Basis for operations involving negative numbers

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BASIS FOR OPERATIONS INVOLVING NEGATIVE NUMBERS

The following are the rules that we learn in algebra for operating on signed numbers (i.e. positive and negative numbers):

Addition.

1. To add two numbers with like signs: 1) add their absolute values and 2) prefix this sum with the common sign.

Examples:

3 + 4 = 7

(-3) + (-4) = -7

2. To add two numbers with unlike signs: 1) find the difference between their absolute values and 2) prefix the number obtained with the sign of the number with the greater absolute value.

Examples:

17 + (-8) = 9

(-6) + 4 = -2

(-18) + 15 = -3

Subtraction.

To subtract one number b from another number a: change the sign of b and add to a i.e. a - b = a + (-b).

Examples:

12 - (7) = 12 + (-7) = 5

(-9) - (4) = -9 + (-4) = -13

2 - (-8) = 2 + 8 = 10

Multiplication.

1. To multiply two numbers having like signs: 1) multiply their absolute values and 2) prefix this product obtained with a plus sign (or no sign).

Examples:

(5)(3) = 15

(-5)(-3) = 15

2. To multiply two numbers having unlike signs: 1) multiply their absolute values and 2) prefix this product with a minus sign.

Examples:

(-3)(6) = -18

(2)(-6) = -12

Division.

1. To divide two numbers having like signs: 1) divide their absolute values and 2) prefix the number obtained with a plus sign (or no sign).

Note. Division by zero is not defined.

Examples:

(-6)/(-3) = 2

6/3 = 2

2. To divide two numbers having unlike signs: 1) divide their absolute values and 2) prefix the number obtained with a minus sign.

Examples:

-12/4 = -3

12/(-4) = -3

Where did these rules come from? What is the basis for them? How were they arrived at? What assumptions underlie them? Why is a negative number times a positive number a negative number? Why is a negative number times a negative number a positive number? Which of these operations involving negative numbers have a natural, intuitive, easily understood meaning and which don't? Does the idea of multiplying anything by a negative number make any sense?

The concept of a negative number makes sense when viewed as representing something like debt or a point below zero on some scale such as one finds on a thermometer. One thinks of the signed numbers that are used in algebra as points on a line with a zero point marked and positive numbers to the right of zero and negative numbers to the left of zero. When it comes to operations on them, they are perhaps best thought of as vectors. For example, +5 can be viewed as a vector extending 5 units to the right of zero and and a -5 can be viewed as a vector extending 5 units to the left of zero --- the two vectors being mirror images of each other and acting as negators or annihilators of each other on the operation of addition in the sense that +5 + (-5) = 0.

If we view "-1" as a unit vector extending in the negative direction it is natural to view -2 as the sum of two -1's , -3 as the sum of three -1's, -n as the sum of n “-1's”, and -a = a×(-1). Moreover, it is natural and intuitive to view the sum of a positive number and a negative number as being equal to the difference in their absolute values with the sign of the larger. Thus the rules for the addition of signed numbers given above seem natural and what we would expect in regarding them as vectors. How about the operation of subtraction on signed numbers? Note that the rule above reduces subtraction to an operation of addition --- adding the additive inverse of the subtrahend. Does the operation give what one would intuitively expect? What would one intuitively expect the result to be from the operation 9 - 2? Our rule gives 7. That makes sense. What about -5 - (-6)? What might we expect that to be? Our rule gives +1. Does intuition give us any help on what this result should be? Can one give an intuitive explanation to that result? Does it make any sense? Could we have arrived at the result by intuition?

How about the concept of multiplying a negative number by a positive number? It is natural to expect this to produce a scaling effect on the negative number in the same way multiplying a positive number by a positive number produces a scaling effect and this is just what the rule above does. So here again the rule makes sense and gives what we would expect.

Now how about the idea of multiplying any number by a negative number? That idea just doesn't make sense. There is no intuitive guide on what the product of a negative number times any other number should be.

So, in summary, we see that the sum of two positive or negative numbers, or the sum of a negative number and a positive number, make sense and have a natural, intuitive interpretation as does the product of a positive number times a negative number. But in the case of a negative number times a positive number or the product of two negative numbers they don’t.

So how were the rules giving the sums and products of signed numbers arrived at?

Q. When the number system was extended to include the negative numbers why was the product of a positive number times a negative number defined to be a negative number? Why was the product of a negative number times a negative number defined to be a positive number?

A. The answer is that the operations had to be defined that way. There was no choice. Consider the following.

Q. Why is a negative number times a positive number negative?

A. First, assume the existence of an additive inverse i.e. that for every number a there is a number a* such that a + a* = 0 [or, said differently, for every number a there is an inverse "-a" such that a + (-a) = 0].

Let a and b be two positive numbers. By the definition of inverse

1) a + (-a) = 0

Now multiply equation 1) by b to get the equivalent equation

2) ab + (-a)b = 0

Equation 2) shows that (-a)b is the inverse of ab (by the definition of inverse) and because ab is positive then (-a)b must be negative. We assume here that the right distributive law holds when we go from equation 1) to equation 2).

We could illustrate this by using actual numbers for a and b. Let a = 2 and b = 3. Then

3) 2 + (-2) = 0

Multiplying equation 3) by 3 gives

4) (2)3 + (-2)3 = 0

Thus since (2)3 is positive, then (-2)3 must be negative for equation 4) to be true.

Why is a negative number times a negative number positive?

Again let a and b be two positive numbers.

5) a + (-a) = 0

Now multiply equation 5) by -b

6) a(-b) + (-a)(-b) = 0

Equation 6) shows that a(-b) and (-a)(-b) are inverses of each other so if one is negative the other must be positive. Now since a(-b) is negative (as we have just proved), (-a)(-b) must be positive.

Again we could illustrate this by using actual numbers for a and b. Let a = 2 and b = 3. Then

7) 2 + (-2) = 0

Multiplying equation 3) by 3 gives

8) (2)(-3) + (-2)(-3) = 0

Thus since (2)(-3) is negative, then (-2)(-3) must be positive for equation 8) to be true.

A number of laws obtain for the positive numbers i.e. the associative laws and commutative laws for both addition and multiplication and the left and right distributive laws. These laws must continue to be obeyed for a system that includes negative numbers. That is a requirement for any new system that we might wish to build that includes negative numbers. Once we have postulated the existence of an additive inverse in the form of a negative number, the rules for operating on our newly postulated signed numbers are forced upon us i.e. they must be as they have been defined in order for the laws for the positive rationals to be obeyed.