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SquaringNumbers

Introduction

Squaring a number simply means multiplying a number by itself. This page teaches how to mentally square numbers in your head.

Squaring Numbers Up To 25

The first 25 squares should simply be practiced and remembered, either by rote, or by using NumberSystems. These are relatively simple, but will be used in calculation in the later steps, and are therefore basic building blocks. Here are the numbers from 1 to 25, and their squares:

X X^2^
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625

Simple Numbers to Square

There are some numbers that are easy to square by their very nature.

Squaring Numbers Ending in 0

When given a number ending in 0, there is an extremely easy shortcut you can take to get the square of the number:

1) Drop the 0 at the end of the number. 2) Mentally multiply the remaining number by itself. 3) Put "00" on the end of the number.

For example, let's take 20. We drop the 0, leaving us with 2. Square 2 and get 4. Putting 00 on the end of the 4, we get 400.

Let's try this with a higher number, like 120. Dropping the 0 gives us 12. The square of 12 gives us 144. Tacking 00 on the end gives us 14,400, which is 120^2^!

Squaring Numbers Ending in 5

When given a number ending in 5, there is a very easy shortcut you can take to get the square of the number:

1) Drop the 5 at the end of the number. 2) Mentally multiply the remaining number by a number 1 higher than itself. (This can also be done by squaring the number and then adding one more of the number.) 3) Put "25" on the end of the number.

For example, let's take 15. We drop the 5, leaving us with 1. 1 higher than 1 is 2, so we multiply 1 * 2 and get 2. Putting a 25 on the end of the 2, we get 225!

Let's try this with a higher number, like 85. Dropping the 5 gives us 8. 8 times 1 greater than itself (9) gives us 72. Tacking a 25 on the end gives us 7,225, which is 85^2^!

Now an even higher number, like 805. Dropping the 5 gives us 80. 80 times 81 is also 80^2^ plus 80 or 6400 plus 80 or 6480. Tacking 25 on the end gives 648,025 which is 805^2^.

Squaring Numbers Near Numbers With Known Squares

The methods of squaring other numbers aren't as simple as that of numbers ending in 0 or 5. However, if you do know, or can easily calculate, the square of one number nearby, you can take advantage of this closeness. Numbers just 1 or 2 away from easier-to-square numbers have their own approaches, which are detailed in this section.

Since every number that doesn't end in 0 or 5 is only 1 or 2 away from a number ending in 0 or 5, these approaches are very useful to learn.

Squaring Numbers 1 Greater Than Numbers With Known Squares

If you know N^2^ (say 120^2^ which is 14,400) and want to calculate (N+1)^2^ you can use the fact that (N+1)^2^ is N^2^+2N+1 or (often simpler and gives the same result) N^2^+N+(N+1).

For example, let's take 121. We know (or can quickly determine) that the square of 120 is 14,400. Then either double N and add one (twice 120 is 240 plus one yields 241), or add N to N+1 (120 plus 121 gives 241) and add that to N^2^ (14,400) to get 14,641 which is 121^2^!

Let's try this with a higher number, like 351. First we need to know 350 squared, which is 35^2^ or 1225 with 00 added on the end to give 122,500. Then add 350 plus 351 (or twice 350 plus one) which is 701 which gives 123,201 as 351^2^.

Squaring Numbers 1 Less Than Numbers With Known Squares

As in the previous example, (N-1)^2^ is N^2^-2N+1 or N^2^-N-(N-1).

So 349^2^ is 350^2^ (122,500) minus 350 minus 349 (or twice 350 minus one) 699 which taken from 122,500 gives 121,801 which is 349^2^. Note that it is often simpler to subtract twice N (700) and then add one for the final result.

Squaring Numbers 2 Greater Than Numbers With Known Squares

As in the previous two examples, (N+2)^2^ is N^2^+4N+4 or N^2^+4(N+1).

So 352^2^ is 350^2^ (122,500) plus four times 351 (1404) which added to 122,500 gives 123,904 which is 352^2^. Note that it is often simplest to double N twice (double 350 is 700 and double that is 1400) added to 122,500 give 123,900 and then add four for the final result of 123,904.

Squaring Numbers 2 Less Than Numbers With Known Squares

As in the previous example, (N-2)^2^ is N^2^-4N+4 or N^2^-4(N-1).

So 348^2^ is 350^2^ (122,500) minus four times 349 (easily calculated as the double double of 350 minus four: 1396) which taken from 122,500 gives 121,104 which is 348^2^. Note that it is often simplest to subtract double double of N (122,500 minus 1400 is 121,100) and then add four for the final result of 121,104.

Squaring Numbers From 1 To 125

Here in this section, you'll learn a comprehensive method for squaring numbers from 1 through 125!

Squaring Numbers Up To 25

As discussed in an earlier section, you should know the first 25 squares from memory before proceeding.

Squaring Numbers from 25 to 50

Here are the steps to calculate squares of numbers from 25 to 50. They may seem tricky at first, but they're easily done in your head after regular practice:

1) Figure out the difference between your number and 50, and call that difference N.
2) Subtract 2500 - (100 * N).
3) Add N^2^ the previous total. You should know N^2^ automatically from section 2.

The result will be the square of the number from 25 to 50!

Let's try 44^2^. The difference between 44 and 50 is 6, so we mentally do 2500-600, giving us 1900. 6^2^, as we already know, is 36. Adding this to our previous total, we get 1,936!

As another example, farther away from 50, we'll try to figure out 32^2^. The difference between 32 and 50 is 18, so we subtract 2500-1800, giving us 700. 18^2^, as you know, is 324. 700+324=1024, which is 32^2^!

Practice this method and get comfortable with it, before moving on.

Squaring Numbers from 50 to 75

If you're comfortable with the technique in section 4, then squaring numbers from 50 to 75 won't be much of a problem:

1) Figure out the difference between your number and 50, and call that difference N.
2) Add 2500 + (100 * N).
3) Add N^2^ the previous total. Again, you should know N^2^ automatically from section 2.

The process is very similar, but note that we add in step 2, instead of subtracting. That's the only difference.

Let's use this process to figure out the answer for 57^2^. The difference between 57 and 50 is 7.
2500+700=3200
7^2^=49
3200+49=3249

Trying this with 72, we see the difference between 72 and 50 is 22.
2500+2200=4700
22^2^=484
4700+484=5184

Squaring Numbers from 75 to 100

The process changes here. With numbers from 25 to 75, we've focused on their difference from 50. As we're getting close to 100, instead. Here's the process to use for numbers from 75 to 100:

1) Figure out the difference between your number and 100, and call that difference N.
2) Subtract N from the number to be squared.
3) Multiply that number by 100 (just add two zeros at the end)
4) Add N^2^ to that number.

To help clarify things, let's try to figure out 97^2^, using this process:
1) 100-97=3, so N=3
2) 97-3=94
3) 94*100=9400
4) 9400+(3^2^)=9400+9=9409

Trying this with a number farther away, like 81^2^ gives us:
1) 100-81=19, so N=19
2) 81-19=62
3) 62*100=6200
4) 6200+(19^2^)=6200+361=6561

Squaring Numbers from 100 to 125

If you've been following the steps, and practicing, you probably won't be surprised to know that this section only involves one minor change from the previous one:

1) Figure out the difference between your number and 100, and call that difference N.
2) Add N to the number to be squared.
3) Multiply that number by 100 (just add two zeros at the end)
4) Add N^2^ to that number.

Once again, step two simply changes from subtracting to adding.

Let's try and figure 106^2^ using this method:
1) 106-100=6, so N=6
2) 106+6=112
3) 112*100=11,200
4) 11,200+(6^2^)=11,200+36=11,236

If you were able to do that mentally, keep in mind that you just did 106^2^ in your head!

How about 124^2^?
1) 124-100=24, so N=24
2) 124+24=148
3) 148*100=14,800
4) 14,800+(24^2^)=14,800+576=15,376

Just to be complete, what about 125^2^?
1) 125-100=25, so N=25
2) 125+25=150
3) 150*100=15,000
4) 15,000+(25^2^)=15,000+625=15,625

Don't forget, though, that there is a much easier way to do 125^2^! Remember?
1) Dropping the 5, we get 12
2) 12+1=13
3) 12*13=156
4) Tacking the 25 on the end, we get 15,625.

Another method for squaring two digit numbers

This is best shown through an example.

Let us say that we wanted to square 43.

The first step is separate it into a problem where we are multiplying two numbers as such. 43 * 43

The next step is to round down to the nearest multiple of ten on one of the numbers, then add up on the other one the same amount we rounded down. In this case we are rounding down to 40, subtracting 3, and adding 3 to the other side, giving us 40 * 46. We must also remember the value that we rounded by.

We now multiply 40 times 46 in our head. The simple way to do this is to drop the zero, then multiply from left to right. 4 * 4 = 16, plus a zero on the end equals 160, and 4 * 6 = 24, and now add the two for 184, and finally add the zero back for 1840.

We now take the number we rounded by earlier, square it, and add it to the total. In this case 3^2 = 9, so our answer is 1840 + 9, or 1849.

With practice this can be done very quickly. Lets go over one more to make sure the process is clear.

56 ^ 2 - Initial Problem

56 * 56 - Split the problem

60 * 52 rounding 4 - Round the numbers

3120 rounded 4 - Multiply the rounded numbers

3120 + 16 - Square the number we rounded by.

3136 - Add the two numbers for the result.

The only thing that might slow you down on this method is multiplying the rounded numbers. But with practice you can do it nearly instantaneously.

Tips

Every square of any number ending with n has the same last digit as n^2^.

For example, 123456789^2^ and 98789^2^ must both end with 1 because 9^2^ is 81. Likewise, 1234567^2^ and 797^2^ must both end with 9 because 7^2^ is 49.