Two families of numbers
Every number on the number line is either rational or irrational.
| Type | Can be written as a fraction of integers? | Decimal form | Examples |
|---|---|---|---|
| Rational number | Yes | Terminates or repeats | 3, −5, 1/2, 0.75, 0.333… |
| Irrational number | No | Never ends, never repeats | √2, π, √5 |
Rational numbers
"Rational" comes from "ratio". A rational number can be written as a/b, where a and b are integers and b isn’t 0.
- 7 = 7/1
- −0.5 = −1/2
- 0.25 = 1/4
- 2 1/3 = 7/3
Their decimals either terminate (stop) or repeat forever:
| Fraction | Decimal | Type |
|---|---|---|
| 3/4 | 0.75 | terminating decimal |
| 1/8 | 0.125 | terminating |
| 1/3 | 0.333… | repeating decimal |
| 5/11 | 0.454545… | repeating |
Irrational numbers
An irrational number’s decimal goes on forever without a repeating pattern.
- √2 = 1.41421356… The square root of any whole number that isn’t a perfect square is irrational.
- π (pi) = 3.14159265… It describes every circle.
- √3, √10, √50
We often use approximations such as π ≈ 3.14 or √2 ≈ 1.41, but these are not exact.
Is the square root rational?
| Number | Perfect square? | Rational? |
|---|---|---|
| √16 | Yes (4 × 4) | Rational: 4 |
| √25 | Yes | Rational: 5 |
| √20 | No | Irrational |
| √0.49 | Yes (0.7 × 0.7) | Rational: 0.7 |
Placing them on a number line
Irrational numbers have exact spots on the number line even though their decimals never end. √2 lies between 1.41 and 1.42. π lies between 3.14 and 3.15.
Worked examples
Example 1: Is √2 rational or irrational?
2 isn’t a perfect square: irrational.
Example 2: Is 0.121212… rational?
It repeats "12": rational (it equals 12/99 = 4/33).
Example 3: Is 0.1010010001… rational?
The pattern grows (one more 0 each time) but never repeats a fixed block: irrational.
Common mistakes
- Thinking every decimal is rational. Non-repeating, non-ending decimals are irrational.
- Thinking π = 3.14 exactly. 3.14 is an approximation.
- Thinking all square roots are irrational. √9 = 3 is rational.
Try it yourself
Rational or irrational?
- −8
- √36
- √7
- 0.666…
- 22/7
Answers: 1) rational 2) rational (6) 3) irrational 4) rational (2/3) 5) rational (it’s a fraction; it’s only an approximation of π)
Quick recap
- Rational numbers can be written as fractions; their decimals terminate or repeat.
- Irrational numbers can’t; their decimals never end or repeat.
- √ of a non-perfect square and pi are irrational.