## How does Arthur Benjamin square 5 digit numbers in his head?

Last month I blogged about mathemagician Arthur Benjamin and his amazing mental Math feats. Benjamin is a master of doing arithmetic in his head with lots of digits involved. In particular, he's able to square a 5-digit number without writing down partial results. How does he do it? I picked up a copy of Benjamin's Secrets of Mental Math to learn how. Here are the steps Benjamin provides for squaring 46,792.

1. First, Benjamin breaks the number into 46,000 + 792.

2. Then he does a little algebra. If a=46,000 and b=792, then (a+b)^2 = a^2 + 2ab + b^2 = (46,000)^2 + 2(46,000)(792) + 792^2.

3. This simplifies a bit to 1,000,000(46^2) + 2,000(46)(792) + 792^2.

4. Then Benjamin sets out to do the middle product: 2,000(46)(792).

5. He's not worried about the factor of 2,000 since he'll take the product of 46 and 792 and multiply it by 2 and add 3 zeros at the end.

6. At this point, Benjamin needs to decide how to multipy 46 x 792 in his head. Building up to this point in the book, Benjamin has shown a variety of methods to streamline multiplication of numbers of with different numbers of digits and he makes the point that you need to spend a few moments looking at the problem at hand and pick the method that will strain your brain the least.

7. For this example, Benjamin notices that 792 is close to 800 so he calculates 46(800-8) = 46x8x100 - 46x8.

8. He calculates that 46x8 = 368, adds two zeros to get 36,800 then subtracts 368 from that to get 36,432.

9. Now he has to multiply 36,432 by 2,000. He doubles 36,432 to get 72,864 then adds three zeros to get 72,864,000.

10. Now things get more interesting. Benjamin has just calculated the 2ab part of the a^2+2ab+b^2 equation. He's got two squares left to compute and he's got to add those two squares to the number he just computed. How is he going to remember this all?

11. Benjamin uses a phonetic code where he translates digits into syllables. The idea is that strings of syllables are easier to remember than strings of digits. I believe this is also how people memorize long strings of arbitrary digits quickly. He memorizes the first two digits, 72, and encodes 864 as "fischer". Then, Benjamin says, out loud, "72 fischer", a couple of times to anchor the digits and phonetic code into his memory. There are 3 zeros at the end. He'll use those later.

12. Now, onto computing 1,000(46^2). Benjamin doesn't say how he does this calculation but I imagine he notices that 46 is close to 50 and calculates 46(50-4) = 46x50 - 46x4. Multiplying by 50 is the same as multiplying by 100 and dividing by 2. So, 46x50 = 2,300. Subtract 46x4 = 184 from 2,300 gives us 2,116 which is 46^2.

13. Benjamin adds 6 zeros to 2,116 to get 2,116,000,000 which is 46,000^2. He adds 2,116,000,000 to the 72,864,000 that he memorized with that phonetic code "72 fischer" and gets 2,188,864,000.

14. Benjamin doesn't want to remember this new number, 2,188,864,000 so first he says "2 billion aloud" since that digit won't change in the subsequent calculation.

15. Next, Benjamin want to says "188 million" but before doing that he needs to see if adding the rest of the product, namely 792^2 will generate a carry. 800 is close to 800 so 792^2 is close to 800^2 which is 640,000. Adding 640,000 to 864,000 will definitely generate a carry so Benjamin says "189 million". Now Benjamin doesn't need to memorize the 2 or the 188 anymore and he's left with 864,000 to add 792^2 to.

15. Benjamin has already noticed that 792 is close to 800 in step 7 so he uses the fact that (a+b)(a-b) = a^2 - b^2, or (a+b)(a-b) + b^2 = a^2 to determine that 792^2 = (792+8)(792-8) + 8^2 = 800 x 784 + 64. Benjamin has, earlier, in the book, given techniques for multiplying single digit numbers by multiple digit numbers so he quickly determines that 800 x 784 + 64 = 627,264.

16. Now, all Benjamin has left to do is add 627,264 to 864,000, which he memorized as "fischer". He adds 864 to 627 to get 1,491. He has already used the 1 as a carry previously so he just says "491 thousand" aloud.

17. All that's left now is "264" which he says aloud as the rest of the answer.

There you have it. It takes quite a bit of mental Math effort to do this kind of calculation in your head but my intent, and Benjamin's, is to demystify mathematics to show you how you could do this kind of arithmetic in your head if you really wanted to.

Secrets of Mental Math goes into more detail than I provided and shows a number of clever Math magic tricks. This is a great book to have if you want to impress your friends with mental Math tricks.

If you enjoy mental Math tricks you may also enjoy these articles:

- How to square large numbers quickly (part 1)
- Impress your friends with mental Math tricks
- Quick multiplication by 12: A gentle introduction to Trachtenberg speed mathematics
- The algebra of cross-multiplication
- Mental Math and slowing down aging
- Mental Math magic by Arthur Benjamin
- Vedic multiplication using bases: an introduction
- How fast can you do mental Math?

You might also enjoy these videos.

pat ballewJanuary 31st, 2008 - 11:57

Not sure how Benjamin does it (he still amazes me) but I think 46^2 is easier to do as (45+1)^2… I would be a bundle of paper towels that Benjamin knows that 45^2 = 2025, so all he has to do is add 91 to get the answer.

My entire Alg II class can square two digit numbers ending in five mentally, so extending this to 46 is pretty easy…

Pat

George BellFebruary 1st, 2008 - 12:07

It’s one thing to work out the steps in front of your keyboard, and quite another to do it in front of hundreds of people! I know that I lose all my ability to manipulate numbers in front of a large audience. Benjamin certainly has much practice, and he doesn’t seem to be bothered about the audience at all. It is funny how he makes calculators look slow. I want to see him beat a calculator in calculating sin(17 degrees)!!

SolFebruary 1st, 2008 - 14:18

@Pat: Yes, I agree. I bet Benjamin would notice that 46 is one more than 45 and take advantage of the easy trick for squaring numbers that end in 5 and then know that (a+1)^2 = a^2+2a+1 so 46^2=45^2+91.

@George: I agree. Doing this kind of arithmetic under pressure takes a lot of hard work to get to. My intent, though, is to take the mystery out of his method so that kids who see Benjamin in action will realize that they too could do what he does. Most of us, myself included, would need to start with two-digit squares and work up from there.

Lim Ee HaiFebruary 2nd, 2008 - 08:53

Mental math is one area of arithmetic that challenges the conventional method of doing it. The benefit that it has on the mind of the learners makes it suitable to be included in math syllabus in primary school. It needs effort though when the numbers get big. The mental square rooting part is still dawning for me. I have read the recommended book and find it nice to have.

QuanFebruary 6th, 2008 - 02:24

nice website here. I’m a math fan myself.

I recall benjamin doing some amazing things in his head throughout this class he was teaching when I was an undergrad at Harvey Mudd College back in 01. Funny guy, good professor. Ridiculously smart.

GarethMarch 3rd, 2008 - 05:30

Interesting post! I’m fascinated with people who can do mental arithmetic like this – particularly as I’m so bad at it!!! I’ve got Benjamin’s book, which is a really good read.

On the subject of 46^2, an even easier way of thinking about it that I learned last night is as follows:

N^2 = (25 + (N – 50)) * 1000 + (N – 50)^2

46 is 4 less than 50. Take 4 from 25 = 21. Square the 4 and add the result to the end = 2116. So simple!

37 is 13 less than 50. Take 13 from 25 = 12. 13^2 = 169. Add result = 1369.

57 is 7 more than 50. Add 7 to 25 = 32. 7^2 = 49, add result = 3249.

Ed FarisMarch 10th, 2008 - 08:53

Do you know of any way to contact Mr.Benjamin? I’m sitting here watching one of his videos (http://static.videoegg.com/ted/flash/fullscreen.html?v=/ted/movies/ARTHURBENJAMIN-2005&cid=/ted/movies), I’m about halfway through, and he has done two calculations incorrectly.

457 sqaured = 208,849

He says it = 205,849

722 sqaured = 521,284

He says it = 513,284

Both wrong calculations were said to be correct by the same volunteer on stage.

SolMarch 15th, 2008 - 08:51

@Lim Ee Hai: Thanks for visiting. It’s nice to see you here again. I completely agree with you re the value of mental Math. Keep up the great work on your blog. I enjoy it and also enjoy that you and I have the same blog theme among all the themes in the universe!

@Gareth: Thanks for sharing yet another mental Math trick!

@Ed: Mr. Benjamin is a Math professor at Harvey Mudd College. You can Google for the Math department there and likely track him down that way.

AnonymousJune 23rd, 2008 - 21:32

could you please help me out in this :

how do you find the 3-digit numbers and the product in a multiplication of 2 , 3 digit numbers ,if the numbers are replaced by alphabets ??

ex: B H I * C A F

————————-

B A J C

D D E E +

J G H D + +

———————-

J A B G G C

—————–

HERE each number is replaced by an alphabet and no two alphabets represent the same number .

SantoshMarch 11th, 2009 - 02:30

Res

Any 4 digits numbers Suares details

THNK

Ian MallettApril 16th, 2009 - 11:04

Squaring 46 can be done easily another way.

Using Benjamin’s “up and down” method, you get:

46**2 = 42*50 + 4**2

42*50 = 42*100/2, so you get 4,200/2 = 2,100

Add 4**2 = 16 to get 2,116

primalAugust 5th, 2010 - 00:29

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ElthanMarch 19th, 2011 - 16:42

Hi All,

I find the above method much too complicated to execute in my head, but perhaps I am so stuck in my own method that any other means of squaring numbers is confusing. I have a method of squaring numbers which allows me to square 4-digit numbers in my head. Granted it takes some time (about a minute), but it’s possible. I haven’t found it mentioned anywhere on the Internet, so I thought I would share it here.

Let’s start with an easy number: 12. We all know the answer is 144, but to demonstrate I start by squaring the individual digits.

1 * 1 = 01

2 * 2 = 04

Both answers should be given two spaces, so I end up with 0104. Obviously, we drop the first zero.

Now I multiply the 1 by the 2, and continue to multiply by the magic number 20. I don’t know why 20, but it works, and it is universal. So I have:

1 * 2 * 20 = 40.

Now I add the 40 to the 104 to get 144.

Let’s try something a little less familiar: 83.

Sqaure the individual numbers to get 6409.

8 * 3 = 24.

24 * 20 = 480.

6409 + 480 = 6889 -> Our final answer.

Three- and four-digit numbers requires a couple of extensions of the above rules, but no more difficult. Any thoughts?

Ian MallettMarch 19th, 2011 - 17:05

I love that method!

I can square huge numbers in my head; like in the trillions: (4*10^13)^2 = 1.6*10^27. They just have be one or two digits xD

The largest (actual) squaring problem I’ve done is four digits, but then I don’t memorize any digits with a code, and it does take a while.

ElthanMarch 19th, 2011 - 17:33

So let’s look at three- and four-digit numbers.

We’ll try squaring 734. Using the method in my previous post, I start with 490916 (squaring each digit individually). Now:

7 * 34 = 238

238 * 200 = 47600

Note that we’ve multiplied by 200 rather than 20 – because we’re one column to the left. Finally:

3 * 4 = 12

12 * 20 = 240

So the square of 734 equals:

490916 +

47600

240

=538756

And now a four-figit number randomly picked out 8126. Again, square the digits individually to get 64010436. Then:

8 * 126 * 2000 = 2016000

1 * 26 * 200 = 5200

2 * 6 * 20 = 240

So 8126^2 equals:

64010436 +

2016000

5200

240

=66031876

I actually do all of this in reverse when I am doing it in my head, squaring the digits individually last. I remember the smaller numbers first and build on them.

Hope this is helpful to anyone who needs to square numbers!

ahmedDecember 10th, 2011 - 05:36

In his book Think Like a Mathematical Genius, Arthur Benjamin has, indeed shown the methods employable to square 3 , 4 and 5 digit numbers , mentally.

I know that it can be done. One also knows that some can do it fast.

MY PROBLEM IS : ” How to hold numbers in ones head, in the first place ?? ” Calculations of the bits is easy.

Any hints about how ordinary persons should learn these processes ?

MichaelMarch 14th, 2014 - 10:02

I did some research and found out how he stores such large numbers in his head. He uses words.

In order to remember complex numbers like 19475 he has devised a system which compacts the number into a simple word which he can then string together as a phrase.

In math we have seen this method used to take large numbers and break them down into something more legible.

For example a binary string ( a string of 1’s and 0’s) can be extremely hard to remember or write down. But what if the base is converted into something that is easier to use such as Hexadecimal.

Using Hex let’s practice taking this impossibly long string and making it easy

1110111100010111001

Group them by 4 digits

0111 0111 1000 1011 1001

Now directly convert them to their hex equivalent (0 – F)

7 7 8 B 9

See how the hex string 778B9 is easier to remember.

Now imagine that a word is used to remember the complex string.

This is how he is storing the complex numbers.

varunMarch 16th, 2014 - 23:03

For 46^2 , Thats not the method he would use, he would do as follows

50 is nearest multiple of ten to 46

{(46+4)*(46-4) }* {4^2}

which breaks to an easy (50*42) + 16 = 2116

It takes 3 to 5 sec if u practice it well

This method was explained in his book

mahri sherlockJanuary 1st, 2015 - 16:49

29106 squared=847159236.

15963 squared= ?another nine digit number without zeros and nothing repeated

JeromeJuly 26th, 2015 - 21:32

I recommend buying his book if you haven’t already. It is awesome. It teaches you the 5 digit square and much more. The title is Secrets of Mental Math ISBN is 978-0-307-33840-2. I got it from the Barnes and Nobles I used to work at for $14