Why Do Two Negatives Make a Positive?

Multiplying two negative numbers results in a positive number because the product of two negative numbers can be described as the additive inverse of a positive number, according to the University of Toronto Mathematics Network. This concept is also explained conceptually by movements on a basic number line.

When a number is multiplied by a negative number, the result is the additive inverse of the answer if there had been no negative. This means that the numbers sum to zero. For example, five times negative two is negative 10 and five times two is 10. Adding negative 10 and 10 yields zero. By following this logic, multiplying two negative numbers results in the additive inverse of a positive number being taken twice. Because the first additive inverse is negative, the second additive inverse must be positive.

Another way to visualize this principle is with the standard number line, centered at zero, where numbers on the right side of zero are positive and numbers on the left are negative. By moving a certain number of steps a certain number of times, multiplication problems are modeled. Multiplying two negative numbers together is analogous to facing the left direction, but taking steps backwards. This movement results in a final answer that is positive.