Cool math shortcuts

[Morgan O'Mally]

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While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

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square 37 for me using your shortcut

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Was waiting for this.

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It still works and serves its purpose of making it such easier to do in your head. Normally if I had to do this in my head I'd do 900 + 210 + 259, but Homer's way is much faster and simpler.

GoT

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My boy is Wicked Smart!

[BradyC]

The only one I remember you probably already know... Quick way to multiply 11 x any two-digit number... just add the two digits together and put the remainder in the middle. Ex: 11 x 23... 2 + 3 = 5 put the five in the middle 253... and if the middle number is two digits just carry it over to the left number like 11 x 58... 5 + 8 = 13 put the 3 in the middle and carry the one over the 5 which gives you 638. I know it's basic but I am noobish.

[Josh W]

I had a math prof. in college who was faster at arithmetic than I was. He's the only person I've ever met who was. He's published books, had infomercials, and calls himself the "Mathemagician". Seeing him mulitply 5-digit number is a 20 seconds is wicked.

And humbling.

Anyways, Homer, your trick was one I figured out for myself in 7th grade. In 6th grade, I was really dumb. See, I sorta figured out then that:

1^3 + 2^3 + 3^3 + ....N^3 = (1+2+3+...n)^2

But I never tried to prove it in 6th grade. I just kept making n one increment larger, and seeing if it still held true. I got up to like 73, then decided it must be true for all numbers. I had a buddy of mine, Nick Harris, do one half of the equation, and I'd do the other half.

The 'trick' you mention is kinda funny, because I've always had n^2 memorized up to n = 50. So it was always just the opposite for me. If I wanted to multiply 41*53, I'd square 47 then subtract 36 for 2173. (although, given those two numbers now, I'd go (53*40) + 53....).

I thought if I could ever memorize the squares up to 100^2, I'd be able to multiply any two digit numbers within maybe 4 seconds.

But then, I made an even greater discovery.

Women.

(they are even tougher to figure out...if you want a real challenge...)

Josh

[Jeff W]

Simple one, but you can use the thinking to devise other tricks.

Divide any number by 9:

1/9=.1111_
2/9=.2222_
3/9=.3333_
...
9/9=.9999_=1(They are equivalent)
10/9=9/9+1/9=1.1111_
11/9=9/9+2/9...
...

Not sure how useful it is. My physics advisor is the U.S. physics olympiad coach and knows a ton of these tricks. I will ask him to recommend a book or something on this.

[JaBlue]

kind of interesting but i utilize the distributive property much faster in my head when I'm doing these kind of calculations.

[gumpzilla]

There's a segment in one of the Feynman books - I think it's the Los Alamos from Below section of Surely You're Joking - that talks about Feynman trying to compete with Hans Bethe at speedy mental arithmetic. A technique very similar to this is mentioned, I believe for squaring numbers near 50.

One thing that strikes me is that fast approximate multiplication might have been somewhat easier in the past when people actually used logarithm tables.

[DrPublo]

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While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

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square 37 for me using your shortcut

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37^2 = 40*34 + 9
= 1360+9 = 1369

Right?

The Doc

[incognito]

Funny, I was just going through my old Netscape bookmarks file yesterday and found this:

http://www.math.hmc.edu/funfacts/allfacts.shtml

There are a few tricks of the type you're looking for in there.

[Rotating Rabbit]

Interesting, I'd do it 900+210+210+49. I'm curious that you sum the last two already.

[partygirluk]

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