The decimal tells you if a number is rational
Look only at a number's decimal. Can you tell if the number is a fraction?
The rule
Yes, when you know how the decimal continues. Check whether it stops, repeats, or does neither.
Rule: A decimal that stops or repeats is rational. A decimal that never ends and never repeats is irrational.
Rational means the number is a fraction. Irrational means it is not.
A fraction always stops or repeats. Long division leaves a remainder of 0, or a remainder comes back. A decimal that does neither cannot come from a fraction.
| Number | Decimal | What it does | Kind |
|---|---|---|---|
| 7/16 | 0.4375 | stops | rational |
| 4/11 | 0.3636… | repeats, block 36 | rational |
| 22/7 | 3.142857142857… | repeats, block 142857 | rational |
| √144 | 12 | stops | rational |
| √2 | 1.41421356… | never repeats | irrational |
| √20 | 4.47213595… | never repeats | irrational |
| 0.1211211121111… | ones grow by one | never repeats | irrational |
√144 = 12, so its decimal stops. √20 is not a whole number, because 4² = 16 and 5² = 25. A square root of a whole number is whole, or irrational. So √20 is irrational. Its decimal never repeats.
A few digits cannot prove that a decimal repeats. You need the number's rule, such as a fraction or a root.
See the digits. Does the decimal stop, repeat, or never repeat?
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