Perfect Squares
A perfect square is not a number with a special property you have to look up. It is a number of things that can be laid out as an actual square - and everything else follows from that.
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"Perfect square" sounds like a label somebody assigned to a list of numbers you now have to memorise. It is not. The name is a description of a shape, and once you can see the shape, the list stops needing to be memorised at all.
Move the slider. Watch what gets added each time.
Move the slider one step. The shaded L is what got added - count it, and you have the gap between two consecutive squares.
The name is literal
6^2 = 36 does not mean "6 has been through the squaring operation". It means: take 36 unit squares and you can lay them out as a square with 6 on each side. Take 35 and you cannot - you will always have a hole or an overhang. That is the whole of the definition, and it is why the word is "square" rather than something abstract.
Read the same picture backwards and you have square roots. sqrt(36) = 6 is the question "if 36 squares make a square, how long is its side?" A number has a whole-number square root exactly when it is a perfect square, which is why the two ideas are always taught together.
The list worth knowing
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Twenty entries covers essentially every school question, because problems are built backwards from answers people can check.
The gaps are the odd numbers
This is the fact the slider is really for. Push it from 5 to 6 and count the shaded L that appears: five squares down the new side, five along the new top, and one in the corner. Eleven.
In general, going from n^2 to (n+1)^2 adds 2n+1 squares - always odd. So the differences between consecutive squares run 3, 5, 7, 9, 11, 13 …, and adding up the odd numbers in order gives the squares in order:
1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16.
That is also a genuinely fast way to extend the table in your head. If you know 14^2 = 196 and want 15^2, add 2*14 + 1 = 29: 225.
Spotting one without a calculator
Two filters get you most of the way.
Look at the last digit. Square any number and it ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8. So 4,398 is not a perfect square and you did not have to work at all. This is not a coincidence: the last digit of a product depends only on the last digits of what you multiplied, and squaring 0 through 9 only ever produces those six endings.
Then bracket it. 200 sits between 14^2 = 196 and 15^2 = 225, so there is no whole number left for it to be the square of. Any number strictly between two consecutive squares is not one.
Why they keep turning up
Perfect squares are the reason some quadratics factor neatly and others do not. x^2 - 49 pulls apart as a difference of two squares because 49 is one; x^2 - 50 does not. They are also what makes a square root simplify: sqrt(75) becomes 5sqrt(3) precisely because 25 is a perfect square hiding inside 75.
Try one
And one that goes the other way - an equation rather than a lookup. Notice that it has two answers, which is the thing square roots are most often wrong about:
Both 7 and -7 square to 49, so both are solutions. The sqrt(49) symbol means only the positive one - that is a convention about the symbol, not a fact about the equation.
Common questions
- What is a perfect square?
- A whole number you get by multiplying some whole number by itself. 9 is a perfect square because 3 times 3 is 9; 10 is not, because no whole number multiplied by itself gives 10. The name is literal: exactly that many unit squares can be arranged into a square.
- What are the perfect squares from 1 to 100?
- 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 - the squares of 1 through 10. Carrying on past 100 gives 121, 144, 169, 196, 225, 256, 289, 324, 361 and 400, which takes you to 20 squared and covers almost everything school questions ask for.
- How can I tell if a number is a perfect square?
- Two quick filters, then one check. A perfect square never ends in 2, 3, 7 or 8, so those are ruled out instantly. If it survives that, find the nearest square you know and work outwards - 200 sits between 196 and 225, so it is not one. Only when a number passes both do you need to take its square root.
- Why are the gaps between perfect squares the odd numbers?
- Growing a square from n on a side to n+1 means adding a strip down one side, a strip along the top and one corner square: n + n + 1, which is always odd. So the differences run 3, 5, 7, 9 and so on, and adding the odd numbers in order gives you the squares in order.
- Is 0 a perfect square?
- Yes. 0 times 0 is 0, so it fits the definition exactly. Some school lists start at 1 because a square with no side length is hard to picture, but nothing in the definition excludes it, and treating 0 as a square keeps later algebra tidy.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
Want to run this on your own numbers?
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