A number is rational if it can be written exactly as a fraction p/q of two integers, with q ≠ 0. This covers all integers, all terminating decimals and all recurring decimals. A number is irrational if it cannot: its decimal goes on forever without repeating. Most surds (like √2) and π are irrational, while a surd that simplifies to a whole number (like √9 = 3) is rational.
The confusion usually starts with surds. Students assume every square root sign means the number is irrational, and every fraction means rational, without actually checking. Both assumptions break down: √16 looks irrational but equals 4, and 0.3333… looks messy but is exactly ⅓, a rational number.
The fix is to stop guessing from appearance and apply one consistent test: can the number be written as an exact fraction of two integers. Everything else in this topic follows from that single question.
Rational if
Integer, terminating decimal, or recurring decimal
Irrational if
Non-terminating, non-recurring decimal
Surd check
Is the number under the root a perfect square?
Special case
π is always irrational, in any form
Classify each of these as rational or irrational: √25, √7, 0.6, 0.121121112…, π, ⅔.
Solution
√25 = 5. 25 is a perfect square, so this is a whole number. Rational.√7. 7 is not a perfect square, so √7 cannot be written as an exact fraction. Irrational.0.6 is a terminating decimal, equal to 3/5. Rational.0.121121112… has a pattern that keeps changing rather than repeating exactly, so it never settles into a fixed recurring block. Irrational.π = 3.14159… never terminates and never recurs. Irrational.⅔ is already a fraction of two integers. Rational.Quick check: for any square root, ask whether the number underneath is one of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. If it is, the root is a whole number and rational. If it is not, the root is irrational.
Show that 0.4̅5̅ (0.454545…, recurring) is rational by writing it as a fraction.
Solution
x = 0.454545…100x = 45.454545…100x − x = 45.454545… − 0.454545…, giving 99x = 45.x: x = 45/99 = 5/11.5/11 is a fraction of two integers, so 0.454545… is confirmed rational.The step students get wrong
Students sometimes call any long decimal "irrational" without checking whether it actually recurs. A decimal is only irrational if the digits never settle into a repeating block, no matter how far you go. If you can spot a repeating pattern, even a long one, the number is rational: it can always be converted back to a fraction using the multiply-and-subtract method above.
Adding, subtracting, multiplying or dividing two rational numbers always gives a rational result (as long as you are not dividing by zero). Mixing rational and irrational numbers is where it gets interesting.
Two irrational numbers can combine to give a rational result, which is the part students find surprising. √2 × √2 = 2, and (3 + √5) + (2 − √5) = 5. The irrational parts cancel out.
Exam tip: if a question asks you to show that a combination of surds is rational, look for a way to make the irrational parts cancel, such as multiplying conjugate surds like (a + √b)(a − √b), which always removes the root.
Rational and irrational numbers together make up the real numbers. Every real number is one or the other, never both. Natural numbers, whole numbers and integers are all subsets of the rational numbers, since every integer n can be written as the fraction n/1, the same integers you break down when finding HCF and LCM by prime factorisation.
Is 0 rational or irrational?
Rational. Zero can be written as 0/1, a fraction of two integers, so it satisfies the definition directly.
Are all square roots irrational?
No. Only square roots of numbers that are not perfect squares are irrational. √4, √9 and √100 are all rational because they simplify to whole numbers.
Is 22/7 the same as π?
22/7 is a rational approximation used to make calculations easier, but it is not exactly equal to π. π itself is irrational and its decimal never terminates or recurs, so no fraction can represent it exactly.
How does this topic connect to the rest of E-Maths?
Recognising irrational numbers matters whenever an exact answer is required instead of a rounded one, such as leaving a length as √5 cm rather than rounding it to 3 significant figures as 2.24 cm. It is one of the foundational ideas that keeps showing up across topics, which is part of why weak number foundations cause trouble later in the syllabus. It also sits alongside the surd and index skills covered in the difference between E-Math and A-Math.
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