Negative Numbers
On a freezing morning the thermometer can read three degrees below zero, and the lift
down to a car park stops at floors below the ground. To describe anything below zero, we need a
new kind of number — the negative numbers.
So far the number line started at 0 and grew to
the right. But it doesn't have to stop at zero — it keeps going to the
left into the negative numbers:
-1, -2, -3, \dots Each one wears a little
minus sign to say "I live on the cold side of zero".
Going left is just
subtraction
that keeps going past zero. When you count down — 3, 2, 1, 0 —
most numbers would stop. Negative numbers just carry on:
3,\ 2,\ 1,\ 0,\ -1,\ -2,\ -3,\ \dots
Zero is the doorway. Step through it and every number grows a minus sign.
Numbers below zero are everywhere once you look. The top of a mountain is
high above sea level — a big positive. But the deepest valleys and
the ocean floor are below sea level, so we measure them with
negative numbers. Sea level itself is the 0 that
splits "up" from "down".
Where do negatives show up?
Three everyday places where numbers dip below zero:
-
Temperature. Above zero it is warm; below zero water
freezes. A frosty morning of -4 degrees is
colder than -1 degree.
-
Floors of a building. The ground floor is
0. Go up: floors 1, 2, 3. Take the lift
down to the car park and you reach floor
-1, then -2 — the
basements below ground.
-
Money. 5 means five coins in
your pocket. -5 means five you owe — a
debt. A bigger debt is a smaller balance.
When a rocket launches, the team counts down: "three, two, one,
zero — lift-off!" That zero is the moment of launch. You can keep the same
idea afterwards: one second after launch is
+1, and one second before launch was
-1. Counting down naturally walks you right
through zero into the negatives.
The number line, stretched to the left
Picture the number line as a ruler that now reaches in both
directions. Zero sits in the middle. The further right you
go, the bigger the number; the further left,
the smaller:
-3 < -2 < -1 < 0 < 1 < 2 < 3
This is the surprising part. With ordinary counting numbers,
3 is bigger than 1. But
flip them negative and the order flips too:
-3 < -1, because -3 sits
further left. A bigger debt is a smaller amount of money!
Try it yourself. A marker has landed somewhere in the cold, negative half of
the line. Read off its value, then press Play to check.
Press Refresh for a brand-new spot.
Worked examples
Example 1 — which is smaller? Compare -2 and -5.
Walk to each on the line. -5 is further left than
-2, so it is the smaller one:
-5 < -2. (Even though "5" feels bigger than "2",
the minus sign turns it around.)
Example 2 — a cold morning. It is 3 degrees, then the temperature drops by 7. What is it now?
Start at 3 and count down 7 steps: 3, 2, 1, 0,
then through zero to −1, −2, −3. We land on
3 - 7 = -4 degrees.
Example 3 — counting down. What comes next: 2,\ 1,\ 0,\ ?
Keep stepping left by one. After 0 comes
-1.
Two traps that catch everyone at first:
-
With negatives, the bigger the digit after the minus, the
lower the number. So -5 is
smaller than -2, not bigger —
it sits further left.
-
Zero is neither positive nor negative. It is the
boundary between the two sides — the doorway you count through, not a
number with a sign.
You can write -0, but it is just
0 again — there is nothing to the left or right
of zero at zero. Zero is the one number that doesn't need a sign,
because it is exactly where positive and negative meet.
Here is the same idea as an animation. Press play: a marker starts up where
it's warm and hops down the scale, crossing zero into the cold negatives —
each value read aloud.
Khan Academy introduces negative numbers here: