Here is the equation:

I was very bored today, and messed around with my calculator until I found this. Completely uninteresting huh :D ?Code:`high * high - high = low * low + low`

where high and low are any real number, and high = low + 1

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- 11-03-2003LurkerWow I found a useless equation!!
Here is the equation:

Code:`high * high - high = low * low + low`

where high and low are any real number, and high = low + 1

- 11-03-2003joshdick
Proof:

Code:`Given: x = y + 1. Now, prove that x^2 - x = y^2 + y.`

First, square both sides: x^2 = (y + 1)^2

Expand: x^2 = y^2 + 2y + 1

Subtract x from both sides: x^2 - x = y^2 + 2y + 1 - x

We were given that x = y + 1, so substitute on the right side of the equation: x^2 - x = y^2 + 2y + 1 - (y + 1)

Simplify: x^2 - x = y^2 + y

- 11-03-2003Lurker
Well at least its one of the first equations that wasn't burned to the ground on these forums :D .

- 11-03-2003joshdick
And now because I'm taking a course called Techniques of Math Proof, here's another proof. This one uses case analysis:

Code:`x^2 - x ?= y^2 + y`

Factor: x (x - 1) ?= y (y+1)

Substitute: x (x - 1) ?= xy

Case 1: x != 0

Divide both sides by x: x - 1 ?= y

Add 1 to both sides: x = y + 1

Case 2: x = 0

Given that x = y + 1 and x = 0, y = -1.

x^2 - x ?= y^2 + y

Now, just plug in x = 0 and y = -1:

0 = (-1)^2 + -1

0 = 1-1

- 11-03-2003joshdickQuote:

*Originally posted by Lurker*

**Well at least its one of the first equations that wasn't burned to the ground on these forums :D .**

- 11-03-2003XSquared
For any x, y and a, where they are all elements of R, and y = x - a, x^2 - ax = y^2 + ay.

Code:`x^2 - ax = (x - a)^2 + a(x-a)`

x^2 - ax = x^2 - 2ax + a^2 - a^2 + ax

x^2 - ax = x^2 - ax

- 11-03-2003akirakun
a lot of discoveries in the world of mathematics are useless, so don't feel too bad. ;)

- 11-04-2003major_small
look at it this way: you can use it as a really long way to find out if there are two consecutive numbers... or to prove that high-low=1, if high is low+1 and they're both real numbers :D

- 11-04-2003LurkerQuote:

*Originally posted by major_small*

**look at it this way: you can use it as a really long way to find out if there are two consecutive numbers... or to prove that high-low=1, if high is low+1 and they're both real numbers :D**