Negative numbers

A negative number is a number less than zero. It is usually written with a minus sign (−) placed before the number, for example −5, −1, or −3.7. Numbers greater than zero are called positive numbers, while zero itself is neither positive nor negative.

Negative numbers make the number line extend in two directions: zero lies at the center, positive numbers extend to the right, and negative numbers extend to the left. For this reason, subtraction is always possible: 3 − 7 = −4.

Definition and properties

Every positive number, for example a, has an additive inverse called −a, which when added to a gives zero: a + (−a) = 0. The absolute value of a negative number is its distance from zero, as in |−5| = 5. Negative numbers are part of the integers, rational numbers, and real numbers.

Rules of operations

Multiplication and division follow the sign rule: when two negative numbers are multiplied, the result is positive ((−2) × (−3) = 6), but when a negative and a positive number are multiplied, the result is negative ((−2) × 3 = −6). When a negative number is subtracted, it is equivalent to adding its opposite: 5 − (−3) = 5 + 3 = 8. The square root of a negative number does not exist among the real numbers; this led to the discovery of imaginary numbers.

Brief history

Negative numbers were used systematically in early Chinese and Indian mathematics, where scholars such as Brahmagupta wrote rules concerning assets and debt. Many European mathematicians long resisted recognizing negative numbers as genuinely real numbers, but by the late modern centuries they became fully integrated into the foundations of mathematics.

Everyday use

Negative numbers are used to measure temperatures below the freezing point of water, depths below sea level, bank account balances when money is owed, and percentage change when a value falls. In all these cases, the minus sign indicates a direction opposite to the positive, not merely a smaller quantity.