What it is
A negative number is a number less than 0. On the number line it sits to the left of 0, and it is written with a minus sign in front: −3. Numbers greater than 0 are positive; 0 itself is neither positive nor negative.
- −5 °C is five degrees below zero.
- −200 in an account can be a debt of 200 or a balance 200 below zero.
- Floor −1 in an elevator is one storey below ground level.
- 3 − 5 = −2: the answer to a subtraction is negative when you take away more than you have.
Every number has an opposite on the other side of 0: −3 and 3 are each other’s opposites, and −(−3) = 3. The distance from 0 is the number’s absolute value: |−3| = 3.
What is not a negative number:
- 0 is not negative.
- −x is not always negative. If x = −4 then −x = 4. The minus sign means “the opposite of”, not “a number below zero”.
To compare negative numbers: the further left on the number line, the smaller. So −7 < −3 < 0 < 2, and −1 is greater than −100.
Why it matters
Negative numbers are needed whenever something can go two ways from a zero point: temperature, height above and below sea level, balance and debt, profit and loss, time before and after an event. Arithmetic is then extended to them:
- Adding a negative number: 5 + (−3) = 5 − 3 = 2.
- Subtracting a negative number: 5 − (−3) = 5 + 3 = 8.
- Minus times minus: (−2) · (−3) = 6.
- Product with one negative factor: (−2) · 3 = −6.
- Sign rules for division: (−6) / 3 = −2 and (−6) / (−3) = 2.
- Absolute value and its definition: how far a number is from 0, whatever its sign.
- Rules for inequalities: multiplying by a negative number reverses the inequality. −2 < 3, but multiplied by −1 it becomes 2 > −3.
Common misconceptions
- “−5 is greater than −2, because 5 is greater than 2.” It is the other way round: −5 is further left, so −5 < −2.
- “Minus and minus makes minus.” In multiplication it makes plus: (−2) · (−3) = 6. In addition it stays negative: (−2) + (−3) = −5.
- “−3² is 9.” The power is worked out before the minus sign: −3² = −9, whereas (−3)² = 9.
- “You cannot take away more than you have.” You can, and the answer is a negative number: 3 − 5 = −2.
Common questions
Is 0 positive or negative?
Neither. 0 is the boundary between the positive and the negative numbers.
Which is greater, −1 or −100?
−1. It is closer to 0 on the number line and so greater; −100 has the larger absolute value but is the
smaller number.
Can you take the square root of a negative number?
Not among the real numbers: no real number squares to −4, because a square is never negative. The square
root of −4 lives among the complex numbers — see the
imaginary unit.