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Real Number System
Arithmetic Operations
Multiplication and Division
The multiplication of two real numbers is similar to the concept of "repeated addition". This is best illustrated with the multiplication of two positive integers x and y:
x × y = xy = x + x + .... + x where the number of x's to be added together is y = y + y + .... + y where the number of y's to be added together is x = z
In the above equation, x is the multiplicand, y the multiplier, and the result z is the product of x and y. The similarity between multiplication and "repeated addition" is slightly less evident when non-integers and negative numbers are involved. With respect to multiplication of positive non-integers, the non-integer real number can be separated into its integer and fractional parts, and multiplication can then be defined as follows:
Given real numbers a, b, where a > 0, b > 0
b = c + d, c is an integer, 0 £ d < 1
ab = a × (c + d) = ac + ad
using the distributive axiom
ac = a + a + .... + a
where a is added together c times (a and c are positive integers)
ad is the real number whose ratio to a is equal to d,
i.e., is (100 × d) percent of a.
ab is the sum of ac and ad.
When negative numbers are involved in multiplication, the only difference in the result is its sign (positive or negative); the absolute value of the product remains unchanged. Specifically, the product of a positive real number and a negative real number is negative, and the product of two negative real numbers is positive.
The product of one and any real number is the real number itself; if a real number is added together once, the result is the same real number, of course. Similarly, the product of zero and any real number is zero; if a real number is added together zero times, nothing is added together, so the result is zero.
The division of two real numbers can be defined in terms of multiplication. Given any two real numbers a and b,
a / b = a × (1/ b) = c
where c is the result of the division of a by b. In the above equation, a is the dividend, b the divisor, and c the quotient of a and b.
"a / b" can be thought of as partitioning a into b equal parts, and then determining the value of the b equal parts. Since a real number cannot be divided up into zero equal parts, it is not plausible to consider the expression ( a / b ) when b = 0 (The equation a / 0 = c implies that 0 × c = a for some real number c, but 0 × c = 0 for all real numbers c.). Therefore, division by zero is undefined. However, division by one is acceptable; the quotient of a real number and one is the original real number.
The following are summary statements regarding the operations of multiplication and division:
(1) The product of zero and any real number is zero.
(2) The product of one and any real number is the real number.
(3) The quotient of zero and any real number is zero.
(4) The quotient of any real number and one is the real number.
(5) Division by zero is undefined, i.e., zero cannot be a divisor.
(6) The product of two positive integers or two negative integers is the product of their absolute values.
Ex. 2 × 2 = -2 × -2 = | -2 | × | -2 | = 4
(7) The product of a positive and a negative integer is the negative of the product of their absolute values.
Ex. 2 × -3 = -2 × 3 = - ( | -2 | × | -3 | ) = -6
(8) The quotient of two positive integers or two negative integers is the quotient of their absolute values.
Ex. ( 4 / 2 ) = ( -4 / -2 ) = | -4 | / | -2 | = 2
(9) The quotient of a positive and a negative integer is the negative of the quotient of their absolute values.
Ex. ( 6 / -3 ) = ( -6 / 3 ) = - ( | -6 | / | -3 | ) = -2