Irrational Number
In short: Irrational number labels any real value that no ratio of two integers can equal, which forces its decimal digits to run on forever without ever settling into a loop.
8th GradeNumbers and Operations8.NS.A.1
What is Irrational Number?
An irrational number refuses to fit into a ratio of whole numbers. Its decimal digits stream on forever with no looping block, so 1.4142135 and onward for \(\sqrt{2}\) can only ever be approximated.
Roots of numbers that are not perfect squares, like \(\sqrt{50}\), land in this category, and so does \(\pi\). By contrast, \(\sqrt{49} = 7\), \(0.75\) and \(0.\overline{12}\) are all rational. Add a rational value to an irrational one and the total is still irrational.
How to spot it
- A radical shows up and the quantity under it is not a perfect square.
- A decimal is described as endless with no loop, or the problem leans on the constant pi.
- A calculator spits out a long decimal for a root and you must choose how to write the exact value.
- A value such as the root of 2 must be pinned between two integers on a number line.
The method
- Ask whether the value could be a ratio of integers; decimals that stop or loop pass this test and are therefore rational.
- For a root, look at the radicand: if it is not a perfect square, the root is irrational.
- Estimate by squaring nearby integers to find which two whole numbers it lies between.
- Keep the exact form, like \(\sqrt{50}\), or round only when an approximation is asked for.
Irrational Number: solved example
Problem
Sort these numbers into rational and irrational: \(\sqrt{45}\), \(\sqrt{81}\), \(0.\overline{12}\), \(3.14\).
Solution
The root of 81 equals 9, a whole number, so it counts as rational. The loop in \(0.\overline{12}\) lets it equal \(\frac{4}{33}\), also rational. And \(3.14\) stops, so it equals \(\frac{157}{50}\).
The radicand 45 falls between the squares 36 and 49, so only \(\sqrt{45}\) is irrational, with a value near 6.708.
Try it
Question 1
Between which two consecutive integers does \(\sqrt{20}\) lie?
Show the answer
\(4^2 = 16\) and \(5^2 = 25\), so \(\sqrt{20}\) is between 4 and 5 (about 4.47).
Question 2
Is \(0.75\) irrational?
Show the answer
No. \(0.75 = \frac{3}{4}\), and a terminating decimal is always rational.
Irrational Number vs. Repeating Decimal
Both kinds of decimals go on forever, but a repeating decimal cycles through a block and can be turned into a fraction, so it is rational. An irrational number has digits that never cycle and no fraction matches it. The test is not whether the digits stop, but whether a pattern repeats.
Keep going
Lessons
Worksheets and quizzes
- Rational and Irrational Numbers – Worksheet
- Rational and Irrational Numbers – Quiz
- Square Roots and Cube Roots – Worksheet
- Square Roots and Cube Roots – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Numbers and Operations