Mathematics
What Is a Surd?
A surd is a root that stays irrational, so it is left in root form. Why √20 and ∛5 qualify, √16 does not, and π is irrational but still not a surd.
Surd
A surd is a root of a rational number that is itself irrational, so it cannot be written exactly and is left in root form.
- Also called
- Exact form, Radical (US usage)
- Where students meet it
- Grade 9 or 10 mathematics, and on GCSE Higher tier and IGCSE 0580 Extended papers, where answers are often required 'in the form a√b'.
The short answer
A surd is a root of a number that cannot be written exactly as a fraction or a terminating decimal, so it is left in root form. √2, √20 and ∛5 are surds. √16 is not, because it equals 4. The test is simply whether the root comes out to a rational number.
An example
√16 = 4 — not a surd √20 = 2√5 — a surd ∛5 — a surd
√16 lands exactly on 4, a whole number, so nothing is irrational and there is no surd. √20 is 4.472135… with no pattern and no end; simplifying it to 2√5 tidies it but does not change what it is. ∛5 is 1.709975… — a cube root rather than a square root, and still a surd, because the definition is about the answer, not the root sign.
The sorting test
Work out the root. If it lands on a whole number or a fraction, it is not a surd. If it runs on forever without repeating, it is. That single question sorts every root you will meet at school.
√9 gives 3, so no. √(9/4) gives 3/2, so no — rational includes fractions, not just whole numbers. √2, √3, √5, √20 and ∛5 all fail to land, so all of them are surds. Simplifying does not change the verdict: 2√5 is still a surd, because there is still a root in it that will not resolve.
The one thing the test does not depend on is the size of the root sign. √1000000 is not a surd, because it is 1000, while √2 is one of the smallest surds there is.
Why leave a root unresolved
Because 2√5 is the exact answer and 4.47 is not. Every time a rounded value is carried into the next line of working, a small error is carried with it, and in a multi-step question those errors accumulate into a final answer that misses.
This is why exam questions say 'give your answer in exact form' or 'in surd form'. Pythagoras' theorem and the exact trigonometric values are the usual sources — a right-angled triangle with legs of 1 and 2 has a hypotenuse of exactly √5, and writing 2.24 throws away the exactness the question asked for.
What a surd is not
Not every irrational number is a surd. π is irrational, but it is not the root of any whole number or fraction, so it does not qualify. All surds are irrational; only some irrationals are surds.
Nor is it a decimal that has been rounded. 2√5 and 4.47 are not two ways of writing the same number: the first is exact and the second is the first, spoiled slightly.
A surd is also not a method. Simplifying √20 into 2√5, and rationalising a denominator to clear a root from the bottom of a fraction, are two separate techniques applied to surds — worth knowing, and not part of what the word means.
Common questions
Is √9 a surd?
No. √9 = 3 exactly, and 3 is a whole number, so nothing is left irrational. The root sign alone never makes a surd — √4, √9, √16, √25 and every other square root of a square number all resolve, which is why recognising square numbers on sight makes surd questions much faster.
Is π a surd?
No. π is irrational, so its decimal never ends and never repeats, but a surd has to be a root of a rational number and π is not the root of anything of that kind. It is the standard counterexample to the assumption that irrational and surd mean the same thing.
Is 2√5 still a surd?
Yes. Multiplying by a whole number does not remove the irrational part — 2√5 is 4.472135… and still does not terminate. It is simply √20 written in the tidier form exams ask for, where the number outside the root is as large as possible and the number inside is as small as possible.
Why do questions say 'give your answer in surd form'?
To ask for exactness rather than a rounded decimal. It signals that the answer will contain a root that does not resolve, and that you should leave it there. It also tells you the working is meant to be done without a calculator's decimal output, which is common on non-calculator papers.
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