Multiplication
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- This article is about multiplication in mathematics. For multiplication in music, see multiplication (music).
- [ \atop n}]
Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = 6/12 = 1/2.
Multiplication can be defined for real and complex numbers, polynomials, matricies and other mathematical quantities as well. The inverse of multiplication is division.
Computation
For several ways to compute products, including very large number, see multiplication algorithms.The standard methods for multiplying numbers using pencil and paper require 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.
Multplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
Terminology
The two numbers being multiplied are formally called the multiplier and the multiplicand (the number multiplied by the multiplier). The difference was important in those numeral systems, such as Roman numerals, in which multiplication is transformation of symbols and their addition. For example, a person multiplying VII by XV changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V) when XV is the multiplier, and XV into LXXV (XV times V) plus XV plus XV (each XV times I) when it is VII. With Arabic numerals, the relations between the terms are all memorized.Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers so they are more commonly referred to as the factors. The result of the multiplication is referred to as the product.
Notation
Multiplication can be denoted in several equivalent ways. All of the following mean, "5 multiplied by 2":
- 5×2
- 5·2
- (5)2, 5(2), (5)(2), 5[2], [5Τ, [5][2]
- 5*2
- 5.2
- 5x and xy
If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written [1 \cdot 2 \cdot \ldots \cdot 99 \cdot 100]. This can also be written with the ellipsis vertically placed in the middle of the line, as [1 \cdot 2 \cdot \cdots \cdot 99 \cdot 100].
Capital pi notation
Alternatively, a product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. This is defined as:
- [ \prod_^ x_ := x_ \cdot x_ \cdot x_ \cdot \cdots \cdot x_ \cdot x_. ]
- [ \prod_^ \left(1 + \right) = \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) = . ]
Infinite products
One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). The product of such a series is defined as the limit of the product of the first [n] terms, as [n] grows without bound. That is:
- [ \prod_^ x_ := \lim_ \prod_^ x_. ]
- [\prod_^\infty x_i := \left(\lim_\prod_^m x_i\right) \cdot \left(\lim_\prod_^n x_i\right),]
Properties
For integers, fractions, real and complex numbers, multiplication has certain properties:
- the order in which two numbers are multiplied does not matter. This is called the commutative property,
- x · y = y · x.
- The associative property means that for any three numbers x, y, and z,
- (x · y)z = x(y · z).
- Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done.
- Multiplication also has what is called a distributive property with respect to the addition,
- x(y + z) = xy + xz.
- Also of interest is that any number times 1 is equal to itself, thus,
- 1 · x = x.
- and this is called the identity property. In this regard the number 1 is known as the multiplicative identity.
- The sum of zero numbers is zero, so
- m · 0 = 0
- Multiplication with negative numbers also requires a little thought. First consider negative one (-1). For any positive integer m:
- (−1)m = (−1) + (−1) +...+ (−1) = −m
- This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s. All that remains is to explicitly define (−1)(−1):
- (−1)(−1) = −(−1) = 1
- Every number x, except zero, has a multiplicative inverse, 1/x, such that x × 1/x = 1.
- Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0, then if b > c then ab < ac.
See also
- Peasant multiplication
- Multiplicative inverse, the reciprocal
- Multiplication table (times table)
- Napier's bones
- Product (mathematics) - lists generalizations
- Slide rule
External links
- [Practicing and Learning Multiplication]
- [Multiplication] and [Arithmetic Operations In Various Number Systems] at cut-the-knot
- [Multiplying Negative Numbers]
- [Modern Chinese Multiplication Techniques on an Abacus]
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