1.
Specials (Unicode block)
–
Specials is a short Unicode block allocated at the very end of the Basic Multilingual Plane, at U+FFF0–FFFF. Of these 16 codepoints, five are assigned as of Unicode 9, U+FFFD � REPLACEMENT CHARACTER used to replace an unknown, unrecognized or unrepresentable character U+FFFE <noncharacter-FFFE> not a character. FFFE and FFFF are not unassigned in the sense. They can be used to guess a texts encoding scheme, since any text containing these is by not a correctly encoded Unicode text. The replacement character � is a found in the Unicode standard at codepoint U+FFFD in the Specials table. It is used to indicate problems when a system is unable to render a stream of data to a correct symbol and it is usually seen when the data is invalid and does not match any character, Consider a text file containing the German word für in the ISO-8859-1 encoding. This file is now opened with an editor that assumes the input is UTF-8. The first and last byte are valid UTF-8 encodings of ASCII, therefore, a text editor could replace this byte with the replacement character symbol to produce a valid string of Unicode code points. The whole string now displays like this, f�r, a poorly implemented text editor might save the replacement in UTF-8 form, the text file data will then look like this, 0x66 0xEF 0xBF 0xBD 0x72, which will be displayed in ISO-8859-1 as fï¿½r. Since the replacement is the same for all errors this makes it impossible to recover the original character, a better design is to preserve the original bytes, including the error, and only convert to the replacement when displaying the text. This will allow the text editor to save the original byte sequence and it has become increasingly common for software to interpret invalid UTF-8 by guessing the bytes are in another byte-based encoding such as ISO-8859-1. This allows correct display of both valid and invalid UTF-8 pasted together, Unicode control characters UTF-8 Mojibake Unicodes Specials table Decodeunicodes entry for the replacement character
2.
International standard
–
International standards are standards developed by international standards organizations. International standards are available for consideration and use worldwide, the most prominent organization is the International Organization for Standardization. International standards may be used either by application or by a process of modifying an international standard to suit local conditions. Technical barriers arise when different groups together, each with a large user base. Establishing international standards is one way of preventing or overcoming this problem, the implementation of standards in industry and commerce became highly important with the onset of the Industrial Revolution and the need for high-precision machine tools and interchangeable parts. Henry Maudslay developed the first industrially practical screw-cutting lathe in 1800, maudslays work, as well as the contributions of other engineers, accomplished a modest amount of industry standardization, some companies in-house standards spread a bit within their industries. Joseph Whitworths screw thread measurements were adopted as the first national standard by companies around the country in 1841 and it came to be known as the British Standard Whitworth, and was widely adopted in other countries. By the end of the 19th century differences in standards between companies were making trade increasingly difficult and strained, the Engineering Standards Committee was established in London in 1901 as the worlds first national standards body. After the First World War, similar national bodies were established in other countries, by the mid to late 19th century, efforts were being made to standardize electrical measurement. An important figure was R. E. B, Crompton, who became concerned by the large range of different standards and systems used by electrical engineering companies and scientists in the early 20th century. Many companies had entered the market in the 1890s and all chose their own settings for voltage, frequency, current, adjacent buildings would have totally incompatible electrical systems simply because they had been fitted out by different companies. Crompton could see the lack of efficiency in this system and began to consider proposals for a standard for electric engineering. In 1904, Crompton represented Britain at the Louisiana Purchase Exposition in Saint Louis as part of a delegation by the Institute of Electrical Engineers. He presented a paper on standardisation, which was so well received that he was asked to look into the formation of a commission to oversee the process. By 1906 his work was complete and he drew up a permanent constitution for the first international standards organization, the body held its first meeting that year in London, with representatives from 14 countries. In honour of his contribution to electrical standardisation, Lord Kelvin was elected as the bodys first President, the International Federation of the National Standardizing Associations was founded in 1926 with a broader remit to enhance international cooperation for all technical standards and specifications. The body was suspended in 1942 during World War II, after the war, ISA was approached by the recently formed United Nations Standards Coordinating Committee with a proposal to form a new global standards body. List of international common standards List of technical standard organisations Global Frameworks and standards organized along function lines, accessed 2014 ^ Cordova
3.
Logical conjunction
–
In logic and mathematics, and is the truth-functional operator of logical conjunction, the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is written as ∧ or ⋅. A and B is true only if A is true and B is true, an operand of a conjunction is a conjunct. Related concepts in other fields are, In natural language, the coordinating conjunction, in programming languages, the short-circuit and control structure. And is usually denoted by an operator, in mathematics and logic, ∧ or ×, in electronics, ⋅. In Jan Łukasiewiczs prefix notation for logic, the operator is K, logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true. The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of the expression. In keeping with the concept of truth, when conjunction is defined as an operator or function of arbitrary arity. The truth table of A ∧ B, As a rule of inference, conjunction introduction is a classically valid, the argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction, therefore, A and B. or in logical operator notation, A, B ⊢ A ∧ B Here is an example of an argument that fits the form conjunction introduction, Bob likes apples. Therefore, Bob likes apples and oranges, Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction, therefore, A. or alternately, A and B. In logical operator notation, A ∧ B ⊢ A. falsehood-preserving, yes When all inputs are false, walsh spectrum, Nonlinearity,1 If using binary values for true and false, then logical conjunction works exactly like normal arithmetic multiplication. Many languages also provide short-circuit control structures corresponding to logical conjunction. Logical conjunction is used for bitwise operations, where 0 corresponds to false and 1 to true,0 AND0 =0,0 AND1 =0,1 AND0 =0,1 AND1 =1. The operation can also be applied to two binary words viewed as bitstrings of length, by taking the bitwise AND of each pair of bits at corresponding positions. For example,11000110 AND10100011 =10000010 and this can be used to select part of a bitstring using a bit mask. For example,10011101 AND00001000 =00001000 extracts the fifth bit of an 8-bit bitstring
4.
Logical disjunction
–
In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation, the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is written as ∨ or +. A or B is true if A is true, or if B is true, or if both A and B are true. In logic, or by means the inclusive or, distinguished from an exclusive or. An operand of a disjunction is called a disjunct, related concepts in other fields are, In natural language, the coordinating conjunction or. In programming languages, the short-circuit or control structure, or is usually expressed with an infix operator, in mathematics and logic, ∨, in electronics, +, and in most programming languages, |, ||, or or. In Jan Łukasiewiczs prefix notation for logic, the operator is A, logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a formula that can have one or more literals separated only by ors. A single literal is often considered to be a degenerate disjunction, the disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of truth, when disjunction is defined as an operator or function of arbitrary arity. Falsehood-preserving, The interpretation under which all variables are assigned a value of false produces a truth value of false as a result of disjunction. The mathematical symbol for logical disjunction varies in the literature, in addition to the word or, and the formula Apq, the symbol ∨, deriving from the Latin word vel is commonly used for disjunction. For example, A ∨ B is read as A or B, such a disjunction is false if both A and B are false. In all other cases it is true, all of the following are disjunctions, A ∨ B ¬ A ∨ B A ∨ ¬ B ∨ ¬ C ∨ D ∨ ¬ E. The corresponding operation in set theory is the set-theoretic union, operators corresponding to logical disjunction exist in most programming languages. Disjunction is often used for bitwise operations, for example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged. Logical disjunction is usually short-circuited, that is, if the first operand evaluates to true then the second operand is not evaluated, the logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to both sides, they are evaluated in parallel, and if one terminates with value true
5.
Negation
–
Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, in classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A would therefore be false, the truth table of ¬p is as follows, Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, conversely, one can define F as p & ¬p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false, while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. But in classical logic, we get an identity, p → q can be defined as ¬p ∨ q. Algebraically, classical negation corresponds to complementation in a Boolean algebra and these algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following, In set theory \ is also used to indicate not member of, U \ A is the set of all members of U that are not members of A. No matter how it is notated or symbolized, the negation ¬p / −p can be read as it is not the case p, not that p. Within a system of logic, double negation, that is. In intuitionistic logic, a proposition implies its double negation but not conversely and this marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two and this result is known as Glivenkos theorem. De Morgans laws provide a way of distributing negation over disjunction and conjunction, ¬ ≡, in Boolean algebra, a linear function is one such that, If there exists a0, a1. An ∈ such that f = a0 ⊕ ⊕, another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a logical operator
6.
Complement (set theory)
–
In set theory, the complement of a set A refers to elements not in A. The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. If A and B are sets, then the complement of A in B, also termed the set-theoretic difference of B and A, is the set of elements in B. The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard, if R is the set of real numbers and Q is the set of rational numbers, then R ∖ Q is the set of irrational numbers. Let A, B, and C be three sets, the following identities capture notable properties of relative complements, C ∖ = ∪. C ∖ = ∪, with the important special case C ∖ = demonstrating that intersection can be expressed using only the relative complement operation. If A is a set, then the complement of A is the set of elements not in A. Formally. The absolute complement of A is usually denoted by A ∁, other notations include A c, A ¯, A ′, ∁ U A, and ∁ A. Assume that the universe is the set of integers, if A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3, assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts, let A and B be two sets in a universe U. The following identities capture important properties of complements, De Morgans laws. Complement laws, A ∪ A ∁ = U, if A ⊂ B, then B ∁ ⊂ A ∁. Involution or double complement law, ∁ = A, relationships between relative and absolute complements, A ∖ B = A ∩ B ∁. Relationship with set difference, A ∁ ∖ B ∁ = B ∖ A, the first two complement laws above show that if A is a non-empty, proper subset of U, then is a partition of U. In the LaTeX typesetting language, the command \setminus is usually used for rendering a set difference symbol, when rendered, the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin
7.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
8.
0 (number)
–
0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone
9.
Asterisk
–
An asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star, computer scientists and mathematicians often vocalize it as star. In English, an asterisk is usually five-pointed in sans-serif typefaces, six-pointed in serif typefaces and it can be used as censorship. It is also used on the Internet to correct ones spelling, the asterisk is derived from the need of the printers of family trees in feudal times for a symbol to indicate date of birth. The original shape was seven-armed, each arm like a shooting from the center. In computer science, the asterisk is used as a wildcard character, or to denote pointers, repetition. Origin Adamantius is known to have used the asteriskos to mark missing Hebrew lines from his Hexapla. The asterisk evolved in shape over time, but its meaning as a used to correct defects remained. In the Middle Ages, the asterisk was used to emphasize a part of text. However, an asterisk was not always used, one hypothesis to the origin of the asterisk is that it stems from the five thousand year old Sumerian character dingir,
10.
Integer
–
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
11.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
12.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
13.
Complex number
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
14.
Exponential function
–
In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
15.
Base (exponentiation)
–
In exponentiation, the base is the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b and it is more commonly expressed as the nth power of b, b to the nth power or b to the power n. For example, the power of 10 is 10,000 because 104 =10 ×10 ×10 ×10 =10,000. The term power strictly refers to the expression, but is sometimes used to refer to the exponent. When the nth power of b equals a number a, or a = bn, for example,10 is a fourth root of 10,000. The inverse function to exponentiation with base b is called the logarithm to base b, for example, log1010,000 =4
16.
Logarithm
–
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
17.
Binary logarithm
–
In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value n. That is, for any number x, x = log 2 n ⟺2 x = n. For example, the logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2. The binary logarithm is the logarithm to the base 2, the binary logarithm function is the inverse function of the power of two function. As well as log2, alternative notations for the binary logarithm include lg, ld, lb, and log. Binary logarithms can be used to calculate the length of the representation of a number in the numeral system. In computer science, they count the number of steps needed for binary search, other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other software packages. The integer part of a binary logarithm can be using the find first set operation on an integer value. The fractional part of the logarithm can be calculated efficiently, the powers of two have been known since antiquity, for instance they appear in Euclids Elements, Props. And the binary logarithm of a power of two is just its position in the sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544 and his book Arthmetica Integra contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these allow them to be interpreted as tables of binary logarithms. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm, virasenas concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two. This definition gives rise to a function that coincides with the logarithm on the powers of two, but it is different for other integers, giving the 2-adic order rather than the logarithm. The modern form of a logarithm, applying to any number was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their more significant applications in information theory, as part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy. Alternatively, it may be defined as ln n/ln 2, where ln is the natural logarithm, using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers
18.
Natural logarithm
–
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit. Parentheses are sometimes added for clarity, giving ln, loge or log and this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. The natural log of e itself, ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 =1. The natural logarithm can be defined for any real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, like all logarithms, the natural logarithm maps multiplication into addition, ln = ln + ln . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, for instance, the binary logarithm is the natural logarithm divided by ln, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity, for example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest, by Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number. The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and their work involved quadrature of the hyperbola xy =1 by determination of the area of hyperbolic sectors. Their solution generated the requisite hyperbolic logarithm function having properties now associated with the natural logarithm, the notations ln x and loge x both refer unambiguously to the natural logarithm of x. log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages, in some other contexts, however, log x can be used to denote the common logarithm. Historically, the notations l. and l were in use at least since the 1730s, finally, in the twentieth century, the notations Log and logh are attested. The graph of the logarithm function shown earlier on the right side of the page enables one to glean some of the basic characteristics that logarithms to any base have in common. Chief among them are, the logarithm of the one is zero. What makes natural logarithms unique is to be found at the point where all logarithms are zero. At that specific point the slope of the curve of the graph of the logarithm is also precisely one
19.
Common logarithm
–
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
20.
Circumference
–
The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of importance in geometry and trigonometry. Informally circumference may also refer to the edge rather than to the length of the edge. The circumference of a circle is the distance around it, the term is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics and this constant, pi, is represented by the Greek letter π. The numerical value of π is 3.141592653589793, pi is defined as the ratio of a circles circumference C to its diameter d, π = C d Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference, the use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius C / r =2 π also has uses in mathematics, engineering. These uses include but are not limited to radians, computer programming, the Greek letter τ is sometimes used to represent this constant, but is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph, arc length Area Caccioppoli set Isoperimetric inequality Pythagorean theorem Volume Numericana - Circumference of an ellipse Circumference of a circle With interactive applet and animation
21.
Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
22.
Diameter
–
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
23.
Imaginary unit
–
The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
24.
Electrical engineering
–
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone. Subsequently, broadcasting and recording media made electronics part of daily life, the invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineers typically hold a degree in engineering or electronic engineering. Practicing engineers may have professional certification and be members of a professional body, such bodies include the Institute of Electrical and Electronics Engineers and the Institution of Engineering and Technology. Electrical engineers work in a wide range of industries and the skills required are likewise variable. These range from basic circuit theory to the management skills required of a project manager, the tools and equipment that an individual engineer may need are similarly variable, ranging from a simple voltmeter to a top end analyzer to sophisticated design and manufacturing software. Electricity has been a subject of scientific interest since at least the early 17th century and he also designed the versorium, a device that detected the presence of statically charged objects. In the 19th century, research into the subject started to intensify, Electrical engineering became a profession in the later 19th century. Practitioners had created an electric telegraph network and the first professional electrical engineering institutions were founded in the UK. Over 50 years later, he joined the new Society of Telegraph Engineers where he was regarded by other members as the first of their cohort, Practical applications and advances in such fields created an increasing need for standardised units of measure. They led to the standardization of the units volt, ampere, coulomb, ohm, farad. This was achieved at a conference in Chicago in 1893. During these years, the study of electricity was considered to be a subfield of physics. Thats because early electrical technology was electromechanical in nature, the Technische Universität Darmstadt founded the worlds first department of electrical engineering in 1882. The first course in engineering was taught in 1883 in Cornell’s Sibley College of Mechanical Engineering. It was not until about 1885 that Cornell President Andrew Dickson White established the first Department of Electrical Engineering in the United States, in the same year, University College London founded the first chair of electrical engineering in Great Britain. Professor Mendell P. Weinbach at University of Missouri soon followed suit by establishing the engineering department in 1886
25.
Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
26.
Complex conjugate
–
In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of ρ e i ϕ is ρ e − i ϕ and this can be shown using Eulers formula. Complex conjugates are important for finding roots of polynomials, according to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients, so is its conjugate. The complex conjugate of a number z is written as z ¯ or z ∗. The first notation avoids confusion with the notation for the transpose of a matrix. The second is preferred in physics, where dagger is used for the conjugate transpose, If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the conjugate of a previous known number is abbreviated as c. c. A significant property of the conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero. The conjugate of the conjugate of a number z is z. The ultimate relation is the method of choice to compute the inverse of a number if it is given in rectangular coordinates. Exp = exp ¯ log = log ¯ if z is non-zero If p is a polynomial with real coefficients, thus, non-real roots of real polynomials occur in complex conjugate pairs. In general, if ϕ is a function whose restriction to the real numbers is real-valued. The map σ = z ¯ from C to C is a homeomorphism and antilinear, even though it appears to be a well-behaved function, it is not holomorphic, it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R and this Galois group has only two elements, σ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation. Similarly, for a fixed complex unit u = exp, the equation z − z 0 z ¯ − z 0 ¯ = u determines the line through z 0 in the direction of u
27.
Sign function
–
In mathematics, the sign function or signum function is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is represented as sgn. The signum function of a number x is defined as follows. Any real number can be expressed as the product of its value and its sign function. The numbers cancel and all we are left with is the sign of x, D | x | d x = sgn for x ≠0. The signum function is differentiable with derivative 0 everywhere except at 0, using this identity, it is easy to derive the distributional derivative, d sgn d x =2 d H d x =2 δ. The signum can also be using the Iverson bracket notation. The signum can also be using the floor and the absolute value functions. For k ≫1, an approximation of the sign function is sgn ≈ tanh . Another approximation is sgn ≈ x x 2 + ε2, which gets sharper as ε →0, note that this is the derivative of √x2 + ε2. This is inspired from the fact that the above is equal for all nonzero x if ε =0. See Heaviside step function – Analytic approximations, the signum function can be generalized to complex numbers as, sgn = z | z | for any complex number z except z =0. The signum of a complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠0, sgn = e i arg z and we then have, csgn = z z 2 = z 2 z. At real values of x, it is possible to define a generalized function–version of the function, ε such that ε2 =1 everywhere. This generalized signum allows construction of the algebra of generalized functions, absolute value Heaviside function Negative number Rectangular function Sigmoid function Step function Three-way comparison Zero crossing Modulus function
28.
Matrix (mathematics)
–
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
29.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
30.
Cylindrical coordinate system
–
The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The origin of the system is the point where all three coordinates can be given as zero and this is the intersection between the reference plane and the axis. The third coordinate may be called the height or altitude, longitudinal position and they are sometimes called cylindrical polar coordinates and polar cylindrical coordinates, and are sometimes used to specify the position of stars in a galaxy. The three coordinates of a point P are defined as, The radial distance ρ is the Euclidean distance from the z-axis to the point P. The azimuth φ is the angle between the direction on the chosen plane and the line from the origin to the projection of P on the plane. The height z is the distance from the chosen plane to the point P. As in polar coordinates, the point with cylindrical coordinates has infinitely many equivalent coordinates, namely and. Moreover, if the radius ρ is zero, the azimuth is arbitrary, in situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be non-negative and the azimuth φ to lie in a specific interval spanning 360°, such as. The notation for cylindrical coordinates is not uniform, the ISO standard 31-11 recommends, where ρ is the radial coordinate, φ the azimuth, and z the height. However, the radius is often denoted r or s, the azimuth by θ or t. In concrete situations, and in many illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height. The cylindrical coordinate system is one of many coordinate systems. The following formulae may be used to convert between them, the arcsin function is the inverse of the sine function, and is assumed to return an angle in the range =. These formulas yield an azimuth φ in the range, for other formulas, see the polar coordinate article. Many modern programming languages provide a function that will compute the correct azimuth φ, in the range, given x and y, for example, this function is called by atan2 in the C programming language, and atan in Common Lisp. In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements, the line element is d r = d ρ ρ ^ + ρ d φ φ ^ + d z z ^. The volume element is d V = ρ d ρ d φ d z, the surface element in a surface of constant radius ρ is d S ρ = ρ d φ d z. The surface element in a surface of constant azimuth φ is d S φ = d ρ d z, the surface element in a surface of constant height z is d S z = ρ d ρ d φ
31.
Spherical coordinate system
–
It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, the use of symbols and the order of the coordinates differs between sources. In both systems ρ is often used instead of r, other conventions are also used, so great care needs to be taken to check which one is being used. A number of different spherical coordinate systems following other conventions are used outside mathematics, in a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes, the polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon, the spherical coordinate system generalises the two-dimensional polar coordinate system. It can also be extended to spaces and is then referred to as a hyperspherical coordinate system. To define a coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows, the inclination is the angle between the zenith direction and the line segment OP. The azimuth is the angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane. The sign of the azimuth is determined by choosing what is a sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate systems definition, the elevation angle is 90 degrees minus the inclination angle. If the inclination is zero or 180 degrees, the azimuth is arbitrary, if the radius is zero, both azimuth and inclination are arbitrary. In linear algebra, the vector from the origin O to the point P is often called the vector of P. Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of to denote radial distance, inclination, and azimuth, respectively, is common practice in physics, and is specified by ISO standard 80000-2,2009, and earlier in ISO 31-11
32.
Euclidean vector
–
In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
33.
Boldface
–
In typography, emphasis is the exaggeration of words in a text with a font in a different style from the rest of the text—to emphasize them. It is the equivalent of prosodic stress in speech, the most common methods in Bold fall under the general technique of emphasis through a change or modification of font, italics, boldface and small caps. Other methods include the alteration of letter case and spacing as well as color, the human eye is very receptive to differences in brightness within a text body. Therefore, one can differentiate types of emphasis according to whether the emphasis changes the “blackness” of text. With one or the other of these techniques, words can be highlighted without making them out much from the rest of the text. This was used for marking passages that have a different context, such as words from languages, book titles. By contrast, a bold font weight makes text darker than the surrounding text, for example, printed dictionaries often use boldface for their keywords, and the names of entries can conventionally be marked in bold. Small capitals are used for emphasis, especially for the first line of a section, sometimes accompanied by or instead of a drop cap. If the text body is typeset in a typeface, it is also possible to highlight words by setting them in a sans serif face. It is still using some font superfamilies, which come with matching serif and sans-serif variants. In Japanese typography, due to the legibility of heavier Minchō type. Of these methods, italics, small capitals and capitalisation are oldest, with bold type, the house styles of many publishers in the United States use all caps text for, chapter and section headings, newspaper headlines, publication titles, warning messages, word of important meaning. Capitalization is used less commonly today by British publishers. All-uppercase letters are a form of emphasis where the medium lacks support for boldface, such as old typewriters, plain-text email, SMS. Culturally all-caps text has become an indication of shouting, for example when quoting speech and it was also once often used by American lawyers to indicate important points in a legal text. Another means of emphasis is to increase the spacing between the letters, rather than making them darker, but still achieving a distinction in blackness and this results in an effect reverse to boldface, the emphasized text becomes lighter than its environment. This is often used in typesetting and typewriter manuscripts. On typewriters a full space was used between the letters of a word and also one before and one after the word
34.
Scalar (mathematics)
–
A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector, more generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, a vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part, the term is also sometimes used informally to mean a vector, matrix, tensor, or other usually compound value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, the term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix. The word scalar derives from the Latin word scalaris, a form of scala. The English word scale also comes from scala, according to a citation in the Oxford English Dictionary the first recorded usage of the term scalar in English came with W. R. A vector space is defined as a set of vectors, a set of scalars, and a multiplication operation that takes a scalar k. For example, in a space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k, the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. According to a theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a vector space where the coordinates are elements of K. For example, every vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a function that assigns to every vector v in V a scalar ||v||. By definition. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k, a vector space equipped with a norm is called a normed vector space. The norm is defined to be an element of Vs scalar field K. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four operations, thus the rational numbers Q are excluded
35.
Bessel function
–
The most important cases are for α an integer or half-integer. Bessel functions for integer α are also known as functions or the cylindrical harmonics because they appear in the solution to Laplaces equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates, Bessels equation arises when finding separable solutions to Laplaces equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore important for many problems of wave propagation. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of order, in spherical problems. Because this is a differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, different variations are summarized in the table below, and described in the following sections. Bessel functions of the kind and the spherical Bessel functions of the second kind are sometimes denoted by Nn and nn, respectively, rather than Yn. It is possible to define the function by its series expansion around x =0, Γ2 m + α, where Γ is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is a function if α is an integer. For non-integer α, the functions J α and J−α are linearly independent, on the other hand, for integer order α, the following relationship is valid, J − n = n J n. This means that the two solutions are no longer linearly independent, in this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below. Another definition of the Bessel function, for values of n, is possible using an integral representation. Another integral representation is, J n =12 π ∫ − π π e i d τ and this was the approach that Bessel used, and from this definition he derived several properties of the function. The Bessel functions can be expressed in terms of the hypergeometric series as J α = α Γ0 F1. This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function. In terms of the Laguerre polynomials Lk and arbitrarily chosen parameter t and these are sometimes called Weber functions as they were introduced by H. M. Weber, and also Neumann functions after Carl Neumann. For non-integer α, Yα is related to Jα by, Y α = J α cos − J − α sin
36.
International Organization for Standardization
–
The International Organization for Standardization is an international standard-setting body composed of representatives from various national standards organizations. Founded on 23 February 1947, the organization promotes worldwide proprietary and it is headquartered in Geneva, Switzerland, and as of March 2017 works in 162 countries. It was one of the first organizations granted general consultative status with the United Nations Economic, ISO, the International Organization for Standardization, is an independent, non-governmental organization, the members of which are the standards organizations of the 162 member countries. It is the worlds largest developer of international standards and facilitates world trade by providing common standards between nations. Nearly twenty thousand standards have been set covering everything from manufactured products and technology to food safety, use of the standards aids in the creation of products and services that are safe, reliable and of good quality. The standards help businesses increase productivity while minimizing errors and waste, by enabling products from different markets to be directly compared, they facilitate companies in entering new markets and assist in the development of global trade on a fair basis. The standards also serve to safeguard consumers and the end-users of products and services, the three official languages of the ISO are English, French, and Russian. The name of the organization in French is Organisation internationale de normalisation, according to the ISO, as its name in different languages would have different abbreviations, the organization adopted ISO as its abbreviated name in reference to the Greek word isos. However, during the meetings of the new organization, this Greek word was not invoked. Both the name ISO and the logo are registered trademarks, the organization today known as ISO began in 1926 as the International Federation of the National Standardizing Associations. ISO is an organization whose members are recognized authorities on standards. Members meet annually at a General Assembly to discuss ISOs strategic objectives, the organization is coordinated by a Central Secretariat based in Geneva. A Council with a membership of 20 member bodies provides guidance and governance. The Technical Management Board is responsible for over 250 technical committees, ISO has formed joint committees with the International Electrotechnical Commission to develop standards and terminology in the areas of electrical and electronic related technologies. Information technology ISO/IEC Joint Technical Committee 1 was created in 1987 to evelop, maintain, ISO has three membership categories, Member bodies are national bodies considered the most representative standards body in each country. These are the members of ISO that have voting rights. Correspondent members are countries that do not have their own standards organization and these members are informed about ISOs work, but do not participate in standards promulgation. Subscriber members are countries with small economies and they pay reduced membership fees, but can follow the development of standards
37.
NIST
–
The National Institute of Standards and Technology is a measurement standards laboratory, and a non-regulatory agency of the United States Department of Commerce. Its mission is to promote innovation and industrial competitiveness, in 1821, John Quincy Adams had declared Weights and measures may be ranked among the necessities of life to every individual of human society. From 1830 until 1901, the role of overseeing weights and measures was carried out by the Office of Standard Weights and Measures, president Theodore Roosevelt appointed Samuel W. Stratton as the first director. The budget for the first year of operation was $40,000, a laboratory site was constructed in Washington, DC, and instruments were acquired from the national physical laboratories of Europe. In addition to weights and measures, the Bureau developed instruments for electrical units, in 1905 a meeting was called that would be the first National Conference on Weights and Measures. Quality standards were developed for products including some types of clothing, automobile brake systems and headlamps, antifreeze, during World War I, the Bureau worked on multiple problems related to war production, even operating its own facility to produce optical glass when European supplies were cut off. Between the wars, Harry Diamond of the Bureau developed a blind approach radio aircraft landing system, in 1948, financed by the Air Force, the Bureau began design and construction of SEAC, the Standards Eastern Automatic Computer. The computer went into operation in May 1950 using a combination of vacuum tubes, about the same time the Standards Western Automatic Computer, was built at the Los Angeles office of the NBS and used for research there. A mobile version, DYSEAC, was built for the Signal Corps in 1954, due to a changing mission, the National Bureau of Standards became the National Institute of Standards and Technology in 1988. Following 9/11, NIST conducted the investigation into the collapse of the World Trade Center buildings. NIST had a budget for fiscal year 2007 of about $843.3 million. NISTs 2009 budget was $992 million, and it also received $610 million as part of the American Recovery, NIST employs about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 1,800 NIST associates complement the staff, in addition, NIST partners with 1,400 manufacturing specialists and staff at nearly 350 affiliated centers around the country. NIST publishes the Handbook 44 that provides the Specifications, tolerances, the Congress of 1866 made use of the metric system in commerce a legally protected activity through the passage of Metric Act of 1866. NIST is headquartered in Gaithersburg, Maryland, and operates a facility in Boulder, nISTs activities are organized into laboratory programs and extramural programs. Effective October 1,2010, NIST was realigned by reducing the number of NIST laboratory units from ten to six, nISTs Boulder laboratories are best known for NIST‑F1, which houses an atomic clock. NIST‑F1 serves as the source of the official time. NIST also operates a neutron science user facility, the NIST Center for Neutron Research, the NCNR provides scientists access to a variety of neutron scattering instruments, which they use in many research fields
38.
ISO 2
–
ISO2 is an international standard for direction of twist designation for yarns, complex yarns, slivers, slubbings, rovings, cordage, and related products. The standard uses capital letters S and Z to indicate the direction of twist, the handedness of the twist is the direction of the twists as they progress away from an observer. Thus Z-twist is said to be right-handed, and S-twist to be left-handed, the convention of using these two letters to unambiguously designate twist direction was already used in the cordage industry by 1957. ISO2,1973 Textiles — Designation of the direction of twist in yarns and related products
39.
Preferred number
–
In industrial design, preferred numbers are standard guidelines for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distances, diameters, volumes, preferred numbers serve two purposes, Using them increases the probability of compatibility between objects designed at different times by different people. They therefore help to minimize the number of different sizes that need to be manufactured or kept in stock, preferred numbers represent preferences of simple numbers and their powers of a convenient basis, usually 10. In 1870 Charles Renard proposed a set of preferred numbers and his system was adopted in 1952 as international standard ISO3. Renards system divides the interval from 1 to 10 into 5,10,20, the factor between two consecutive numbers in a Renard series is approximately constant, namely the 5th, 10th, 20th, or 40th root of 10, which leads to a geometric sequence. This way, the relative error is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10. In applications for which the R5 series provides a too fine graduation and it is effectively an R3 series rounded to one significant digit. 0.10.20.51251020501002005001000 and this series covers a decade in three steps. Adjacent values differ by factors 2 or 2.5, unlike the Renard series, the 1-2-5 series has not been formally adopted as an international standard. However, the Renard series R10 can be used to extend the 1-2-5 series to a finer graduation and this series is used to define the scales for graphs and for instruments that display in a two-dimensional form with a graticule, such as oscilloscopes. The denominations of most modern currencies, notably the euro and British pound, the United States and Canada follow the series 5,10,25,50,100, and also $5 and $10 which belong to the same series. However, after that comes $20, not $25, the ¼-½-1 series is also used by currencies derived from the former Dutch gulden, some Middle Eastern currencies, and the Seychellois rupee. However, newer notes introduced in Lebanon and Syria due to follow the standard 1-2-5 series instead. The E-series is another system of preferred numbers and it consists of the E3, E6, E12, E24, E48, E96 and E192 series. Originally introduced in 1948, it was standardized in the international standard IEC63 by the International Electrotechnical Commission in 1963. It works similarly to the Renard series, except that it subdivides the interval from 1 to 10 into 3,6,12,24,48,96 or 192 steps. These subdivisions ensure that when some arbitrary value is replaced with the nearest preferred number, use of the E series is mostly restricted to electronic parts like resistors, capacitors, inductors and Zener diodes. Commonly produced dimensions for other types of components are either chosen from the Renard series instead or are defined in relevant product standards
40.
Film speed
–
Film speed is the measure of a photographic films sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system. A closely related ISO system is used to measure the sensitivity of digital imaging systems, highly sensitive films are correspondingly termed fast films. In both digital and film photography, the reduction of exposure corresponding to use of higher sensitivities generally leads to reduced image quality, in short, the higher the sensitivity, the grainier the image will be. Ultimately sensitivity is limited by the efficiency of the film or sensor. The speed of the emulsion was then expressed in degrees Warnerke corresponding with the last number visible on the plate after development. Each number represented an increase of 1/3 in speed, typical speeds were between 10° and 25° Warnerke at the time. The concept, however, was built upon in 1900 by Henry Chapman Jones in the development of his plate tester. In their system, speed numbers were inversely proportional to the exposure required, for example, an emulsion rated at 250 H&D would require ten times the exposure of an emulsion rated at 2500 H&D. The methods to determine the sensitivity were later modified in 1925, the H&D system was officially accepted as a standard in the former Soviet Union from 1928 until September 1951, when it was superseded by GOST 2817-50. The Scheinergrade system was devised by the German astronomer Julius Scheiner in 1894 originally as a method of comparing the speeds of plates used for astronomical photography, Scheiners system rated the speed of a plate by the least exposure to produce a visible darkening upon development. ≈2 The system was extended to cover larger ranges and some of its practical shortcomings were addressed by the Austrian scientist Josef Maria Eder. Scheiners system was abandoned in Germany, when the standardized DIN system was introduced in 1934. In various forms, it continued to be in use in other countries for some time. The DIN system, officially DIN standard 4512 by Deutsches Institut für Normung, was published in January 1934, International Congress of Photography held in Dresden from August 3 to 8,1931. The DIN system was inspired by Scheiners system, but the sensitivities were represented as the base 10 logarithm of the sensitivity multiplied by 10, similar to decibels. Thus an increase of 20° represented an increase in sensitivity. ≈3 /10 As in the Scheiner system, speeds were expressed in degrees, originally the sensitivity was written as a fraction with tenths, where the resultant value 1.8 represented the relative base 10 logarithm of the speed. Tenths were later abandoned with DIN4512, 1957-11, and the example above would be written as 18° DIN, the degree symbol was finally dropped with DIN4512, 1961-10
41.
British Standard Pipe
–
It has been adopted as standard in plumbing and pipe fitting, except in the United States, where NPT and related threads are the standard used. These can be combined into two types of joints, Jointing threads These are pipe threads where pressure-tightness is made through the mating of two threads together. In the modern standard version, it is simply a size number. For a taper thread, it is the diameter at the length from the small end of the thread. The taper is 1 to 16, meaning that for each 16 units of measurement increase in the distance from the end, for left-hand threads, the letters, LH, are appended