Pointless Mathematical Shortcuts!

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 NumbersFirst, 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^2This 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 = 2209For 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= 8464Once 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= 6889Not too bad though.Squaring Three Digit NumbersOnce 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^2For example:(359)^2= (300+59)^2= 300^2+2(3)(59)+59^2= 90000+354+3481= 93835Not 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 Relationships21^2= 20^2 + 20 + 21= 441This 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= 49So 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.

Thread moved here from the Proposed New Topics forum.

Several observations.First, before calculators and computers became ubiquitous, mental math skills were very important. In one of Feynman's autobiographical books he describes casual games of mathematical mental oneupsmanship that he and colleagues used to play in their spare time. I think Fermi might have been involved at one lunch. So you're participating in a long and noble tradition.Second, since your mind is so facile with math, perhaps naturalist isn't the best profession for you?Third, fix your iPod. There are tons of science (and probably some math, too) podcasts out there.--Percy

Proof by induction:Obviously this holds for the case x = 1.Now, suppose that it is true for x. Then we must prove that it is true for x+1. Now (x+1)² = x²+2x+1, and 2x+1 is the x+1th odd number. Hence as by hypothesis x² is the sum of the first x odd numbers, (x+1)² is the sum of the first x+1 odd numbers.