Until now we have worked with whole numbers (0, 1, 2, 3, ...). But in real life we often need numbers less than zero — for temperatures below 0°C, for depths below sea level, for money owed. These situations require negative numbers.
Integers
The set of integers includes all whole numbers and their negatives:
..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...
Positive integers are to the right of 0 on the number line; negative integers are to the left.
Ordering Integers
On a number line, a number to the right is always greater.
So -1 > -5 (since -1 is to the right of -5).
The smallest negative integer does not exist (they go on forever).
Absolute Value
The absolute value of an integer is its distance from 0, always positive.
|−7| = 7, |5| = 5, |0| = 0.
- Addition of Integers
- Same sign: add and keep the sign. (-3) + (-5) = -8.
- Different sign: subtract the smaller absolute value from the larger, and keep the sign of the larger. (-7) + 4 = -3.
Subtraction of Integers
To subtract, add the opposite:
a - b = a + (-b).
So 5 - (-3) = 5 + 3 = 8.
And -4 - 6 = -4 + (-6) = -10.
Represent -6 on a number line. What integer is 4 units to its right?
-6 + 4 = -2. The integer 4 units to the right of -6 is -2.
Compare -8 and -3. Which is greater?
On the number line, -3 is to the right of -8, so -3 > -8.
Calculate (-15) + 8.
Different signs: |−15| = 15, |8| = 8. Difference = 7. Negative has larger absolute value, so answer is -7.
Calculate 9 - (-4).
= 9 + 4 = 13.
The temperature in Shimla is -5°C in the morning and rises by 12°C by noon. What is the noon temperature?
(-5) + 12 = 7°C.
A submarine is at -350 m (below sea level). It rises 120 m. Where is it now?
-350 + 120 = -230 m.
Arrange in ascending order: -3, 7, -10, 0, 4.
Ascending (smallest to largest): -10, -3, 0, 4, 7.
- Key Facts
- 0 is neither positive nor negative.
- The sum of an integer and its opposite is 0: (-a) + a = 0.
- Negative numbers appear in real contexts: temperature, altitude, bank balance.
Common mistakes
- Do not confuse the sign of a number with the operation: -3 - (-5) = -3 + 5 = 2 (not -8).
- Among negative numbers, the one with larger absolute value is SMALLER: -10 < -2.
- Zero is an integer but is neither positive nor negative.
Summary
Integers extend whole numbers to include negatives. The number line orders integers from left (smaller) to right (larger). Adding and subtracting integers follows sign rules. Integers model real-world situations like temperature, debt, and depth.