Why do we need negative numbers?
Counting numbers are fine for counting marbles, but some situations need numbers less than zero:
A winter night in Leh can be 8 degrees below zero, written −8 °C.
A submarine 50 metres below sea level is at −50 m, while a plane 3000 m above sea level is at +3000 m.
If you owe a friend ₹20, your balance is −₹20.
Numbers with a minus sign, such as −1, −2 and −3, are negative numbers.
What are integers?
The integers are the whole numbers together with their negatives:
… , −3, −2, −1, 0, 1, 2, 3, …
1, 2, 3, … are positive integers (the plus sign is usually not written).
−1, −2, −3, … are negative integers.
0 is neither positive nor negative.
Each positive integer has an opposite, the negative integer at the same distance from 0. The opposite of 5 is −5, and the opposite of −5 is 5.
The number line
Draw a line, mark 0 in the middle, and mark equal steps on both sides. Positive integers go to the right of 0 and negative integers go to the left.
-5 -4 -3 -2 -1 0 1 2 3 4 5
──┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼──Comparing integers
On the number line, a number is always greater than every number to its left.
3 > −1, because 3 is to the right of −1.
−2 > −7, because −2 is to the right of −7.
Every positive integer is greater than 0, and 0 is greater than every negative integer.
The second point surprises many students. Think of temperature: −2 °C is warmer than −7 °C, so −2 is the larger number.
Example. Arrange −4, 6, 0, −9, 2 in increasing order: −9, −4, 0, 2, 6.
Adding integers on the number line
To add a positive number, move right.
To add a negative number, move left.
Example. −3 + 5: start at −3 and move 5 steps right. You land on 2.
Example. 4 + (−6): start at 4 and move 6 steps left. You land on −2.
Rules for adding
Case | Rule | Example |
|---|---|---|
Both positive | add, answer is positive | 4 + 3 = 7 |
Both negative | add the sizes, answer is negative | (−4) + (−3) = −7 |
One of each | subtract the smaller size from the larger, keep the sign of the larger | (−9) + 4 = −5 |
A number and its opposite always add up to zero: 7 + (−7) = 0. They are called additive inverses.
Subtracting integers
To subtract an integer, add its opposite.
5 − 8 = 5 + (−8) = −3
2 − (−6) = 2 + 6 = 8
(−3) − (−5) = (−3) + 5 = 2
Subtracting a negative number makes the answer bigger. If a debt of ₹6 is cancelled, you are ₹6 better off.
Summary
Integers are …, −2, −1, 0, 1, 2, …; zero is neither positive nor negative.
On a number line, numbers increase from left to right.
Adding a negative moves left; to subtract, add the opposite.