Multiplication is one of the four elementary mathematical operations of arithmetic. The multiplication of whole numbers may be thought as a repeated addition; the multiplier can be multiplicand second. A × b = b + ⋯ + b ⏟ a For example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12 Here 3 and 4 are the factors and 12 is the product. One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3: 4 × 3 = 3 + 3 + 3 + 3 = 12 Thus the designation of multiplier and multiplicand does not affect the result of the multiplication; the multiplication of integers, rational numbers and real numbers is defined by a systematic generalization of this basic definition. Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths; the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.
The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is defined for other types of numbers, such as complex numbers, more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is written using the sign "×" between the terms. For example, 2 × 3 = 6 3 × 4 = 12 2 × 3 × 5 = 6 × 5 = 30 2 × 2 × 2 × 2 × 2 = 32 The sign is encoded in Unicode at U+00D7 × MULTIPLICATION SIGN. There are other mathematical notations for multiplication: Multiplication is denoted by dot signs a middle-position dot:5 ⋅ 2 or 5.
3 The middle dot notation, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is standard in the United States, the United Kingdom, other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication. In algebra, multiplication involving variables is written as a juxtaposition called implied multiplication; the notation can be used for quantities that are surrounded by parentheses. This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations. In vector multiplication, there is a distinction between the dot symbols; the cross symbol denotes the taking a cross product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk is still the most common notation. This is due to the fact that most computers were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard; this usage originated in the FORTRAN programming language. The numbers to be multiplied are called the "factors"; the number to be multiplied is the "multiplicand", the number by which it is multiplied is the "multiplier". The multiplier is placed first and the multiplicand is placed second; as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a elementary level and
Addition is one of the four basic operations of arithmetic. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples; this observation is equivalent to the mathematical expression "3 + 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting items, addition can be defined on other types of numbers, such as integers, real numbers and complex numbers; this is part of a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices. Addition has several important properties, it is commutative, meaning that order does not matter, it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 1 is the same as counting. Addition obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks. Addition of small numbers is accessible to toddlers. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign "+" between the terms; the result is expressed with an equals sign. For example, 1 + 1 = 2 2 + 2 = 4 1 + 2 = 3 5 + 4 + 2 = 11 3 + 3 + 3 + 3 = 12 There are situations where addition is "understood" though no symbol appears: A whole number followed by a fraction indicates the sum of the two, called a mixed number. For example, 3½ = 3 + ½ = 3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead; the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration.
For example, ∑ k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands; this is to be distinguished from factors. Some authors call. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is used, both terms are called addends. All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give". Using the gerundive suffix -nd results in "addend", "thing to be added". From augere "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was higher than the addends.
Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus. The Middle English terms "adden" and "adding" were popularized by Chaucer; the plus sign "+" is an abbreviation of the Latin word et, meaning "and". It appears in mathematical works dating back to at least 1489. Addition is used to model many physical processes. For the simple case of adding natural numbers, there are many possible interpretations and more visual representations; the most fundamental interpretation of addition lies in combining sets: When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity, it is useful in higher mathematics. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. One possible fix is to consider collections of objects that can be divided, such as pies or, still bet
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables; this use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers; the use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions.
For example, the expression 3 x 2 − 2 x y + c has the following components: 1: Exponent, 2: Coefficient, 3: term, 4: operator, 5: constant, x, y: variables A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable. A term is an addend or a summand, a group of coefficients, variables and exponents that may be separated from the other terms by the plus and minus operators. Letters represent constants. By convention, letters at the beginning of the alphabet are used to represent constants, those toward the end of the alphabet are used to represent variables, they are written in italics. Algebraic operations work in the same way as arithmetic operations, such as addition, multiplication and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols are omitted, implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 × x 2 is written as 3 x 2, 2 × x × y may be written 2 x y.
Terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is omitted; when the exponent is one. When the exponent is zero, the result is always 1; however 0 0, being undefined, should not appear in an expression, care should be taken in simplifying expressions in which variables may appear in exponents. Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. For example, exponents are formatted using superscripts, e.g. x 2. In plain text, in the TeX mark-up language, the caret symbol "^" represents exponents, so x 2 is written as "x^2". In programming languages such as Ada, Perl and Ruby, a double asterisk is used, so x 2 is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, it must be explicitly used, for example, 3 x is written "3*x".
Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general numbers. This is useful for several reasons. Variables may represent numbers. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P the problem can be described algebraically as C = P + 20. Variables allow one to describe general problems, without specifying the values of the quantities that are involved. For example, it can be stated that 5 minutes is equivalent to 60 × 5 = 300 seconds. A more general description may state that the number of seconds, s = 60 × m, where m is the number of minutes. Variables allow one to describe mathematical relationships between quantities. For example, the relationship between the circumference, c, diameter, d, of a circle is described by π = c / d. Variables allow one to describe some mathematical properties. Fo
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. The result of a subtraction is called a difference. Subtraction is signified by the minus sign. For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, a total of 3 apples. Therefore, the difference of 5 and 2 is 3, that is, 5 − 2 = 3. Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, irrational numbers, decimals and matrices. Subtraction follows several important patterns, it is anticommutative. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters; because 0 is the additive identity, subtraction of it does not change a number. Subtraction obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond.
General binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks. Subtraction of small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. In advanced algebra and in computer algebra, an expression involving subtraction like A − B is treated as a shorthand notation for the addition A +. Thus, A − B contains two terms, namely A and −B; this allows an easier use of commutativity. Subtraction is written using the minus sign "−" between the terms; the result is expressed with an equals sign. For example, 2 − 1 = 1 4 − 2 = 2 6 − 3 = 3 4 − 6 = − 2 There are situations where subtraction is "understood" though no symbol appears: A column of two numbers, with the lower number in red indicates that the lower number in the column is to be subtracted, with the difference written below, under a line.
This is most common in accounting. Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend; the result is the difference. All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, in turn a compound of sub "from under" and trahere "to pull". Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted". From minuere "to reduce or diminish", one gets "minuend", "thing to be diminished". Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c; this movement to the right is modeled mathematically by addition: a + b = c. From c, it takes b steps to the left to get back to a; this movement to the left is modeled by subtraction: c − b = a. Now, a line segment labeled with the numbers 1, 2, 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3, it takes 2 steps to the left to get to position 1, so 3 − 2 = 1.
This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number. From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0, but 3 − 4 is still invalid. The natural numbers are not a useful context for subtraction; the solution is to consider the integer number line. From 3, it takes 4 steps to the left to get to −1: 3 − 4 = −1. Subtraction of natural numbers is not closed; the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number; such a case uses one of two approaches: Say that 26 cannot be subtracted from 11. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15. Subtraction of real numbers is defined as addition of signed numbers. A number is subtracted by adding its additive inverse.
We have 3 − π = 3 +. This helps to keep the ring of real numbers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it. A ring has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything. Subtraction is anti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers a − b = −. Subtraction is non-associative. Should the expres
Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b denoted by LCM, is the smallest positive integer, divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define LCM as 0 for all a, the result of taking the LCM to be the least upper bound in the lattice of divisibility; the LCM is the "lowest common denominator" that can be used before fractions can be added, subtracted or compared. The LCM of more than two integers is well-defined: it is the smallest positive integer, divisible by each of them. A multiple of a number is the product of an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2; because 10 is the smallest positive integer, divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of − 2 as well.
In this article we will denote the least common multiple of two integers b as lcm. Some older textbooks use; the programming language J uses a*.b What is the LCM of 4 and 6? Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76...and the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72... Common multiples of 4 and 6 are the numbers that are in both lists: 12, 24, 36, 48, 60, 72.... So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12; when adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, 2 21 + 1 6 = 4 42 + 7 42 = 11 42 where the denominator 42 was used because it is the least common multiple of 21 and 6. Suppose there are two meshing gears in a machine, having m and n teeth and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear.
When the gears begin rotating, we can determine how many rotations the first gear must complete to realign the line segment by making use of LCM. The first gear must complete LCM/m rotations for the realignment. By that time, the second gear will have made LCM/n rotations. Suppose there are three planets revolving around a star which take l, m and n units of time to complete their orbits. Assume that l, m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after LCM units of time. At this time, the first and third planet will have completed LCM/l, LCM/m and LCM/n orbits around the star; the following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor known as the greatest common factor: lcm = | a ⋅ b | gcd . This formula is valid when one of a and b is 0, since gcd = |a|. However, if both a and b are 0, this formula would cause division by zero.
There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, lcm = 21 ⋅ 6 gcd = 21 ⋅ 6 gcd = 21 ⋅ 6 3 = 126 3 = 42; because gcd is a divisor of both a and b, it is more efficient to compute the LCM by dividing before multiplying: lcm = ⋅ | b | = ⋅ | a |. This reduces the size of one input for both the division and the multiplication, reduces the required storage needed for inte
Arithmetic is a branch of mathematics that consists of the study of numbers the properties of the traditional operations on them—addition, subtraction and division. Arithmetic is an elementary part of number theory, number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra and analysis; the terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory. The prehistory of arithmetic is limited to a small number of artifacts which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed; the earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system influence the complexity of the methods.
The hieroglyphic system for Egyptian numerals, like the Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the sexagesimal system for Babylonian numerals and the vigesimal system that defined Maya numerals; because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs.
For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, their relationships to each other, in his Introduction to Arithmetic. Greek numerals were used by Archimedes and others in a positional notation not different from ours; the ancient Greeks lacked a symbol for zero until the Hellenistic period, they used three separate sets of symbols as digits: one set for the units place, one for the tens place, one for the hundreds. For the thousands place they would reuse the symbols for the units place, so on, their addition algorithm was identical to ours, their multiplication algorithm was only slightly different. Their long division algorithm was the same, the digit-by-digit square root algorithm, popularly used as as the 20th century, was known to Archimedes, who may have invented it, he preferred it to Hero's method of successive approximation because, once computed, a digit doesn't change, the square roots of perfect squares, such as 7485696, terminate as 2736.
For numbers with a fractional part, such as 546.934, they used negative powers of 60 instead of negative powers of 10 for the fractional part 0.934. The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra; the ancient Chinese used a positional notation similar to that of the Greeks. Since they lacked a symbol for zero, they had one set of symbols for the unit's place, a second set for the ten's place. For the hundred's place they reused the symbols for the unit's place, so on, their symbols were based on the ancient counting rods. It is a complicated question to determine when the Chinese started calculating with positional representation, but it was before 400 BC; the ancient Chinese were the first to meaningfully discover and apply negative numbers as explained in the Nine Chapters on the Mathematical Art, written by Liu Hui. The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing 0.
This allowed the system to represent both large and small integers. This approach replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number and determined the results for multiplication, division and subtraction of zero and all other numbers, except for the result of division by 0, his contemporary, the Syriac bishop Severus Sebokht said, "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols." The Arabs learned this new method and called it hesab. Although the Codex Vigilanus described an early form of Arabic numerals by 976 AD, Leonardo of Pisa was responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202, he wrote, "The method of the Indians surpasses any known method to compute.
It's a marvelous method. They do their computations using nine figures and symbol zero". In the Middle Ages, arithmetic was one of the seven
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, 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. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia