52-485: Multiplicative may refer to: Multiplication Multiplicative function Multiplicative group Multiplicative identity Multiplicative inverse Multiplicative order Multiplicative partition Multiplicative case For the multiplicative numerals once, twice, and thrice, see English numerals Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
104-593: A = sup x ∈ A x {\displaystyle a=\sup _{x\in A}x} and b = sup y ∈ B y , {\displaystyle b=\sup _{y\in B}y,} then a ⋅ b = sup x ∈ A , y ∈ B x ⋅ y . {\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.} In particular,
156-461: A multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"): In some countries such as Germany , the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with
208-411: A rectangle (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths . The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property. The product of two measurements (or physical quantities ) is a new type of measurement, usually with a derived unit . For example, multiplying the lengths (in meters or feet) of
260-591: A list of the first twenty multiples of a certain principal number n : n , 2 n , ..., 20 n ; followed by the multiples of 10 n : 30 n 40 n , and 50 n . Then to compute any sexagesimal product, say 53 n , one only needed to add 50 n and 3 n computed from the table. In the mathematical text Zhoubi Suanjing , dated prior to 300 BC, and the Nine Chapters on the Mathematical Art , multiplication calculations were written out in words, although
312-461: A multiplication sign (such as ⋅ or × ), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language. The numbers to be multiplied are generally called the "factors" (as in factorization ). The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier". Usually, the multiplier is placed first, and
364-567: A nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their positive amounts , combined with the sign derived from the following rule: × + − + + − − − + {\displaystyle {\begin{array}{|c|c c|}\hline \times &+&-\\\hline +&+&-\\-&-&+\\\hline \end{array}}} (This rule
416-431: A pentagon. The unique ratio of side lengths is a b = 0.815023701... {\displaystyle \displaystyle {\frac {a}{b}}=0.815023701...} . A crossed quadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of
468-511: A professor of mathematics at Princeton University , wrote the following: These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci in the 13th century. Grid method multiplication , or the box method, is used in primary schools in England and Wales and in some areas of
520-406: A rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex. A crossed quadrilateral is sometimes likened to a bow tie or butterfly , sometimes called an "angular eight". A three-dimensional rectangular wire frame that is twisted can take
572-434: A rectangle is a rhombus , as shown in the table below. A rectangle is a rectilinear polygon : its sides meet at right angles. A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position (comprising two of translation and one of rotation ), one for shape ( aspect ratio ), and one for overall size (area). Two rectangles, neither of which will fit inside
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#1732771800805624-435: A rectangle. A parallelogram with equal diagonals is a rectangle. The Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A , B , C , and D , for any point P on the same plane of a rectangle: For every convex body C in
676-414: A unique rectangle with sides a {\displaystyle a} and b {\displaystyle b} , where a {\displaystyle a} is less than b {\displaystyle b} , with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and
728-432: Is a convex quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which
780-506: Is a rectilinear convex polygon or a quadrilateral with four right angles . It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square . The term " oblong " is used to refer to a non- square rectangle. A rectangle with vertices ABCD would be denoted as [REDACTED] ABCD . The word rectangle comes from
832-430: Is a consequence of the distributivity of multiplication over addition, and is not an additional rule .) In words: Two fractions can be multiplied by multiplying their numerators and denominators: There are several equivalent ways to define formally the real numbers; see Construction of the real numbers . The definition of multiplication is a part of all these definitions. A fundamental aspect of these definitions
884-987: Is both a multiple of 3 and a multiple of 5. The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions. The product of two natural numbers r , s ∈ N {\displaystyle r,s\in \mathbb {N} } is defined as: r ⋅ s ≡ ∑ i = 1 s r = r + r + ⋯ + r ⏟ s times ≡ ∑ j = 1 r s = s + s + ⋯ + s ⏟ r times . {\displaystyle r\cdot s\equiv \sum _{i=1}^{s}r=\underbrace {r+r+\cdots +r} _{s{\text{ times}}}\equiv \sum _{j=1}^{r}s=\underbrace {s+s+\cdots +s} _{r{\text{ times}}}.} An integer can be either zero,
936-491: Is often written using the multiplication sign (either × or × {\displaystyle \times } ) between the terms (that is, in infix notation ). For example, There are other mathematical notations for multiplication: In computer programming , the asterisk (as in 5*2 ) is still the most common notation. This is due to the fact that most computers historically were limited to small character sets (such as ASCII and EBCDIC ) that lacked
988-394: Is that every real number can be approximated to any accuracy by rational numbers . A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation ; for example, π {\displaystyle \pi }
1040-421: Is that the magnitudes are multiplied and the arguments are added. The product of two quaternions can be found in the article on quaternions . Note, in this case, that a ⋅ b {\displaystyle a\cdot b} and b ⋅ a {\displaystyle b\cdot a} are in general different. Many common methods for multiplying numbers using pencil and paper require
1092-477: Is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with
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#17327718008051144-412: Is the least upper bound of { 3 , 3.1 , 3.14 , 3.141 , … } . {\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.} A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations , and, in particular, with multiplication. This means that, if a and b are positive real numbers such that
1196-406: Is used in many periodic tessellation patterns, in brickwork , for example, these tilings: A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size,
1248-746: The Latin rectangulus , which is a combination of rectus (as an adjective, right, proper) and angulus ( angle ). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram , and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical , elliptic , and hyperbolic , have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling
1300-464: The Marchant , automated multiplication of up to 10-digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand. Methods of multiplication were documented in the writings of ancient Egyptian , Greek, Indian, and Chinese civilizations. The Ishango bone , dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in
1352-653: The Upper Paleolithic era in Central Africa , but this is speculative. The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus , was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42 , 4 × 21 = 2 × 42 = 84 , 8 × 21 = 2 × 84 = 168 . The full product could then be found by adding
1404-428: The cross symbol × , by the mid-line dot operator ⋅ , by juxtaposition , or, on computers , by an asterisk * ) is one of the four elementary mathematical operations of arithmetic , with the other ones being addition , subtraction , and division . The result of a multiplication operation is called a product . The multiplication of whole numbers may be thought of as repeated addition ; that is,
1456-492: The discrete Fourier transform reduce the computational complexity to O ( n log n log log n ) . In 2016, the factor log log n was replaced by a function that increases much slower, though still not constant. In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of O ( n log n ) . {\displaystyle O(n\log n).} The algorithm, also based on
1508-568: The factors , and 12 is the product . One of the main properties of multiplication is the commutative property , which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3: Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication. Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers. Multiplication can also be visualized as counting objects arranged in
1560-515: The United States to help teach an understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows: and then add the entries. The classical method of multiplying two n -digit numbers requires n digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on
1612-418: The appropriate terms found in the doubling sequence: The Babylonians used a sexagesimal positional number system , analogous to the modern-day decimal system . Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables . These tables consisted of
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1664-583: The early Chinese mathematicians employed Rod calculus involving place value addition, subtraction, multiplication, and division. The Chinese were already using a decimal multiplication table by the end of the Warring States period. The modern method of multiplication based on the Hindu–Arabic numeral system was first described by Brahmagupta . Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry Burchard Fine , then
1716-485: The fast Fourier transform, is conjectured to be asymptotically optimal. The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than 2 bits). One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as: When two measurements are multiplied together,
1768-416: The first digit of the multiplier: Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators , such as
1820-413: The fundamental idea of multiplication. The product of a sequence, vector multiplication , complex numbers , and matrices are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers. In arithmetic , multiplication
1872-419: The multiplicand is placed second; however, sometimes the first factor is considered the multiplicand and the second the multiplier. Also, as the result of multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithms , such as the long multiplication . Therefore, in some sources,
1924-473: The multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand , as the quantity of the other one, the multiplier ; both numbers can be referred to as factors . For example, the expression 3 × 4 {\displaystyle 3\times 4} , phrased as "3 times 4" or "3 multiplied by 4", can be evaluated by adding 3 copies of 4 together: Here, 3 (the multiplier ) and 4 (the multiplicand ) are
1976-418: The other, are said to be incomparable . If a rectangle has length ℓ {\displaystyle \ell } and width w {\displaystyle w} , then: The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter , the square has the largest area . The midpoints of the sides of any quadrilateral with perpendicular diagonals form
2028-399: The others. Thus, 2 × π {\displaystyle 2\times \pi } is a multiple of π {\displaystyle \pi } , as is 5133 × 486 × π {\displaystyle 5133\times 486\times \pi } . A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and
2080-469: The outside and exceed 180°. A rectangle and a crossed rectangle are quadrilaterals with the following properties in common: [REDACTED] In spherical geometry , a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in
2132-517: The plane by rectangles or tiling a rectangle by polygons . A convex quadrilateral is a rectangle if and only if it is any one of the following: A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular . A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length . A trapezium
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2184-439: The plane, we can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and 0.5 × Area ( R ) ≤ Area ( C ) ≤ 2 × Area ( r ) {\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)} . There exists
2236-405: The product is of a type depending on the types of measurements. The general theory is given by dimensional analysis . This analysis is routinely applied in physics, but it also has applications in finance and other applied fields. A common example in physics is the fact that multiplying speed by time gives distance . For example: Rectangle In Euclidean plane geometry , a rectangle
2288-416: The product of two positive real numbers is the least upper bound of the term-by-term products of the sequences of their decimal representations. As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in § Product of two integers . The construction of
2340-505: The real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations. Two complex numbers can be multiplied by the distributive law and the fact that i 2 = − 1 {\displaystyle i^{2}=-1} , as follows: The geometric meaning of complex multiplication can be understood by rewriting complex numbers in polar coordinates : Furthermore, from which one obtains The geometric meaning
2392-417: The same vertex arrangement as isosceles trapezia). A rectangle is cyclic : all corners lie on a single circle . It is equiangular : all its corner angles are equal (each of 90 degrees ). It is isogonal or vertex-transitive : all corners lie within the same symmetry orbit . It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°). The dual polygon of
2444-511: The sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry. In elliptic geometry , an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. In hyperbolic geometry , a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length. The rectangle
2496-403: The shape of a bow tie. The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A crossed rectangle may be considered equiangular if right and left turns are allowed. As with any crossed quadrilateral , the sum of its interior angles is 720°, allowing for internal angles to appear on
2548-405: The term "multiplicand" is regarded as a synonym for "factor". In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in 3 x y 2 {\displaystyle 3xy^{2}} ) is called a coefficient . The result of a multiplication is called a product . When one factor is an integer, the product is a multiple of the other or of the product of
2600-464: The tiling is imperfect . In a perfect (or imperfect) triangled rectangle the triangles must be right triangles . A database of all known perfect rectangles, perfect squares and related shapes can be found at squaring.net . The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search. A rectangle has commensurable sides if and only if it
2652-495: The title Multiplicative . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Multiplicative&oldid=947842846 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Multiplication Multiplication (often denoted by
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#17327718008052704-454: The two sides of a rectangle gives its area (in square meters or square feet). Such a product 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. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1. Several mathematical concepts expand upon
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