Mathematical Fallacy: Prove 0 = 2

Yesterday, my roommate and I were discussing on his friend's post for a proof that 0 = 2.
Really? Well, let's check this out and examine.

0 = 2

Step 1:     0 = (100 - 100) / (100 -100)

Step 2:     0 = (10(*square) - 10(*square))  / ( (10)(10) - (10)(10) )

*means 10 squared - 10 squared

Factoring out numerator through "Difference of 2 Squares" gives us:

Step 3:     0 = ( (10 + 10)(10 - 10)) / ( (10)(10) - (10)(10) )

Step 4:     Using the distributive property into the denomator, it gives us:

                0 = ( (10 + 10)(10 - 10)) / ( 10 * (10 - 10) )

Step 5:     Thus divide out (10 - 10), gives us:

                 0= ( 10 + 10) / ( 10 )

Step 6:     0 = 20 / 10

Answer:   0 = 2


Haha...As we can see, it seems right. Where is the problem?

The problem is in Step 5, it is invalid since "division by zero" is undefined.
0/0 has no value and is this is called an indeterminate form.

If you carefully examine Step 5, it would turn out that 0 is equal to any number, which is very wrong.
Check this out using another representation of Step 5.

Equation. 1:    0 = (2 * 0) / (1 * 0)
Equation. 2:    0 = (3 * 0) / (5 * 0)

Both equation are correct but if using the invalid process in Step 5, it would be the same as:

Equation. 1:    0 = (2 )(0) / (1)(0)
             
                      Divide out 0 in the denominator

                      0 =  2/1

                      0 =  2

Equation. 2:    0 = (3)(0) / (5)(0)  

                      Divide out 0 in the denominator

                      0 =  3/5

Now, we've seen where the problem is. :)