I have a short-cut for mentally calculating the square of any two-digit number.


For numbers where the second digit is 5 or less, proceed as in this example:

73 x 73

Think: (70 + 3) x (70 + 3)

Square the larger number (70 x 70): 4900
Multiply the two numbers in a bracket (70 x 3 = 210) and double it: 420
Add this on to get: 5320
Square the smaller number (3 x 3): 9
Add this on to get final result: 5329


For numbers where the second digit is 6 or more, proceed as in this example:

46 x 46

Think: (50 - 4) x (50 - 4)

Square the larger number (50 x 50): 2500
Multiply the two numbers in a bracket (50 x 4 = 200) and double it: 400
Take this off to get: 2100
Square the smaller number (4 x 4): 16
Add this on to get final result: 2116


There's nothing particularly special about the above, except that it is a way of looking at the task that makes it easier to do in your head.

Do you have any other short-cuts for other mental math tasks?

I do two-digit squares a bit differently, so that there's only two numbers to add, instead of three.

First, I learn the squares of the numbers 1-25 so that I can just look at them and see their square.

Now, for numbers from 26-50, I'm going to use their distance from 50 as a reference:

37 * 37?

37 is halfway between 50 and . . . 24 (50 is 37 + 13, so we do 37-13 to get 24).
Take 24 * 50, which is 1,200 (This is why I use 50, it's so easy to multiply by! Just take half the other number and add 2 zeroes!)
We've moved 13 each way, so what is 13 squared? 169 (Remember, you need to know the squares from 1-25 by heart).
Take the 1,200, add 169, and you've got 1,369!

How about 42 squared?
That's 34 * 50 (8 each way), which is 1,700.
8 squared is 64.
1,700 + 64 = 1,764!

For numbers from 50 to 75, you use the same process.

56 squared?

56 is halfway between 62 and 50, so 62 * 50=3,100
We went 6 each way, and 6 squared is 36.
3,100 + 36 = 3,1,36!

From 75-100, We use the same process again, but with 100 as the reference number (100 is veeeeery easy to multiply by!)

84 squared?

84 is halfway between 68 and 100. 68 * 100 = 6,800!
16 each way, means 16 squared, which is 256,
6,800 + 256 = 7,056!

Finally, you can use this method with numbers from 100-125, as well!

How about 119 squared?

119 is halfway between 100 and 138, and 138 * 100 = 13,800.
19 each way means 19 squared, which is 361.
13,800 + 361 = 14,161!

Do I have any other mental math shortcuts? Mental math is a favorite hobby of mine, so here are a few goodies for you:
Figuring Cube, Fifth and Square Roots
Doug Canning's "Mental Shopper"
Multiplying two 3-digit numbers together
Arthur Benjamin's "Mathemagics" Course
BEATCalc (Over 600 mathematical shortcuts!)

If you explore my site, linked in the signature below, you'll find plenty of mathematical wonders!

(a + b)(a + b) = a*a + 2*a*b + b*b

In the example in the first post, a = 70, and b = 3.

I think it's fun to try to reshape that expression to get further shortcuts when the number to be squared fullfills certain criterias. For example it's good to write the number A to be squared as 10a+b so that a and b are the digits of the number A.

A^2 = (10a + b)^2 = 10a(A + b) + b^2

which gives us a remarcably easy expression to do in the head when a is small, like 1 or 2. Let's assume that a=1 and that expression simply means:

Add the first digit, b, to the whole expression, then write the square of b to the right of it. Just remember that if b^2 is larger than 9 you have to carry the left digit to the number you already wrote.

12^2 means that you write 12 + 2 first then write 2^2 to the right of it. 144 in other words. 17^2 means that you write 17+7=24 first, then write 49 to the right of it. The 4 in 49 however needs to be carried to the first number giving us 289.


This is also a good one for when a=2 but I would brobably not use it for a=3 even though you could. For a=2 the expression says:

Add the last digit to the whole number and _double_ it. Write the square of the last digit to the right of it (and remember to carry if it goes above 9). So 26^2 means that we add 6 to 26 to get 32 which we double to get 64. 36 (square of 6) is written to the right of it but the 3 is carried to get 676.



The original expression can be written differently if we only go for number ending in 5.

A^2 = (10a + 5)^2 = 100a(a + 1) + 25

which simply means: Take the left digit times a number one higher than itself and write this down. Write 25 to the right of it. Never any special cases since 25 takes up the two rightmost digits and the a(a+1) is multipled by 100 to never interfere.

Example: 35^2 means that we take 3*4 to get 12. Write 25 to the right of it for the result 1225.

65^2 means 6*7 = 42 and you write 25 to the right of it to get 4225.



Yet another way to reshape the expression is to work around the number 50:

A^2 = (50 + d)^2 = 100(25 + d) + d^2

where d doesn't have to mean a digit, but is the distance from 50, which may also be negative. The expression means: see what distance from 50 the number is (might be negative) and add it to 25. Write the square of the distance to the right of it to fill the two rightmost positions. You see that there are no special cases as long as d^2 < 100 so this strategy is very easy for numbers from 41 to 59.

Example: 56^2 means that we are 6 from 50. Add 6 to 25 to get 31 and write 6^2=36 to the right of it and the result is 3136.

For 48^2 we have the distance -2 from 50 which we add to 25 to get 23. Put 2^2 to the right of it (need to fill two positions) to get 2304.



For numbers around 100 you can look at the first expression again, which turns into

A^2 = (100 + d)^2 = 100(A + d) + d^2

which means that you add the distance (might be negative) from 100 to the original number then write the square of the distance to the right of it to fill two digits.

Example: 108^2 means that we are 8 from 100 so we add 8 to the original to get 116. Write 8^2=64 to the right of it to get 11664.

91^2 has distance -9 from 100. So we subtract 9 from 91 to get 82 and write 9^2=81 to the right of it to get 8281.


Other ideas for squaring numbers that fullfill certain criterias? I know this is going the opposite direction from the nice and general approach Scott uses.

/Tomas

You might want to check my short-cut methods for squaring two-digit numbers, particularly those ending with a 5, in MATHEMATICAL WIZARDRY - or not. Best - HARRY LORAYNE.

Does your method for squaring numbers ending in 5 differ from what I wrote above? I'm very interested in reading any more shortcuts if you have any for those and other numbers.

/Tomas

Tomas,

I've read MATHEMATICAL WIZARDRY and can answer the question you asked
Harry Lorayne.

The answer is "NO."

I don't think Mr. Lorayne read your post. If he did, his follow-up post
doesn't make any sense.

There is not a single post where he doesn't plug his own books/videos...

Hi: Yes, the method for the squaring of any 2-digit number that ends in 5 is the same. Obviously, I didn't realize it because I scanned it. I scanned before I got to that part because I DON'T KNOW ALGEBRA (I have only one year of high school), and when I saw all those =s and +s and >s and <s, I SCANNED.

So, yes, sorry, it's the same method, it's JUST THAT I TEACH IT BETTER - without the =s and +s and >s and <s - for people who are as UNEDUCATED OR AS STUPID AS I AM, or who will simply scan when they see all those symbols, and just won't learn it. So, again, sorry - it's the same. The DIFFERENCE is that people (most of them, anyway) LEARN it when I teach it. (Incidentally, I didn't, don't, understand the rest of that "algebraic" post, so I don't know if any of it is similar to what I teach (in "plain" language) in MATHEMATICAL WIZARDRY. (Oops; so sorry, Nir Dahan; please, please, forgive.) It may very well be.

Nir Dahan: Thank you so much for your insightful post - and I'll keep on doing it until YOU BUY AT LEAST ONE!!! Or, I'll keep doing it until YOU plug one for me! (Incidentally, you are, of course, WRONG.)

Best - HARRY LORAYNE.

Harry - I got several of your books and think highly of them - I just think it is not the place to plug them all over the place.

best,

Nir

p.s. I still don't have mathematical wizardry though...

Nir: You are, of course, entitled to your opinion, as am I. It just bothers me a bit when I see a post like your original, that has absolutely nothing to do with the question at hand, doesn't forward the discussion at all, and makes a flat, all-out statement that simply is not so. Forgive me, but it falls perfectly into the "who asked you" category.

I'm sure you don't want someone (I certainly won't do it) to go over all my posts over the years and point out the "single posts" that do not mention a book/DVD of mine. In other words, your flat statement is untrue. And when I do mention one or the other, it's usually to try to help someone. Believe me, I don't have to plug my stuff - it sells much more than most. So, to repeat, you may think I "plug them all over the place;" so many others thank me for mentioning them, telling people about them. So, Nir, I'll keep doing what I think is right and you can go on complaining about it.

I am, however, pleased that you think highly of my books. And, who knows?, if I hadn't mentioned MATHEMATICAL WIZARDRY (oops!) you may not have known about it - and YOU plugged it here, not me. Auf wiedersehn, HL.

Harry,

let's close this issue here.

Nir

p.s. I am originally from Israel, so SHALOM would be more fitting... ;-)

Sure, Nir. Since you started the issue, you apparently have the right to close it when you think its time. I accept your apology (hah!). Shalom, zahr mir gesindt, etc., whatever fits the bill. HL

I think Harry Lorayne has every right to mention his books in this thread. He mentions it as a refernece, no one has to buy it ( them ).

I was about to pass this thread by as I was lost on all the math symbols. Then I saw Harry's beautiful post and I smiled and read the whole thread.

Thank you Harry!!

I want to make a small addition to this topic. It is not the non plus ultra but nice and not so well known.

This shortcut works always, when the front numbers are the same and the last digits totals to 10! (1+9 or 2+8 or 3 + 7 or 4+6 or as you already know 5+5)

Example:

32 x 38 = ?

3 x (3+1) = 12 and behind this write 2 x 8 = 16 , so it is 1 2 1 6

other example:

114 x 116 = ?

11 x 12 = 132 and behind this 24 (because of 4 x 6) so: 1 3 2 2 4


last example:

71 x 79 = ?

7 x 8 = 56 and behind this: 1 x 9 = 9 - so: 5609

I hope somebody like it!

Angelo

Very cool, Angelo. Thanks!

(10a + b)(10a + 10-b) = 100a(a+1) + b(10-b)

I'd never have spotted that.

/Tomas