√ Where Are Square Roots on the Number Line?

If you know the area of a square field, can you find its side? Today we explore the square root: square numbers, the number line, simplifying surds and an exam-style question!

A square field with an area of 25 m² has a side of 5 m, because 5 × 5 = 25. The side of a square field is the square root of its area: √25 = 5.

Numbers like 1, 4, 9, 16, 25 and 36, which come from multiplying a whole number by itself, are called square numbers. The square root of a square number is a whole number: √36 = 6.

A field with an area of 31 m² is bigger than the 25 square and smaller than the 36 square. Since 25 < 31 < 36, we know 5 < √31 < 6. So √31 is between 5 and 6.

So which is it closer to? 31 is 6 units from 25, but only 5 units from 36. 31 is closer to 36, so √31 is closer to 6 (√31 ≈ 5.57).

On the number line, we put √31 between 5 and 6, closer to 6. Check: 5.5 × 5.5 = 30.25, which is less than 31, so √31 is bigger than 5.5.

Simplifying surds: split the number under the root into “a square number × another number”. Pick the biggest square factor: 48 = 16 × 3. √16 = 4 comes outside the root: √48 = 4√3.

Watch out, a trap! √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. So √(9 + 16) ≠ √9 + √16. If there is an addition inside the root, add first, then take the root.

Exam-style question: A square garden has an area of 50 m². Which two consecutive whole numbers is the length of one side between? The answer is C: since 49 < 50 < 64, we know 7 < √50 < 8.

Summary: the side of a square field is the square root of its area. Put a number that is not a square number between two neighbouring square numbers, and its root is between their roots. With simplifying surds, √48 = 4√3. A root does not spread over an addition. Now try it in Square Root Garden!