Multiplication is a fundamental operation of arithmetic whose use stretches far back into antiquity. Ancient civilizations found the need to 'tally up' the quantities of various goods that were gathered together to be stored, sold, or bartered. Hence the need arose for arithmetic. The most basic and earliest arithmetic operation used would have been for the combining of numbers (i.e., addition). Given large numbers of transactions, it should not be too difficult to imagine how early mankind developed a notation to indicate the repetitive addition of numbers, thereby devising multiplication. That is to say, multiplication was (and is) a kind of shorthand notation for the operation of addition. For instance, the multiplication of 6 by 5 indicates the successive addition of "6" five times (6 + 6 + 6 + 6 + 6); the resulting number "30" is called the product of "6 times 5". Over the course of time, multiplication tables and other techniques were devised to aid in finding the product of numbers. (Today, children still refer to multiplication cards or tables in order to memorize the products of various numbers.) The word multiply is from the Middle English "multiplien", derived from the Old French "multiplier", which in turn was based on the Latin "multiplicare". The first known appearance of the word multiply was in 1390 in the Confessio ameantis n by John Gower.
Various symbols are used to indicate the multiplication of two numbers. For numbers "a" and "b", "a x b", "a ⋅ b","(a)(b)", and "ab" are equivalent ways of expressing the multiplication of "a times b". However, "x" is most commonly used in arithmetic expressions, while the other three multiplicative notations typically appear in higher mathematics, such as in algebraic expressions. In the expression "a x b", the numbers a and b are both called factors.
As mentioned above, multiplication is a fundamental operation of arithmetic (altogether there are four basic arithmetic operations: addition, subtraction, multiplication and division). By calling multiplication an operation, it is meant that any two numbers multiplied together results in a third unique number. As pointed out earlier, the process of multiplication stretches back into antiquity. It would have originally been used for the counting numbers (i.e., the whole numbers {1, 2, 3, 4,...}). Over time, the use of multiplication was expanded to included fractions, negative numbers, irrational numbers, etc., on up to the so-called real number system. For all these number systems the following rules are valid for the operation of multiplication. (The laws are given in terms of real numbers "a, b, and c", but the laws are also valid for the number systems discussed previously, like the rational numbers. Also, a bracketed operation, like "(b x c)", is to be carried out before an unbracketed operation):
Law 1: "a x b" is a unique real number, closure law;
Law 2: "a x b = b x a", commutative law;
Law 3: "a x (b x c) = (a x b) x c", associative law;
Law 4: "a x 1 = a", identity law;
Law 5: for every number "a" (except zero) there exists a corresponding and unique number "1 / a" such that "a x (1 / a) = 1", multiplicative inverse law; and
Law 6: "a x (b + c) = (a x b) + (a x c)", distributive law of multiplication.
It should be noted that law 5 above does not apply for the counting or natural numbers nor for the set of integers, since by definition those number systems do not encompass reciprocals. By using the multiplicative laws above one can derive additional rules, or theorems, governing multiplication of the real numbers. For example, the rule of 'equality for numbers' states that if "a = b" and "c = d", then "a x c = b x d" is valid. The (above) multiplicative laws and the 'laws of equality' demonstrate the validity of this statement:
Step 1: "a x c" is a real number, closure law;
Step 2: "a x c = a x c", reflexive rule of equality (i.e., any number equals itself);
Step 3: "a x c = b x c", substitution rule of equality (i.e., "b" substituted for "a");
Step 4: "a x c = b x d", substitution rule of equality (i.e., "d" substituted for "c") [QED].
Multiplication of fractions is encountered in the study of arithmetic. Briefly, the product of two fractions is equal to the product of the two numerators over the product of the two denominators, and the resulting fraction is reduced to lowest terms, as in the following example: "(2 / 3) x (5 / 6) = (2 x 5) / (3 x 6) = 10 / 18 = 5 / 9".
Multiplication is also defined as an operation on complex numbers, vectors, functions, matrices, tensors, and so forth. For these mathematical objects the laws of multiplication given previously for the real numbers, rational numbers, etc., may or may not hold. By way of highlighting some of the deviations from the ordinary rules of multiplication, the multiplication of matrices is described below.
The multiplication of matrices is valid only if the order of matrix A is (i x j) and the order of matrix B is (j x k); that is multiplication is only valid if the column (j) of matrix A is equal in order to the row (j) of matrix B. In addition, the commutative law of multiplication is not valid for matrix multiplication. The product of matrix multiplication is "C = A x B" and the order of the resulting matrix C is of order (i x k) and its elements are cik = (aijbjk), where j is summed over for all possible values of i and k.
The study of multiplication is taken to a deeper level in the branch of mathematics known as abstract algebra. Beginning in the nineteenth century, in an attempt to further the theory of solutions to algebraic equations, certain patterns or rules (like the multiplicative laws stated previously) were related to the operations of addition and multiplication as applied to the real numbers. These investigations eventually led to applications in disparate fields of mathematics (like algebra and topology), and became important in their own right. For example, it turns out that the non-zero rational, real and complex numbers form what are called infinite Abelian groups under the operation of multiplication. Similarly, the dual arithmetic operations of multiplication and addition form what are called fields upon various sets, such as the set of real numbers.
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