Squaring Numbers: UP-DOWN Method

With this method, you will be able to square any number irrespective of digit’s value, place or number very quickly. With a little practice you can do it in your head, or you can do it on paper and still impress others with your math skills.

Step-1:   Round up or down the given number to the nearest multiple of 10 or 100 i.e. determine the closest number to the given number that ends in a ‘0’ to be easy to work with.

Step-2:  Find the difference between the number and the closest number with the zero.

Step-3:   If the difference is +ve add it to the number, if the difference is –ve subtract it from the number.

Step-4:    Multiply it with the base (determined round) number.

Step-5:    Add the square of the difference to the product.

Let’s see an example squaring the number 27.

-     First, determine the closest number to the given number that ends in a zero. In this example, the number is 30. 

-      Next, determine the difference between the given number and the closest number with the zero. In this case, it will be (30 – 27) = 3. 

-      Subtract the result from the given number; 27 - 3 = 24. 

-      Now, multiply the number with the zero by the sum of the given number and the difference we determined: 30 x 24 = 720

-      Now square the difference we determined before, and add it to the result above: 720 + 3² = 720 + 9 = 729. This is the square of the number!

An example of squaring a 3-digit number let’s say 788.

-      Round up or down the given number to the nearest multiple of 100 and easy to work with. In this example, the number is 800. 

-     Find the difference between the given number and the rounded up number. In this case, it will be (800 – 788) = 12. 

-      Subtract the result from the given number; 788 - 12 = 776. 

-    Multiply the rounded number by the subtraction of the given number and the difference we determined: 800 x 776 = 620800

-      Square the difference, and add it to the result above: 620800 + 12² = 620800 + 144 = 620944. This is the square of the number!

Now let’s see how this works for 2-digit numbers:

41² =?

To square 41, subtract 1 to obtain 40 and add 1 to obtain 42. Next multiply 40 × 42. Don’t panic! This is simply a 2-by-1 multiplication problem (specifically, 4 × 42) in disguise. Since 4 × 42 = 168; 40 × 42 = 1680. Almost done! All you have to add is the square of 1 (the number by which you went up and down from 41), giving you 1680 + 1 = 1681.

It works whether you initially round down or round up. For example, let’s examine 77², working it out both by rounding up and by rounding down:

Try to round up or down to the nearest multiple of 10. So if the number to be squared ends in 6, 7, 8, or 9, round up, and if the number to be squared ends in 1, 2, 3, or 4, round down. (If the number ends in 5, you do both!) With this strategy you will add only the numbers 1, 4, 9, 16, or 25 to your first calculation.

Let’s try another problem. Calculate 56²

Squaring numbers that end in 5 is even easier. Since you will always round up and down by 5, the numbers to be multiplied will both be multiples of 10. However, for squaring numbers ending in 5, you should have no trouble beating someone with a calculator, by following the tricks mentioned earlier in this book under ‘Multiplying numbers ending in 5’.

Now let’s see how this method works for a 3-digit number:

Squaring three-digit numbers is an impressive feat of mental prestidigitation. Just as you square two-digit numbers by rounding up or down to the nearest multiple of 10, to square three digit numbers, you round up or down to the nearest multiple of 100. Take 193:

By rounding up to 200 and down to 186, you’ve transformed a 3-by-3 multiplication problem into a far simpler 3-by-1 problem. After all, 200 × 186 is just 2 × 186 = 372 with two zeros attached. Almost done! Now all you have to add is 7² = 49 to arrive at 37,249.

 

Now try squaring 314:

For this three-digit square, go down 14 to 300 and up 14 to 328, then multiply 328 × 3 = 984. Tack on two 0s to arrive at 98,400. Then add the square of 14, which is 196 to arrive at 98,596.

The farther away you get from a multiple of 100, the more difficult squaring a three-digit number becomes. For instance, try squaring 636:

                                               

The hard part comes in adding 1,296 to 403,200. Do it one digit at a time, left to right, to arrive at your answer of 404,496.

Here’s an even tougher problem, 863²:

The first problem is deciding what numbers to multiply together. Clearly one of the numbers will be 900, and the other number will be in the 800s. But what number? You can compute it two ways:

1. The hard way: the difference between 863 and 900 is 37 (the complement of 63). Subtract 37 from 863 to arrive at 826.

2. The easy way: double the number 63 to get 126, and take the last two digits to give you 826.

Here’s why the easy way works. Because both numbers are the same distance from 863, their sum must be twice 863, or 1726. One of your numbers is 900, so the other must be 826. You then compute the problem like this:

With practice this can easily be done in your head.