Mathematics.

numbers

Negative Numbers

Pre-Algebra20 minDifficulty2 out of 10

You should know: variables and expressions

Overview

A negative number is a number less than zero, written with a minus sign (e.g. -3), representing a quantity in the opposite direction or sense from a positive value — debt instead of credit, below zero instead of above, leftward instead of rightward. Negative numbers, together with zero and the positive numbers, form the integers, and they follow consistent rules for addition, subtraction, and multiplication that extend ordinary arithmetic.

Formal Definition

Definition

For every positive number a, there is a corresponding negative number -a, defined as the additive inverse of a: the unique number satisfying a + (-a) = 0.

a+(−a)=0a + (-a) = 0
Additive inverse
(−a)(−b)=ab(-a)(-b) = ab
Product of two negatives
(−a)(b)=−(ab)(-a)(b) = -(ab)
Product of a negative and a positive

Properties

Sign of a product

(−a)(−b)=ab,(−a)(b)=−ab(-a)(-b) = ab, \quad (-a)(b) = -ab

Subtracting a negative

a−(−b)=a+ba - (-b) = a + b

Worked Examples

  1. Adding a positive to a negative moves toward zero from -7.

    −7+3=−4-7 + 3 = -4

Answer: -4

Practice Problems

Difficulty 2/10

Compute (-5) × (-4).

Common Mistakes

Common Mistake

Assuming subtracting a negative makes a number smaller, e.g. thinking 5 - (-3) = 2.

Subtracting a negative is the same as adding its positive counterpart: 5 - (-3) = 5 + 3 = 8.

Summary

  • A negative number -a is the additive inverse of a, satisfying a + (-a) = 0.
  • Multiplying two negatives gives a positive; multiplying a negative and a positive gives a negative.
  • Subtracting a negative number is equivalent to adding its positive counterpart.

References

  1. WebsiteWikipedia — Negative number