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Squaring 2-Digit Numbers Mentally

Published on Tuesday, January 18, 2011 in , , , ,

Introduction

In this post, you'll learn how to square numbers from 1-100 in your head!

As a refresher, squaring a number simply means to multiply it by itself. For example, 4 squared is 16 because 4 times 4 is 16. You should know the squares of all the numbers from 1 through 10 by heart already.

Multiples of 10

If you already know your squares of the numbers 1 through 10, the multiples of 10 are easy. When a number from 1-100 ends in zero, simply drop the ending 0, square the remaining number, and then add 2 zeroes. For example, to work out 20 squared, drop the zero leaving the 2, square it to get 4, then tack on 2 zeroes to that 4, resulting in 400.

70 squared? 4900, because 7 squared is 49, and the two zeroes added make it 4900. 100 is trickier, but uses the same approach. 100 with the final zero dropped gives us 10. 10 squared is 100, and adding 2 zeroes gives us 10,000.

Multiples of 5

Multiples of 5 are almost as easy. You do need to make sure you know your multiplication tables up to at least 10 times 10. The method taught here is also taught in the root extraction tutorial, as well.

When given a number ending in 5, simply take the 10s digit, and multiply by a number one higher than itself. Take that answer, take a "25" on the end, and you've got the answer!

For example, let's say you're asked what 35 squared is. Take the 3 (the 10s digit), and multply it by 4 (which is one higher than 3), and you get 12. Tack a 25 on the end, giving you 1225. Simple, isn't it?

Let's try a higher number, like 75 squared. 7 times 8? 56. Tacking on the 25, gives us 5625!

Here's a slideshow to help explain this procedure in more detail:


To quiz yourself on squaring multiples of 10 and 5, click here. To learn the mental math approach to squaring the remaining numbers, click here. To learn the memorization approach to squaring the remaining numbers, click here.

Numbers from 1-25

For the approach using pure mental math, you'll need to know your squares from 1-25 by heart. From 1 to 10 you should already know, and from the techniques on the first page, 15, 20, and 25 will be easily handled, as well. That leaves these squares to learn by heart:

Number Square of Number
11 121
12 144
13 169
14 196
16 256
17 289
18 324
19 361
21 441
22 484
23 529
24 576

The must be known by heart, because the method we're going to use to work out the remaining numbers requires that you can give the above numbers quickly.

Numbers from 26-50

To work out the numbers from 26 to 50, we're going to use an approach in which we multiply by 50.

Multiplying any number by 50 is easy – all you have to do is divide the number by 2, and add 2 zeroes (more accurately, you would move the decimal 2 places to the right). 48 times 50? Half of 48 is 24, and two zeroes added results in 2400, which is the correct answer. This method only involves multiplying even numbers by 50, so you won't have to worry about dealing with numbers like 24.5 (Half of 49).

When given any number from 26-50, you're first going to work out how far that number is from 50, then subtract that distance from the given number. For example, if you're given the number 47, it's easy to work out that it's only 3 away from 50. Subtracting that 3 from 47, we get 44.

Instead of solving 47 times 47, then, we're going to work out the much easier problem of 44 times 50, which is 2200. However, this isn't the same as the answer to 47 squared, so we need to make an adjustment.

From 47, we both moved up 3 to 50 and down 3 to 44. So, we square this 3 to get 9, and add that to the other answer we worked out, 2,200, to get a total of 2,209. This is the answer to 47 times 47!

So, when given a number, you work out how far the given is from 50, and find a number that's equally far below the given number (the “low” number), and also remember this difference. Multiply 50 times the “low” number, adjust it by squaring the difference you moved, and adding that amount, and the result will be the square!

Let's try this with 44, to help make this clearer. 44 is 6 away from 50, so we figure out that 44 - 6 = 38. 38 times 50 is easy, 1,900. We moved a difference of 6 in both directions, so we add 36 (6 squared) to 1900, to get 1,936!

How about 39 squared? That's 11 away from both 50 and 28. 28 times 50? 1,400. 11 squared is 121, and adding that to 1,400, we get 1,521!

How about 35? Trick question! That's made easier by the multiples of 5 technique from the first page. Don't forget to use the easier techniques in the easier cases.

Numbers from 51-75

The same technique is going to be used for numbers from 51 to 75, but with one minor change. You'll be moving down to 50, and up to another number (Previously, you moved up to 50, and down to another number). Other than that, the process is basically the same.

Let's try 56 squared. 56 is 6 away from 50 and 62. 50 times 62 is 3,100, plus 36 (6 squared) gives us 3,136!

How about 67? The distance makes this a little more challenging, but the process is still the same. 67 is 17 away from 50 and 84, so we multiply those two numbers to get 4,200. 17 squared is 289, so we work out 4,200 plus 289 to get our final answer of 4,489.

Numbers from 76-100

As easy as multiplying by 50 has been, multiplying by 100 is even easier – just add 2 zeroes!

For numbers from 76-100, we're going to adjust upward to 100, as opposed to using 50 as we have been. Wait until you see how easy this makes the process!

Let's try working out 98 squared. 98 is 2 away from 100 and 96, and multiplied together, that gives us 9,600. 2 squared is 4, and 9600 plus 4 is 9,604. That's 98 squared!

How about 91? We start out with 8,200 (do you see why?), and add 81 (again, do you see why?), to get 8,281.

The more you practice each of these stages, the more you'll get a feel for certain patterns. This will allow you to speed up your calculations.

Numbers from 100-125

By now, you've probably figured out that you can go up to 125 with just a minor adaptation, similar to that we used when going above 50.

What's 103 squared? It's between 106 and 100, so we multiply those to get 10,600. 3 squared is 9, so that added in gives us 10,609!

What about a toughie, like 124? That's between 100 and 148, so we start with 14,800. 24 squared is 576, so we add those together to get 15,376.

With a little practice, you should have this process down in a faster time than you may have ever thought possible.

To practice squaring numbers, click here. To learn an alternative approach using memorization for squaring the numbers from 1 to 100, click here.

Memorizing the Squares

It was Alabama math and science teacher Jim Wilder who first suggested the idea of memorizing the squares to me. The process is similar to the one I use for memorizing 400 digits of Pi.

First, you should learn the the multiples of 10 and 5 techniques from the first page, and you should still know the squares from 1-25 by heart, as those are still quicker than the memory approach. This also helps minimize the amount of links needed.

Prerequisites:

Link System
Major System

Links

With the exception of the multiples of 10 and 5, Jim Wilder put in some amazing work developing major system mnemonics for all the squares from 26 to 99 (Thanks again, Jim, for both your work and willingness to share it with us!):

Number Square of Number Number Mnemonic Square Mnemonic
26 676 iNCH SHaKiSH
27 729 kNocK Key, NaP
28 784 kNiFe CoVeR
29 841 kNoB FoRT
31 961 MaiD PuSHeD
32 1,024 MooN DoSe NeaR
33 1,089 MuM ToSS FiB
34 1,156 MaRRy TighT LeaSH
36 1,296 MaTCH DowN PuSH
37 1,369 MoCHa DaMn, CHeaP
38 1,444 huMVee TiRe RoaR
39 1,521 MoP TaiL kNoT
41 1,681 RaT TouCH, FiT!
42 1,764 RuN TaKe SHaRe
43 1,849 RaM TuFF RoPe
44 1,936 RoaR TiP MatCH
46 2,116 ReaCH NoT TouCH
47 2,209 RoCK NoN SouP
48 2,304 ReeF NaM, SiR!
49 2,401 RiB uNRaiSeD
61 3,721 SHaDow MaKe NighT
62 3,844 CHaiN MoVe ReaR
63 3,969 CHuM MoP SHiP
64 4,096 CHaiR RiSe, PuSH
66 4,356 CHeeCH RuM LuSH
67 4,489 CHeCK RaRe FiB
68 4,624 CHeF ReaCH NeaR
69 4,761 CHiP RocK SHeeT
71 5,041 KiT LooSe, RighT?
72 5,184 CaN LeaD FeaR!
73 5,329 GuM LiMe NuB
74 5,476 CaR LuRe CaSH
76 5,776 CoaCH LuCK, CoaCH
77 5,929 CoKe LeaP, NaP
78 6,084 CaVe CHooSe FiRe
79 6,241 CaP SHiN, Right?
81 6,561 FaT JeLLo JeT
82 6,724 FiN CHiC NoiR
83 6,889 FoaM CheF FiB
84 7,056 FouR CaSe LatCH
86 7,396 FiSH CoMb PuSH
87 7,569 FaKe CoaL CHiP
88 7,744 FiFe KicK ReaR
89 7,921 FiB CaP NoD
91 8,281 PaT FuN FighT
92 8,464 PaN VeRy CHaR'd
93 8,649 PaM FiSH RuB
94 8,836 PouR ViVa MuCHo!
96 9,216 PiTCH BaNNeD SHow
97 9,409 PuCK PooR SOB
98 9,604 PuFF PuSHeS aiR
99 9,801 PaPa PuFFS iT

Memorizing the 50s

Like multiples of 10 and 5, the squares of numbers in the 50s have their own trick that is easy to remember.

Number Square of Number
51 2,601
52 2,704
53 2,809
54 2,916
56 3,136
57 3,249
58 3,364
59 3,481

When given a number in the 50s, simply take the ones digit and add it to 25. Next, take the square of the digit in the ones place, and tack that on to the right of the previous answer.

For an example, let's use 53. The ones digit is a 3, so we add 25 to get 28. 3 squared is 9, so we add 09 to the end of the other digit to get 2,809.

57? 7 plus 25 is 32. 7 squared is 49. Therefore, 57 squared is 3,249. Once the pattern clicks, you'll find that these are quick and easy.

To practice squaring numbers, click here. To learn an approach using mathematics for squaring the numbers from 1 to 125, click here.

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8 Response to Squaring 2-Digit Numbers Mentally

November 20, 2011 at 10:34 AM

I'm afraid at my age memorizing anything new seems hard, and since any two digit number is within three of a multiple of ten or five, it seems easiest to know how to square (x+/- a) where a is less than or equal to three and you would be done... 77 ^2 = (75+2)^2 = 5625 + 300 + 4

December 23, 2012 at 7:32 PM

ok i have an idea on most a formula type thing on most of 11-25.

11 squared. 11 times 11. take one of the elevens and add a 0 to the end. so now it looks like 110 times 11. now take the other eleven and make the tens place equivelent to the ones place. in this case. 11 stays the same. then you separate that number. so its 11 plus 1. square the single number ( the ones place that you separated) and then add the 2 numbers back together. in this case. 10 plus 1 squared is still 11. so you take that number that you have after doing all that and add it to the first number (110) 11 plus 110 is 121.

12 squared. 12 times 12 120 times 12 120 times 22 120 times 20 plus 2 120 times 20 plus 2 squared 120 times 24 120 plus 24 144

13 squared. 13 times 13 130 tiems 13 130 times 33 130 times 30 plus 3 130 times 30 plus 3 squared 130 times 30 plus 9 130 times 39 130 plus 39 169

14 squared (trickier) 14 times 14 140*14 140*44 140*40+4 140*40+16 140*56 140+56 196

15 squared 150*15 150*55 150*50+5 150*50+25 150*75 150+75 225

16 squared 160*16 160*66 160*60+6 160*60+36 160*96 160+96 256

17 squared 170*17 170*77 170*70+7 squared 170*70+49 170*119 289

18 squared 180*18......180*88....180*80+8 squared.....180+80+64........260+64....324

19 squared 190*19....190*99....190*90+9 squared .....190*90+81....190+90+81...280+81......361


this doesnt work for anything past 20 or anything below 11. another trick i use (im a freshman, and we do alot of squaring in my algebra honors class) is if i know what the square of the number below the number im trying to square, exapmle: if i need to square 21, and i dont have a calculator and i dont want to do it on paper, and i know 20 squared is 400, than i take 20. multiply it by 2, and add one. so 441 is 21 squared.
now that i know what 21 squared is i can find 22 squared....... 441 + 21*2+1 441+43.....484

also ive been taught that if its just easier to do, lets say im squaring 88, so (90-2)^2 instead of having to memorize your formula, becuase (90-2)^2 is just 90^2 - 2(90*2)+ -2^2....... based of of (x+/- a)^2 is equivelent to x^2 +/- 2ax + a^2. I do like the 5 formula of yours, that will be helpful. lol ive been typing for like 3 hours now and i forgot how i even got here.

January 14, 2014 at 10:38 PM

This is awesome.

January 27, 2014 at 10:25 AM

I uploaded a video to find the square for numbers ending with 5
https://www.youtube.com/watch?v=g9VQ5TbVqN8

January 12, 2016 at 1:41 AM

Not sure if this will help some folks, but:

Squaring two digit numbers that end in 6 (16, 26, 36, etc.):

- Square the nearest '5' using the '5' method (15, 25, 35, etc.)

- Add to that two times the number you just squared + 1

Example:

16 squared = 15 squared + 31 (15* 2 + 1) = 256

76 squared = 75 squared + 151 (75 * 2 + 1) = 5776

Since the '5's are so easy, this eliminates the addition/subtraction from 50/100. Easier for me because I've been doing it for so long. Your mileage may vary.

January 21, 2016 at 1:29 AM

Another quick tip for squaring 'teens'. May help if you read the list above and thought, 'I'm too lazy to memorize those squares!'

I think of it as 'Add a zero, drop the one, square the last digit', and it works like this:

16 squared

Add a zero = 160
Drop the one = 60
Square the last digit = 36

Add 'em together and viola!, 196

Or 12 (which I hope we already have memorized since 2nd grade)

120 + 20 + 4 = 144



January 23, 2016 at 2:35 AM

Edit (above): 16 squared = 256 (160+60+36). Must've been the water......

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