JENECONKMaths Universe

What Surds Are

Ages 14–18

Difficulty adapts to you

What Surds Are

Start with square roots you already know. √9 = 3 because 3×3 = 9, so 9 is a perfect square. A surd is an exact root that cannot be simplified to a whole number or an ordinary fraction, such as √2 or √5. Surds are irrational numbers: their decimal expansions continue forever without repeating. We keep the root form because it is exact; √2 is more accurate than a rounded decimal such as 1.414.

Your goals

  • ✓You can recognise surds and understand why some roots cannot be written as ordinary fractions.
  • ✓You can use perfect square, surd, irrational number correctly.
  • ✓You can test whether the number under the root is a perfect square before calling it a surd.

Words to know

perfect square:
A number made by multiplying a whole number by itself, such as 25 = 5×5.
surd:
An exact root that cannot be simplified to a whole number or ordinary fraction.
irrational number:
A number whose decimal goes on forever without repeating.

Examples

  • 1√16 = 4 is not a surd because 16 is a perfect square.
  • 2√2 is a surd because no whole number multiplied by itself equals 2.
  • 3√12 is a surd, but it can be simplified later because 12 contains the square factor 4.
  • 4The exact length of a diagonal may be √5 units. Writing 2.236 units is only an approximation.
  • 5A surd is not the same as any number with a root sign: √25 is an exact whole number, while √7 remains a surd.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Perfect-square check

Best for: Deciding whether a root is already exact
  1. 1.List nearby perfect squares.
  2. 2.Check whether the number is one of them.
  3. 3.Keep a non-perfect square in root form.

Example: 25 is a perfect square because 5×5 = 25; √25 = 5, while √7 stays a surd.

Method 2

Exact versus approximate

Best for: Choosing the correct final form
  1. 1.Keep the root for an exact answer.
  2. 2.Use a decimal only when requested.
  3. 3.Do not round during the working.

Example: √2 is exact; 1.414 is only an approximation.

Check these

  • • Do not call every square root a surd or replace an exact root with a rounded decimal.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own what surds are example, solve it, and explain how you checked it.

Spark

Spark

Creativity & Innovation

“Challenge yourself: can you explain this to a friend?”