Products and Quotients
Products and quotients are the results obtained by applying the operations of multiplication and division, respectively, to particular mathematical objects. Both will be examined; products first, followed by quotients.
Product is defined in mathematics as the quantity obtained by multiplying one quantity or expression by another quantity or expression. For instance, the expression "5 x 6" is the product of the whole numbers "5" and "6". Products are encountered in elementary arithmetic: children typically learn by rote the products of various whole numbers. Beyond whole numbers, there are also products of integers, rational numbers, real numbers, and complex numbers, as well as products of vectors, functions, matrices, tensors, and other 'higher' mathematical objects.
The word product first appeared in use during the first half of the 15th century. Product is from the Middle English and the Medieval Latin word "productum" (something produced), which was derived from the neuter of the Latin "productus" and the past participle of the Latin "producere" (to bring forth). In its simplest form the product (the result of multiplying one quantity by another) is found by multiplying a number "a" by a number "b". The number "a" is called the multiplicator and "b" is called the multiplicand. Both the multiplicator and multiplicand are called factors because the product is identical if the multiplicator and multiplicand are switched.
Various symbols are used to indicate the product of two numbers. The symbols "x" (lying cross), "⋅" (raised dot), and "*" (asterisk) are all used; so for numbers "a" and "b", "a x b", "a ⋅ b", and "a * b" are all called the product of "a" and "b". The expressions "(a)(b)" and "ab" (called 'bracket' notation and 'juxtaposition' notation, respectively) also denote the product of numbers "a" and "b".
As previously pointed out, besides the products of whole numbers, integers, and real numbers, there are products of complex numbers, vectors, functions, matrices, tensors, etc. The symbols described previously to denote multiplication of integers and real numbers (i.e., the cross, dot, asterisk, brackets, and juxtaposition) are also used to indicate products of functions, complex numbers, etc. Different symbols, however, may denote different types of products. Consider the products of vectors. The lying cross "x" indicates the cross product of two vectors, which is itself another vector. In contrast, the dot symbol "⋅" indicates the dot product of two vectors, which is a scalar quantity (i.e., a real number).
The powers and factorials of integers are compact notations for the expression of products. Powers are used to denote the product of equal terms (normally having integer values). The nth power of a, denoted as "an", is the product of n factors of a. That is to say: "a2 = a ⋅ a", "a5 = a ⋅ a ⋅ a ⋅ a ⋅ a", and so forth. In the expression "an", a is called the basis and n the exponent. As indicated above, both "a" and "n" customarily have integer values only.
The factorial of a positive integer n also denotes a product. Factorial, represented as "n!", is the product of all positive integers from 1 to n. It is written out as "n! = 1 ⋅ 2 ⋅ 3 ⋅ ... ⋅ (n-1) ⋅ n", where by convention, 0! = 1. For example "5! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 120".
Quotient is defined in mathematics as the quantity obtained by dividing one quantity or expression by another quantity or expression. The process of determining a quotient is called division. For instance, the expression "30 / 5" is the quotient of "30" divided by "5". One may form quotients of whole numbers, integers, rational numbers, real numbers, complex numbers and functions.
The word quotient first appeared in use during the first half of the 15th century. It was derived from the Middle English "quocient", an alteration of the Latin "quotiens" (how many times), which was derived from the Latin "quot" (how many). In its simplest form the quotient is found by dividing a number "a" by a number "b". The number "a" is called the dividend and "b" is called the divisor. The divisor must have a non-zero value. We can see the reason for this with an example: assume "6 / 0 = n", where "n" is some number. Multiplying both sides of the equation by "0" results in the equation "6 = n x 0 ", which violates the multiplicative rule that the product of zero and any number equals zero. Quotients are most often represented by the symbols "/" (called diagonal or solidus) and "÷" (obelus) so that for numbers "a" and "b" (with a non-zero value of "b") the quotient may be expressed "a / b" or "a ÷ b".
For real numbers (and for whole numbers, integers, and rational numbers, all of which are contained within the set of real numbers) the product of two quotients is the quotient of the product of the dividends (or numerators) and the product of the divisors (or denominators). Symbolically this is written "(a / b) ⋅ (c / d) = (a ⋅ c) / (b ⋅ d)", where b and d are non-zero. The quotient of two quotients may be rewritten as the product of the 'top' quotient and the inverted 'bottom' quotient, and the result reduced to lowest terms. Symbolically this process is "(a / b) / (c / d) = (a / b) ⋅ (d / c) = (a ⋅ d) / (b ⋅ c)", where b, c, and d are non-zero. For example "(2 / 3) / (5 / 6) = (2 / 3) ⋅ (6 / 5) = 12 / 15 = 4 / 5".
Finally, it should be noted that for the rational, real, and complex numbers, division may be defined as the inverse operation of multiplication, where "a / b = a ⋅ (1 / b)". Therefore, the quotient "6 / 2" represents the same number as "6" (the dividend of the quotient) multiplied by the reciprocal of the divisor (i.e., 1 / 2), or written as an equation: "6 / 2 = 6 ⋅ (1 / 2)". Multiplication by the inverse of a quantity is meaningful in many applications where the operation of division is not well defined (for example, matrices).
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