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OPERATIONS ON REAL NUMBERS

    In the preceding section of this article, it was mentioned that the sum of a pair of real numbers is the real number that is the measure of the resultant of the corresponding pair of directed changes. To gain further insight into addition of real numbers it may be convenient to refer to a picture of the number line.

For example:


Notice that we may apply our knowledge of addition of numbers-of-arithmetic when we wish to add a pair of positive numbers [or a pair of negative numbers]; for the arithmetic value of the sum of a pair of positive numbers [or of a pair of negative numbers] is the sum of their arithmetic values:

3 + 2 = 5

If we wish to find the sum of a positive number and a negative number, for example:



or the sum of a negative number and a positive number, for example:

we may apply our knowledge of subtraction of numbers-of-arithmetic:

3 - 2 = 1

In the preceding section of this article, subtraction of real numbers was defined in terms of addition and oppositing. For example:

+3 - +2

is simply a shorthand notation for

+3 + - +2

Since

- +2 = -2

it follows that

+3 - +2 = +3 + - +2 = +3 + -2 = +1

Thus, the result of subtracting +2 from +3 is +1. Similarly, the result of subtracting +3 from +2 is -1:

+2 - +3 = +2 + - +3 = +2 + -3 = -1

Multiplication of real numbers is similarly related to multiplication of numbers-of-arithmetic:

Notice that the arithmetic value of the product of a pair of real numbers is the product of their arithmetic values:

2 - 3 = 6

Notice also that the product of a positive number by a positive number [or of a negative number by a negative number] is a positive number. The product of a positive number by a negative number [or of a negative number by a positive number] is a negative number.

In the preceding section of this article, division of a real number by a nonzero number was defined in terms of multiplication and reciprocation. For example:

+6 / +3

is simply a shorthand notation for

+6 - +1/+3

Let us see what it means to divide +6 by +3. We wish to find the real number that satisfies the open sentence

A real number satisfies this open sentence if and only if it satisfies the following open sentence:



So a real number satisfies the open sentence

if and only if it satisfies the open sentence

We notice that +2 is the real number which satisfies the last open sentence

So it follows that

or, equivalently, that

Here are some other examples to illustrate division of real numbers:

Notice the similarity between division of real numbers and division of numbers-of-arithmetic. For example:

Notice also that the result of dividing a positive number by a positive number [or of dividing a negative number by a negative number] is a positive number. The result of dividing a positive number by a negative number [or of dividing a negative number by a positive number] is a negative number.

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