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Author Topic:   Pointless Mathematical Shortcuts!
JustinC
Member (Idle past 4844 days)
Posts: 624
From: Pittsburgh, PA, USA
Joined: 07-21-2003


Message 1 of 2 (512630)
06-19-2009 3:06 PM


For the last two years I've had a job as a naturalist (more accurately, a hiking guide) in the New England area. My job requires me to drive around alot between the White Mountains, the Adirondacks, and the Poconos area.
During these 4-5 hour jaunts I figured out that the best way to keep my mind occupied (considering my iPod broke and I abhor the radio)is to find random numbers while driving and perform mathematical operations on them. While doing this I've found some shortcuts that I'd like to share. I do not claim these are original or profound.
I've also been inspired by a video on TED by Arthur Benjamin where he does lightning fast calculations. I wanted to see if I could find a way to perform calculations faster.
Squaring Two Digit Numbers
First, memorize the first 25. You should already know most of them so filling in the holes isn't much of a challenge.
Once you know the first 25 it is pretted easy to find the squares of 25 through 75 using the formulas:
(50 + X)^2 = 2500 + 2(50)X + X^2= 2500 + 100X + X^2, and
(50 - X)^2 = 2500 - 2(50)X + X^2= 2500 - 100X + X^2
This makes it extremely easy for the 40's and 50s. For example,
57^2=(50+7)^2=2500+100(7)+49= 2500+700+49= 3249
47^2=(50-3)^2=2500-100(3)+9 = 2500-300+9 = 2209
For the rest of the numbers between 25 and 75 its a bit harder because the hundreds place changes once you add X^2 but I still think it makes it pretty easy. For example,
(71)^2=(50+21)^2= 2500+100(21)+21^2= 2500+2100+441= 5041
[edit] Almost forgot 75-99!
I simply use the formula:
(100-x)^2= 10,000-2(100)x+x^2= 10,000-200x+x^2, for example:
92^2= (100-8)^2= 10,000-1600+64= 8464
Once again, it gets a little more difficult when x is greater than 10 because of the hundredths place changes when x^2 is added. For example:
83^2= (100-17)^2= 10,000-3400+ 289= 6600+289= 6889
Not too bad though.
Squaring Three Digit Numbers
Once you get pretty comfortable with the latter method, three digit numbers become easier also. You simply use the formula:
(x+y)^2= x^2+2ab+y^2
For example:
(359)^2= (300+59)^2= 300^2+2(3)(59)+59^2= 90000+354+3481= 93835
Not the easiest calcuation, but I can usually get it within 20 seconds after a lot of practice. It's amazing that Mr. Benjamin can do it in a split second. I don't think any amount of practice can my time down to that.
Other Curious Square Relationships
21^2= 20^2 + 20 + 21= 441
This is simply a result of the formula
(x+1)^2= x^2+ 2x+1= x^2+ x + (x+1)
So if you know the square above or below the one you are trying to find, it turns into a simple addition problem.
And, I'm sure this is some Fundamental Law of Arithmetic or something, but while driving I realized the cool relationship that x^2 is the sum of the first x odd numbers. For example:
7^2= 1+3+5+7+9+11+13= 49
So I just figured I'd share these "discoveries" here and I hope that you guys can share some of your mathematical shortcuts with me as well.
Coffee House?
Edited by JustinC, : No reason given.

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Message 2 of 2 (512649)
06-19-2009 4:47 PM


Thread copied to the Pointless Mathematical Shortcuts! thread in the Coffee House forum, this copy of the thread has been closed.

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