Take 4 counters and you can arrange them in a 2 by 2 square. Take 9 and you can make a 3 by 3 square. Take 10 and you cannot make a square at all. That picture is where the word comes from.
Squaring and square rooting
| Operation | What it does | Example |
|---|---|---|
| Squaring | Multiply a number by itself | 7² = 7 × 7 = 49 |
| Square rooting | Find the number that was squared | √49 = 7 |
Squaring takes 7 to 49, and square rooting takes 49 back to 7. If you are given the side of a square and want the area, you square. If you are given the area and want the side, you take the root.
The square numbers, and their patterns
The first twelve square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. Learning them is worth the effort, because they turn up constantly.
The differences are exactly the odd numbers in order. To turn a 3 by 3 square into a 4 by 4 square you add a strip of 3 down the side, a strip of 3 along the top and one more in the corner, which is 7 counters.
Every square number ends in 0, 1, 4, 5, 6 or 9, and none ever ends in 2, 3, 7 or 8. So 4327 cannot be a perfect square. But ending in 6 does not prove a number is square, since 26 ends in 6 and is not. The test can rule out but never rule in.
Finding a square root by prime factorisation
Break the number into prime factors. Because a square is a number multiplied by itself, every prime factor appears an even number of times. Take one factor from each pair, multiply them, and that is the root.
If any prime factor is left over without a partner, the number is not a perfect square. That is useful information, not a failure. Multiplying or dividing by that lonely factor produces the nearest perfect square.
Estimating a root that is not whole
To estimate the square root of 50, notice that 49 is 7 squared and 64 is 8 squared, so the root lies between 7 and 8, and much closer to 7 because 50 is only just past 49.
The first is confusing squaring with doubling: 7 squared is 49, not 14. The second is forgetting that the units of an area are square units: a square of side 7 cm has area 49 square centimetres.
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