How to Square Any Number in Your Head

Many think math is useless. No! Wrong! I’m here to show you that math is cool, theory and everything. Squaring a number in your head will blow away the average person, and I guarantee that.

Disclaimer: I AM a professional head-squarer, and what I tell you IS most helpful to ME. This tutorial is reserved only for WHOLE numbers.

Step one is the benchmark numbers. These are your easy multiplication numbers, which are the multiples of 5.

To square a number with a ones place of zero, simply square the preceding digits as its own number, and append two zeros. Here’s a short proof: if we take x to represent the digits above the ones place, then the number we are squaring is 10x. Squaring 10x gives us 100x², which shows that our number squared is made up of the squared collection of digits after ten (x²) and two zeros at the end (multiplying by 100 adds two zeros to the end of the number, the ones and tens digits).

To square a number ending in 5, the last 2 digits are 25 and the first digits are found by multiplying the digits, above the ones place, by itself plus one. We observe the same premise: the number we are squaring would now be 10x + 5. When expanded, (10x + 5)² becomes 100x² + 100x + 25. Since the first two terms of this expanded form do not involve the ones and tens digits (x² and x are both being multiplied by 100, so their values do not affect the ones and tens digits), it is safe to assume that the square of every number ending in 5 ends in 25. For the hundreds digit and beyond, we are left with 100x² and 100x. Now, since we know we are just finding the hundreds and beyond, the 100s can be temporarily ignored out of our remaining terms so that we can generate the value of the additional digits themselves. All that we have now is x² + x, which is equivalent to x(x + 1). Thus, the first digits (the hundreds and above) are found by multiplying the tens digit and above by itself + 1.

If you’re confused, here’s a picture example.

Now that the benchmark numbers have been established, we can find the squares of all the other whole numbers using the difference of squares formula: a² - b² = (a + b)(a - b). All we have to do is take our number, find the closest benchmark number, and add the difference of squares of the two numbers to the square of the benchmark number. To use the difference of squares formula, make “a” equal to the number whose square you want to find and “b” equal to the benchmark number. 

Here is the final example:

I hope you’ve at least taken something away from this article, whether understanding where formulas came from or how there is a pattern in the squaring of numbers. Thanks for reading!