Wow I found a useless equation!!

Here is the equation:

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

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I was very bored today, and messed around with my calculator until I found this. Completely uninteresting huh :D ?

Proof:

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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

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Yup :)

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

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

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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

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Yeah, I dig proofs.

Quote:

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

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Well, there are now two proofs supporting your conjecture, so one can't argue with the truth. It still remains useless as you said.

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

Code:

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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

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

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

Quote:

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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

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:D