacci prime. 3405691582 – hexadecimal CAFEBABE, used as a placeholder in programming,3405697037 – hexadecimal CAFED00D, used as a placeholder in programming. 3735928559 – hexadecimal DEADBEEF, used as a placeholder in programming,3486784401 –3204294836223 – 16th Carol number. 4294967291 – Largest prime 32-bit unsigned integer,4294967295 – Maximum 32-bit unsigned integer, perfect totient number, product of the five prime Fermat numbers. 4294967296 –2324294967297 – the first composite Fermat number,6103515625 –5146210001000 – only self-descriptive number in base 10. 6975757441 –1786983776800 – 15th colossally abundant number, 15th superior highly composite number 7645370045 – 27th Pell number,8589934592 –2339043402501 – 25th Motzkin number. 9814072356 – largest square pandigital number, largest pandigital pure power,9876543210 – largest number without redundant digits
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