Class 10 Maths · Chapter 1 · Topic 1 of 4
What makes a number rational or irrational?
Classify the value, not the way it is written.
Ideas you’ll use
- Rational number
- A number that can be written as an exact fraction a/b. Here a and b are integers (numbers such as −2, −1, 0, 1 and 2, with no fractional part), and b cannot be zero.
- Irrational number
- A number that cannot be written as an exact fraction of integers. Its decimal digits go on forever without a repeating pattern.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
Two kinds of number. One number line.
Each is an exact fraction of integers.
No exact fraction of integers exists.
Write it as a fraction
An integer is already a fraction with denominator 1. A terminating or repeating decimal can be a fraction too.
4 = 4/1; 0.5 = 1/2; 0.333… = 1/3In the picture: Find the rational points on the number line.
Read all the reasoning together
Write it as a fraction
An integer is already a fraction with denominator 1. A terminating or repeating decimal can be a fraction too.
4 = 4/1; 0.5 = 1/2; 0.333… = 1/3Simplify a square root first
A root symbol does not automatically make a number irrational. Work out its value before deciding.
√25 = 5 = 5/1Keep exact and approximate apart
The point for √2 has an approximate position, but √2 is an exact irrational number. A rounded calculator value is not a proof.
√2 ≈ 1.414…
Classify the value, not the way it is written.
Worked example: Simplify first, then decide
√25 = 5 = 5/1√25 is exactly 5, so it is rational. A square-root sign does not automatically mean “irrational”.
Read the supporting explanation
Integers can be written as fractions: 4 = 4/1 and −3 = −3/1. A decimal that ends, such as 0.5, is also a fraction: 0.5 = 1/2. A decimal that repeats a fixed pattern, such as 0.333…, is a fraction too: 0.333… = 1/3.
Go deeper: Why a calculator cannot decide
√2 means the positive number whose square is 2. It is about 1.414…, but 1.414 is only a nearby value. Squaring 1.414 gives 1.999396, not exactly 2.
Even a rational number such as 1/7 has more decimal digits than a calculator can show. To decide whether a number is irrational, we need a mathematical reason, not a long display of digits.
Go deeper: A square-root sign does not decide the answer
√25 = 5, so it is rational. √2 is irrational. Always simplify the expression before deciding.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Classify −9/4, 0.125 and √16.
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