Key takeaway: An irrational number is a point on the number line that no fraction can hit exactly, so its decimal digits continue forever with no repeating block.
Grade 88.NS.A.1Numbers and Operations
Definition of Irrational Number
An irrational number is a number that escapes every fraction. Try any ratio of whole numbers you like; none will land on it. Written as a decimal, its digits run on forever without a repeating block, so on paper you can only approximate it.
The most familiar members are π and the square roots of whole numbers that are not perfect squares, such as \(\sqrt{2}\) or \(\sqrt{30}\). These values still sit at exact spots on the number line. We estimate them by trapping them between two neighboring whole numbers, then tightening the bounds with decimals. The square root of a perfect square, like \(\sqrt{81} = 9\), is rational.
Formula and notation
A number \(x\) is irrational when it cannot be written as
\[x = \frac{a}{b} \quad \text{with integers } a, b \text{ and } b \neq 0\]
Approximations use the \(\approx\) sign: \(\sqrt{30} \approx 5.477\).
Key facts
- Decimals that end or repeat are rational, so irrational decimals do neither.
- Every irrational number sits at an exact spot on the number line, even if its digits never end.
- Together with the fractions, these values fill every gap on the real line.
- Adding a fraction to one of these values still gives an irrational result.
Irrational Number example
Classify each number as rational or irrational: \(\sqrt{81}\), \(\sqrt{30}\), \(0.313131\ldots\) (the block 31 repeats). Then place \(\sqrt{30}\) between two consecutive whole numbers.
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\(\sqrt{81} = 9 = \frac{9}{1}\), so it is rational.
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The decimal 0.313131… repeats, so it equals a fraction (\(\frac{31}{99}\)) and is rational.
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30 is not a perfect square, so \(\sqrt{30}\) is irrational.
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Since \(5^2 = 25 < 30 < 36 = 6^2\), \(\sqrt{30}\) lies between 5 and 6.
Only \(\sqrt{30}\) is irrational, and it lies between 5 and 6.
Common mistake
Watch out
Noor says \(\frac{1}{3} = 0.333\ldots\) is irrational because its decimal goes on forever.
Why it is wrong
Going on forever is not the test for an irrational number. The digits of \(\frac{1}{3}\) repeat the single digit 3, and it is the fraction \(\frac{1}{3}\), so it is rational. An irrational decimal goes on forever with no repeating pattern at all.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations