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dollars equal ten cents
 
 
Theorem: 1$ = 10 cent
Proof:
We know that $1 = 100 cents
Divide both sides by 100
$ 1/100 = 100/100 cents
=> $ 1/100 = 1 cent
Take square root both side
=> squr($1/100) = squr (1 cent)
=> $ 1/10 = 1 cent
Multiply both side by 10
=> $1 = 10 cent

log negative one zero
 
 
Theorem: log(-1) = 0
Proof:
a. log[(-1)^2] = 2 * log(-1)

On the other hand:
b. log[(-1)^2] = log(1) = 0

Combining a) and b) gives:
2* log(-1) = 0
Divide both sides by 2:
log(-1) = 0

n equals n plus one
 
 
Theorem: n=n+1

Proof:
(n+1)^2 = n^2 + 2*n + 1

Bring 2n+1 to the left:
(n+1)^2 - (2n+1) = n^2

Substract n(2n+1) from both sides and factoring, we have:
(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)

Adding 1/4(2n+1)^2 to both sides yields:
(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2

This may be written:
[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

Taking the square roots of both sides:
(n+1) - 1/2(2n+1) = n - 1/2(2n+1)

Add 1/2(2n+1) to both sides:
n+1 = n

one is negative one
 
 
Theorem: 1 = -1
Proof:
1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = 1^ = -1

Also one can disprove the axiom that things equal to the same thing are equal to each other.

1 = sqrt(1)
-1 = sqrt(1)
Therefore 1 = -1

As an alternative method for solving:

Theorem: 1 = -1
Proof:
x=1
x^2=x
x^2-1=x-1
(x+1)(x-1)=(x-1)
(x+1)=(x-1)/(x-1)
x+1=1
x=0
0=1
=> 0/0=1/1=1


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