Multiplication
product law example laws
Multiplication is often described as repeated addition. For example, the product 3 × 4 is equal to the sum, 4 + 4 + 4, of three 4s. The law on which this is based is the distributive law: a(b + c) = ab + ac. In this instance, the law is applied to 4(1 + 1 + 1), which gives 4(1) + 4(1) + 4(1) or 4 + 4 + 4.
When one or both of the multipliers are not natural numbers, the law still applies, 0.4(1.2) = 0.4(1) + 0.4(0.1) + 0.4(0.1), but the terms of the sum are not simply "repeated," and other rules, such as the rules for placing the decimal point, are needed.
Of course one does not go all the way back to the distributive law every time he or she computes a product. In fact, much of the time in the early grades in school is devoted to building a multiplication table and memorizing it for subsequent use. In applying that table to products such as 12 × 23, however, one does make explicit use of the distributive law:
Here a is 12; b is 20; and c is 3.
In talking about multiplication, several terms are used. In 6 × 3, the entire expression, whether it is written as 6 × 3 or as 18, is called the product. The 6 and the 3 are each called multipliers, factors, or occasionally terms. The older words "multiplicand" (for the 6) and "multiplier" (for the 3), which made a distinction between the number that got multiplied and the number that did the multiplying, have fallen into disuse. Now "multiplier" applies to either number.
Multiplication is symbolized in three ways: with an " ×," as in 6 × 3; with a centered dot, as in 6 • 3; and by writing the numbers next to each other, as in 5x, 6(3), (6)(3), or (x + y)(x - y). This last way is usually preferred.
For all numbers a, b, and c | |
ab is a unique number | the closure law |
ab = ba | the commutative law |
a(bc) = (ab)c | the associative law |
a•1 = a | the multiplicative identity law |
If ab = cb and b ≠ | the cancellation law |
From these laws one can derive three more useful laws: | |
a•0 = 0 | multiplication by zero property |
If ab = 0, then a = 0, or b = 0, or both. | nonexistence of zero-divisors |
The factors in a product may be combined in any order. | generalized commutative property |
Multiplication is governed not only by the distributive law, which connects it with addition, but by laws that apply to multiplication alone. These laws appear in the table above.
Since arithmetic is done with natural numbers, some additional laws are needed to handle decimal fractions, common fractions, and other numbers which are not natural numbers:
Decimals: Multiply the decimal fractions as if they were natural numbers. Place the decimal point in the product so that the number of places in the product is the sum of the number of places in the multipliers. For example, 3.07 × 5.2 = 15.964.
Fractions: The numerator of the product is the product of the numerators; the denominator of the product is the product of the denominators. For example, (3/7)(5/4) = 15/28.
Signed numbers: Multiply the numbers as if they had no signs. If one of the two factors has a minus sign, give the product a minus sign. If both factors have minus signs, write the product without a minus sign. For example, (3x)(-2y) = -6xy; (-5)(-4) = 20; (-x)^{2} = x^{2}; and (5x)(x) = 5x^{2}.
Powers of the same base: To multiply two powers of the same base, add the exponents. For example 10^{2} × 10^{3} = 10^{5}, and x^{2} × x = x^{3}.
Monomials: To multiply two monomials, rearrange the factors. For example, (3x^{2}y)(5xyz) = 15x^{3}y^{2}z.
Polynomials: To multiply two polynomials multiply each term of one by each term of the other, combining like terms. For example, (x + y)(x - y) = x^{2} - xy + xy - y^{2 }= x^{2} - y^{2}.
Multiplication is the model for a variety of practical situations. In one we have a number, a, of groups with b things in each group. The product, ab, represents the total number. For example, seven egg cartons hold 7 × 12 eggs in all. In other situations we have "direct variation" or proportionality: y = kx. So many gallons of gasoline at so much per gallon calls for multiplication, as does computing distance as a function of rate and time, D = RT.
Unlike addition, in which a length plus a length is another length, and a length plus a weight, meaningless, the product of two quantities of the same type or of different types is frequently meaningful, and of a type different from both. For example, the product of two one-dimensional measures such as length becomes the two-dimensional measure, area. Multiplying a force, such as gravity, by a distance yields work, which is a change in the amount of energy. Thus, while multiplication is closely tied to addition computationally, in application it accommodates relationships of more complex dimensions.
While multiplication is ordinarily an operation between numbers, it can be an operation between other kinds of mathematical elements as well. Multiplication in the broader sense will obey many of the laws ordinary multiplication does, but not necessarily all of them.
For example in "clock arithmetic" all the basic laws hold except for cancellation. In clock arithmetic, 3 × 4 = 3 × 8 because both leave the hands in the same position, but of course, 4 does not equal 8. In the multiplication of matrices, commutativity does not hold.
A particularly interesting extension of the idea of multiplication is in the Cartesian product of two sets. If A = {1, 2, 3}, and B = {x, y}, then A × B is the set {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)}, formed by pairing each element of A with each element of B. Because sets are sometimes used as the basis for arithmetic, Cartesian products form an important link between sets and ordinary multiplication.
See also Division.
Resources
Books
Dantzig, Tobias. Number, the Language of Science. Garden City, NY: Doubleday and Co., 1954.
Klein, Felix. "Arithmetic." In Elementary Mathematics from an Advanced Standpoint. New York: Dover, 1948.
Smith, David Eugene, and Jekuthiel Ginsberg."From Numbers to Numerals and from Numerals to Computation." In The World of Mathematics. edited by James R. Newman. New York: Simon and Schuster, 1956.
Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.
Periodicals
Jourdain, Philip E. B. "The Nature of Mathematics." The World of Mathematics Edited by James R. Newman. New York: Simon and Schuster, 1956.
J. Paul Moulton
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