Theorem: 1 = 1/2:
Proof:

We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)
+…

as 1/2((1/1 – 1/3) + (1/3 – 1/5) + (1/5 – 1/7) + (1/7 – 1/9) + … ).
All terms after 1/1 cancel, so that the sum is 1/2.

We can also re-write the series as (1/1 – 2/3) + (2/3 – 3/5) + (3/5 – 4/7)
+ (4/7 – 5/9) + …

All terms after 1/1 cancel, so that the sum is 1.

Thus 1/2 = 1.

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Theorem: e=1
Proof:
2*e = f
2^(2*pi*i)e^(2*pi*i) = f^(2*pi*i)
e^(2*pi*i) = 1

Therefore:
2^(2*pi*i) = f^(2*pi*i)
2=f
Thus:
e=1

It is proven that the celebration of birthdays is healthy. Statistics show that those people who celebrate the most birthdays become the oldest. — S. den Hartog, Ph D. Thesis Universtity of Groningen.

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