Three Tricks for Squaring Numbers

(Image by Paul Schadler)

A mental math installment:

While walking to Carnegie Mellon the other day I started thinking about squaring two digit numbers. So instead of hoarding my findings and short amount of research I thought I would write a tutorial on how to square two digit numbers, and then some.

Method 1: If You Know the Previous Square

This method is only marginally helpful, but will come in handy if you know how to easily square the number previous to the number you’re trying to square. Let’s say you’re trying to find x2. If you know what (x-1)2 is already, all you have to do is add x and (x-1) to (x-1)2 to find x2.

Proof:

(x-1)2 + x + (x-1) = (x2 – 2x + 1) + x + x – 1 =
(x2 – 2x + 1) + 2x – 1 =
x2 + (2x – 2x) + (1 – 1) =
x2

q.e.d.

Example:

Let’s find 312 knowing that 302 = 900.

312 = 302 + 31 + 30 = 900 + 31 + 30 = 961.

This method is mostly helpful for squaring numbers which are one more than a multiple of ten, since humans can square multiples of ten without much thought (more on this later). Also, you’re not restricted to squaring two digit numbers with this method, which is quite fantastic.

Method 2: Multiplying One Up, One Down, and then Adding One

In my opinion this method is a bit more fun, but it definitely requires some mental multiplication. In fact, you’ll almost be doing as much work (or more) using this method as you would be in outrightly squaring a number, but it is quite an amusing trick.

Let’s say you want to square a number x. If you multiply (x-1) and (x+1) together, and then add 1, you’ll find x2.

Proof:

(x-1)(x+1) + 1 =
(x2 + x – x – 1) + 1 =
x2 + (x – x) + (1 – 1) =
x2

q.e.d.

Example:

Let’s find 312 again using this trick.

312 = (30)(32) + 1 = 960 + 1 = 961.

Like I said, if you can’t quickly calculate that (30)(32) is 960, then this method isn’t saving you much mental energy, but this averaging method could be a shortcut for some.

Method 3: Squaring Two Easy Numbers Instead of One Hard Number (Plus a Step)

In my opinion, this method is best practice. This explanation will be a bit lengthier, and also a bit harder to write out in plain English, but it’s the most fruitful by far of the three methods.

Like I said in my previous post on mental math, perhaps the most extensive repository most folks have for math is their times tables. With some exception, people are able by their mid-teens to multiply together two one digit numbers with relative ease, all the way up to (9)(9) = 81. Let’s make use of this fact, plus the fact that folks can easily multiply multiples of ten, to square any two digit number easily.

First, one needs to realize that any two digit number is at most five numbers away from a multiple of ten. For instance, 34 is 4 numbers away from 30, and 65 is 5 numbers away from both 60 and 70. So when we’re thinking of squaring large two digit numbers it’s best to think of it as a multiple of ten (e.g. 10, 20, 30, 40, …) plus or minus a number no greater than five.

So when we go to square a number like 74 mentally (yikes!), it’s better for one to imagine this number not as 74 but as (70 + 4), or to imagine the number 66 not as a single number, but as the difference between two, i.e. (70 – 4). It’s a matter of breaking down a difficult single process into several easier ones.

So let’s find 742. Instead of looking at the single number 74, let’s break it down into (70 + 4). Now let’s square this number.

(70 + 4)2 =
(70 + 4)(70 + 4)

At this point in the game we’re multiplying two binomials together. Remember the FOIL (Front, Outer, Inner, Last) method from way back when? I thought so. Continuing with this method we have

702 + (70)(4) + (4)(70) + 42
4900 + 280 + 280 + 16 (a)

We’ve broken the process down into more or less multiplying single digit numbers together and then adding zeroes at the end. We know 72 = 49, (7)(4) = (4)(7) = 28, and 42 = 16, and it’s then easy to multiply these numbers by powers of ten.

Adding these numbers up we get 5476. We’ve taken a difficult multiplication problem and turned it into an easier addition problem (though some would beg to differ, I’m sure). It works the same for a number like 66, which looks like (70 – 4) when we break it down. The only difference is that we’ve traded our plus sign for a negative one. The only part of the arithmetic that changes is that instead of adding 280 twice to expression (a) above, we subtract it twice. So similarly to 742 we have that

662 =
4900 – 280 – 280 + 16 =
4356

And this method, in my opinion, is by far the easiest way to mentally square two digit numbers. A similar process will work with three digit numbers. Perhaps I’ll write on that later.

There you go. Hopefully the next time you need to square a number quickly you’ll be more equipped for usual. So for now, happy squaring, and let me know if anyone has any suggestions or additions.

Note: For all you nitpickers out there let it be known that when I’ve used the word “number” at any point in this post I’ve actually meant this word to mean “positive integer”.

(Image by Paul Schadler)

Have a comment?
  1. hey ian -

    thanks for the lesson.

    i wanted to say hey and let you know i think of you often. it’s funny when i talk to scott; he reminds me a lot of you, and it’s been good to have the opportunity to talk to him as much as i have.

    in any case, i want to hear how life is for you and what’s new and old and obscure and pointless as well. i’m in erie currently (and will be for three years hence), so if you feel like a 2 hour trip for a coffee, beer, and conversation, feel free to come up.

    also, give emily my good wishes.

    peace,
    adam

  2. N L Shraman says:

    There is another very easy metod to make a sqare:

    Suppose you have to find out 43^2, just start writing from the right side
    (i) 3*3=9 (unit multiplied by unit)
    (ii) (3*4)*2=12 write 2 carry one (unit*tens and double it)
    (iii) 4*4=16 add carry 1, write 17 (tens*tens)
    The answer is 17 2 9

  3. Thanks N L Shraman, I appreciate it! I’ll have to keep that in mind for the future…

  4. N L Shraman says:

    Dear Ian, I am a mental math and mental memory trainer of India, you may discuss any problem related to the math and memory. You may visit my site for detail http://unlimitedmemory.tripod.com

  5. N L Shraman says:

    excuse me for some words are not spelled correctly due to poor response n site.

  6. N L Shraman says:

    MULTIPLY THE NUMBERS WHOSE SUM OF UNITS IS 10 AND TENS ARE SAME
    Just multiply unit by unit and tens by ( tens+1) e.g.

    83×87= 8x(8+1)=72 and 3×7=21

    The Ans is 72 21

  7. Dear Ian,
    My full name is Nand Lal Shraman. I am from Kanpur, India. I have already responded your methods of squaring. Today I am going to describe the method for squaring large numbers say three digits in a very easy way. For example :
    What is the square of 425 ? here is the method:
    Step 1 Multiply 5*5 (Unit*Unit)=25 carry 2 leave 5
    Step 2 Multiply 5*2(unit*tense)*2=20 ,add 2=22 carry 2,leave2
    Step 3 Multiply 5*4*2+2*2= 44 add 2=46 carry 4 leave 6
    ( Unit*hundredth)*2 + TENSE*2
    Step 4 Multiply 2*4*2 add 4=20 ,carry 2 leave 0
    (tense*hundredth)*2 add carry
    Step 5 Multiply 4*4 add 2=18 leave 18
    (Multiply hundredth*hundredth add 4
    The answer is 180625
    However it looks somewhat difficult but it as easy as 1,2,3… Let me describe again:
    Multiply U*U
    Multiply U*T double it and add carry leave unit
    Multiply U*H double it + T*T add carry, leave unit
    Multiply T*H double it add carry leave unit
    Multiply H*H add carry leave it.
    Next Time I will decribe how to mutiply or square large numbers of 4,5,6,7,8,9,10,11,12 digits etc.
    It is a fun once you start calculating.
    Yours Own
    N. L. Shraman
    Top trainer of math and memory in India

  8. Aung Khaing says:

    Hi Ian and Nand,

    My name is Aung Khaing, a sophomore student in the University of Arkansas, Fayetteville. I also like to play with numbers like you do. I devised a general formula for any squares. Please see how you like that. Because of the limited way to experess my thought on this web, all I can do is to type into Excel and copy it.

    If you would have a chance to see it in Microsoft word explanation and Microsoft equation editor, you would see more clearly. If you do that on paper you don’t really need to wirte down all the zeros. You can just leave blanks in places of zeros (eg. four blanks for 10^4)and can fill in with nonzero numbers. In this way, you only need 3 lines on paper to square 3 digit number and add them up. For any 2 digit number, you can square it in mind.

    For now, this is the best way I can transfer my thought. You can just copy the following formula into and excel and test it.

    For 2 digit numbers, treat them like AB^2.
    =(((10^2)*(A1^2))+(B1^2)+(2*(10)*(A1*B1)))

    For 3 digits numbers, treat them like ABC^2.
    =(((10^4)*(A1^2))+((10^2)*(B1^2))+(C1^2)

    +(2*(10^3)*(A1*B1))+(2*(10^2)*(A1*C1))+(2*(10)*(B1*C1)))

    For 4 digits numbers, treat it like ABCD^2.
    =(((10^6)*(A1^2))+((10^4)*(B1^2))+((10^2)*(C1^2))+(D1^2)

    +(2*(10^5)*(A1*B1))+(2*(10^4)*(A1*C1))+(2*(10^3)*(A1*D1))

    +(2*(10^3)*(B1*C1))+(2*(10^2)*(B1*D1))+(2*(10)*(C1*D1)))

    For 5 digits numbers, treat it like ABCDE^2.
    =(((10^8)*(A1^2))+((10^6)*(B1^2))+((10^4)*(C1^2))

    +((10^2)*(D1^2))+(E1^2)+(2*(10^7)*(A1*B1))+(2*(10^6)*(A1*C1))

    +(2*(10^5)*(A1*D1))+(2*(10^4)*(A1*E1))+(2*(10^5)*(B1*C1))

    +(2*(10^4)*(B1*D1))+(2*(10^3)*(B1*E1))+(2*(10^3)*(C1*D1))

    +(2*(10^2)*(C1*E1))+(2*(10)*(D1*E1)))

    Sincerely,
    Aung Khaing :)
    Sophomore, University of Arkansas, Fayetteville

  9. Jenna says:

    hey; the first method you have here I’ve wondered, is this a published theorum? I came up with it once myself and showed my precal teacher, who said that yes, she did recognize it. Also, though.. it can be extrapolated further if you have the spare time.

    (a+n)^2= a^2 + n[a + [a+n)]

    22^2 = 484

    20^2 + 2(20+22) = 484

    For any positive nonzero number a, the square of a+n will equal the square of a plus n[a + (a+n)].

  10. Jenna, unfortunately I do believe that all of these methods have been discovered long ago, although I’m sure it’s the case in mathematics that some very simple conjectures have gone a long time without having been stated. Take the Beal Conjecture, for instance. With all the talk of Fermat’s Last Theorem no one had stated the more generalized conjecture until literally hundreds of years later.

  11. Case says:

    I too came up with the (a|n)^2 = a0^2 + n(a|n+a0) as described earlier while on my way to work and I must say that for the hard ones this seems to be the quickest mental way for me to do it… that is to say I can typically solve in 30 seconds or less with it now.

  12. Matt says:

    Just curious if anyone still reads this, i came up with something, but i don’t want to post if no one is going to read it

  13. Jessica says:

    It’s kind of hard to read since you put x2 instead of x^2 or something, but good job.

  14. Yam Dak says:

    I thought of your second 2nd method along time ago too. I improved on it too. It’s (x+a)(x-a)+a^2.
    x^2 – a^2 + a^2 = x^2
    31^2= (32)(30)+1^2=961
    Another example: 55^2=(60)(50)+5^2=3000+25=3025
    Another example: 75^2=(80)(70)+5^2=5600+25=5625
    Same example: 75^2=(100)(50)+25^2=5000+625=5625

  15. Avi Pai says:

    Hey Guys, this is great learning. Thanks for contributing. Any discoveries on how to cube?

  16. tom says:

    You are all thinking WAYYYY too much into this.

    lets take 18squared.

    18 x 18.

    You look at the end of the numbers first 8 x 8. This is 64. So you know that the final number will end up in 4.
    Then you go back to the begining. 10 x 10 = 100
    Now the middle. 10 x 8 = 80. Plus another 80. This is 160.

    Then finish the rest 100 + 160 + 64 = 324.

    THIS WORKS WITH EVERYTHING UP TO 99.

    This is probably really obvious but what is the point in making it harder for yourself? I am 16 years old.

  17. Shraman N L says:

    RECTIFICATION OF MY POST DATED NOV 4, 2005

    There is another very easy metod to make a sqare:

    Suppose you have to find out 43^2, just start writing from the right side
    (i) 3*3=9 (unit multiplied by unit)
    (ii) (3*4)*2=24 write 4 carry 2 (unit*tens and double it)
    (iii) 4*4=16 add carry 2, write 18 (tens*tens)
    The answer is 1849

  18. priyanka says:

    its very interesting

    • ashok says:

      hey let me tel abt the suaring of two digit number ending with 5′s

      for ex take 25*25

      step 1: multiply 5*5=25
      step 2:multiply 2*3=6

      the answer is 625

      first of all wen u see 25*25 nd 35*35 etc wen ever u fing 2 digit numbers ending with 5 you just write 25 atlast and then multiply the first num with the conseqte num as shown above

      if it is 45*45
      u just write last 25
      and multiply 4*5=20
      so answer is 2025

    • ashok says:

      if u want more short cut just mail me once

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