OT: MATH, or, I'm FREEEAKING OUT!

[AirElemental]

Ok,
so I was a demented little kid.
I hated the multiplication tables.
I mean, They just seemed redundant to me.
I decided I'd figure out a way around em.
see, 4x6 seemed an awful lot like 5x5 to me.
I quickly learned the connection, and present it to you below:
(Note: this method was "Discovered" in elementary school, but
I didnt develop the knowledge to explain it till much later.)


4x6 = 24
5x5 = 25. 25-1=24, therefore 5x5-1 = 24 right? follow me?
ok now try it with 3x5. should be 4x4-1 right? good. keep going.
1x3 =2x2-1.
ok so we see a pattern. but what happens when the integers
get farther out?

1x11.
Well, first get the average, or 6 in this case.
6x6... now... 6x6 = 36. thats way off of 11. 36-11 = 25. so 6x6-25=1x11?
Close.. but.. the way it works is like this:

1....6....11
6-1=5. 11-6=5.
hmm.
ok so 6x6-5x5=11.
1x99=50x50-49x49.
or to be more specific,
X*Y=((X+Y)/2)^2 - (y-(X+Y)/2)^2.
try it. you'll be amazed.
then you'll wonder why I went through all the trouble.

300 x 400 = 350x350-50x50
-1 x -3 = (-2x-2) - (-1x-1)
1.5 x 2.5 = 2x2 - .5x.5
now that last one makes it much easier to see the answer is 3.75, huh?
cheers

 

[hong]

1729 is the smallest positive integer that can be expressed as the sum of two different cubes in two different ways (10^3 + 9^3 or 12^3 + 1^3).

 

[Heretic Apostate]

Pretty much, you've proved that X * Y = X * Y.

Here's my little obsession. Sums of the powers.

For instance, 1 + 2 + 3 + ... + n = n * (n + 1) / 2

1 ^ 2 + 2 ^ 2 + 3 ^ 2 + ... + n ^ 2 = n * (n + 1) * (2 * n + 1) / 6

1 ^ 3 + 2 ^ 3 + 3 ^ 3 + ... + n ^ 3 = [n * (n + 1) / 2] ^ 2

And so on.

I do these by hand. I establish the pattern, factor the equation, and then prove the formula for all positive integers, n.

I used to do these when I was working grave shift. It was my way of staying awake. Really upset my bosses, however. There I'd be, plugging away at these problems, without the benefit of even a calculator (and they get VERY complicated when you get to the 7th power and above). They'd yell at me for it. Meanwhile, most of the other people were off gossiping, taking long coffee breaks, or going outside to smoke several times an hour.

But, yeah, math is fun.

 

[Duncan Haldane]

I figured out something similar to this a few years ago, and that is
x^2 = (x-1)^2 + x + (x-1)

for example
5^2 = (5-1)^2 + 5 + (5-1)
5^2 = 4^2 + 5 + 4
25 = 16 + 5 + 4
25 = 25

Now, this doesn't seem very useful, but if you are working with large squares and don't have a calculator, it can be...

eg
21^2 = 20^2 + 20 + 21
21^2 = 400 + 41
21^2 = 441

also, the reverse

x^2 = (x-1)^2 - x - (x-1)

so,

19^2 = 20^2 - 20 - 19
19^2 = 400 - 39
19^2 = 361

for me it means I can calulate the square of a number pretty quickly, even if it is a few away from a value I know.

Duncan

 

[Moe Ronalds]

*brain explodes*

 

[Vaxalon]

x^2 = (x-1)^2 + x + (x-1)

x^2=x^2-2x+2x-1+1 .... add 2x, subtract 2x, add 1, subtract 1

x^2=(x^2-2x+1)+2x-1 .... regroup terms

x^2=(x-1)^2+2x-1 ..........replace square of x-1

x^2=(x-1)^2+x+(x-1) ........regroup again

It's really quite trivial.

Originally posted by AirElemental
4x6 = 24
5x5 = 25. 25-1=24, therefore 5x5-1 = 24 right? follow me?
ok now try it with 3x5. should be 4x4-1 right? good. keep going.
1x3 =2x2-1.
ok so we see a pattern.

So what you're saying is as follows:

x^2-1=(x-1)(x+1)

This, also, is a trivial identity that is taught in high school algebra.

Originally posted by AirElemental
1x11.
Well, first get the average, or 6 in this case.
6x6... now... 6x6 = 36. thats way off of 11. 36-11 = 25. so 6x6-25=1x11?
Close.. but.. the way it works is like this:

1....6....11
6-1=5. 11-6=5.
hmm.
ok so 6x6-5x5=11.

What you seem to be saying is as follows:

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

Again, this is trivial. If you multiply out (x-n)(x+n) you get x^2-xn+xn-n^2. The xn terms cancel out.

So in the end, "Well, yes, of course. Didn't everyone learn this?"

 

[Simon Magalis]

Wow, you guys are doing sooo much to improve the image of gamers. LOL

 

[Crothian]

Visual music.

 

[hong]

Wow, you guys are doing sooo much to improve the image of gamers. LOL

Nah. This is nothing compared to the materials science and wound ballistics threads that crop up now and then in GURPS discussions....

 

[Storminator]

Again, this is trivial.

So in the end, "Well, yes, of course. Didn't everyone learn this?"

Algebraically trivial does not mean computationally worthless. In fact, much of trigonometry is based on replacing difficult to compute expressions with "simplified' ones.

And this will probably hurt your brain Heretic, but you can go to

http://mathworld.wolfram.com/PowerSum.html

and see general solutions to all those power summations.

PS