Algebra - Negative Numbers & Rule of Signs

The Algebraic Definition of the Negative of a Number

Finally, the way we define a negative number in algebra is as follows. −5, for example, is that number which when added to 5 itself, results in 0.
5 + (−5) = 0.

That is, to each number a there corresponds one and only one number −a called its negative. And when we add it to a, we get 0.

a + (−a) = −a + a = 0

Problem 1 What number is the negative of xyz? Why?

SOLUTION: −xyz , because xyz + (−xyz) = 0.

Problem 2 What number is the negative of −q? Why?

SOLUTION: ___________________

Problem 3 If s + t = 0, then what is the relationship between t and s?

SOLUTION: _________________________

Problem 4 If you had to prove that b − a is the negative of a − b,
how would you do it?

SOLUTION: Show that a − b + b − a = 0 .

To prove a fact about anything, whether in mathematics, logic, or the law, we have simply to show that it satisfies the definition of that fact.

An Explanation of the Rule of Signs

To decide how negative numbers should behave, we are not able to copy arithmetic. Rather, we have to respect the either-or, yes-or-no nature of logic.

For example, the introduction of the word not into a statement changes its truth value. If the statement was true, "not" makes it false, and vice-versa. If the statement Today is Monday is true, then Today is not Monday is false. But if we write Today is not not Monday, then that changes its truth value again -- that statement is true

Now in algebra we do not have true or false, but we do have the logical equivalent: positive or negative. Thus if the value of x is positive, then the value of −x must be negative, and vice-versa.

And so since we call the positive or negative value of a number its sign, then we can state the following principle:

A minus sign changes the sign of a number.


Geometrically, a minus sign reflects a number symmetrically about 0 with the number line to illustrate it visually.

We saw that with −(−3) = 3.
(As for 0, it is best to say that it has both signs: −0 = +0 = 0). If we now apply this principle to multiplication:

A negative factor changes the sign of a product.


Thus if ab is positive, then (−a)b cannot also be positive. It must be negative -- it must be the negative of ab.

(−a)b = −ab.

That is, "Unlike signs produce a negative number."

And upon introducing another negative factor, the sign changes back:
(−a)(−b) = ab.

"Like signs produce a positive number."

This same logical principle will apply to division and fractions. Hence we have the Rule of Signs.

To prove that (−a)b is the negative of ab in what some call a rigorous manner, we would have to apply the definition of the negative of a number given above. We would have to prove:

ab + (−a)b = 0.