MathsSeeing how numbers work1 / 10▾

  1. Square roots

    01What is a square root?
  2. 02Perfect squares
  3. 03Heron's method
  4. 04Roots that never end

  5. Probability

    05What is probability?
  6. 06The law of large numbers
  7. 07Two dice
  8. 08Bayes' theorem
  9. 09The bell curve
  10. 10Three meanings of probability

01 / 10 · Square roots

What is a square root?

From an area back to a side

Take a square with sides 3 long. Its area is 3 × 3 = 9. Multiplying a number by itself is called squaring it, and we write it 3² = 9.

A square root runs the same idea backwards. You are given the area, 9, and asked: how long is the side? The answer is 3, so we say the square root of 9 is 3, written √9 = 3.

3² = 3 × 3 = 9
√9 = 3        because 3 × 3 = 9
√20 ≈ 4.47    because 4.47 × 4.47 ≈ 20

Try it · Drag the area slider. The square grows and its side length, the square root, updates. The grid shows how many unit squares fit inside.

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That is all a square root is: the side of a square whose area you already know. Every positive number is the area of some square, so every positive number has a square root, even when it is not a whole number. A square with area 20 has a side of about 4.47.

Strictly, (−3) × (−3) = 9 too, so −3 is also a square root of 9. The √ sign always means the positive one, the one that can be a length.

Key ideas

Words worth remembering

Square (of a number)
The number multiplied by itself: n² = n × n.
Square root
The number that, squared, gives back the original: √a × √a = a.
Radical sign (√)
The symbol for the positive square root.

Think about it

If you double the area of a square, does its side double too? Try area 8 and area 16.