This is a proof that 2 = 1. What's wrong with it?

Question

why is this wrong?

a = x [true for some a's and x's]
a+a = a+x [add a to both sides]
2a = a+x [a+a = 2a]
2a-2x = a+x-2x [subtract 2x from both sides]
2(a-x) = a+x-2x [2a-2x = 2(a-x)]
2(a-x) = a-x [x-2x = -x]
2 = 1 [divide both sides by a-x]

Answer 1

Yeah, those guys got it.

You know I was trying to remember this question for the longest while. Thanks.

Answer 2

a = x [true for some a's and x's]
a+a = a+x [add a to both sides]
2a = a+x [a+a = 2a]
2a-2x = a+x-2x [subtract 2x from both sides]
2(a-x) = a+x-2x [2a-2x = 2(a-x)]
2(a-x) = a-x [x-2x = -x]
2 = 1 [divide both sides by a-x]

First, the proof, as I remember it, is this.
A a = x
B aa = ax (multiply both sides by a)
C aa - xx = ax - xx (subtract xx from both sides)
D aa - xx = (a + x)(a - x) (distributive property)
E ax - xx = x(a - x) (distributivity again)
F (a+x)(a-x) = x(a-x) Substitution D&E back into C
G a+x = x property of multiplicative inverse
H x+x = x Substitution A into G
I 2x = x Definition
J 2 = 1 property of multiplicative inverse
K 1 = 0 Adding -1 to both sides

Either we have revolutionized mathematics...
or there's an error.

In your final step and in my step G, we are multiplying by the multiplicative inverse of a - x. But a = x. So a - x = 0. 0 has no multiplicative inverse, i.e., there is no number by which we can multiply 0 to get 1.

It's like trying to convince somebody that 1x0 = 2x0 so 1 = 2

Answer 3

2(a-x)=a+x here is your mistake
lets say a is actually 3.. and therefore x would be 3 as well since a=x
so wouldn't you preform the obvious next step 2(3-3)=3-3
3-3=0
so you would have 2(0)=0
2*0 is 0 so you are just proving that 0=0 and not 2=1

Answer 4

from the equation a=x, (a-x) would therefore be equal to zero (0). But at your 6th line of equations, you divided both sides by (a-x) which is zero. dividing by zero is not possible. it is a mathematical error. that's why the proof that 2=1, isn't a proof at all.. :)

Answer 5

Starting at step 4 you have zero on both sides. a-x=zero. You can't divide zero by zero and get 2 or 1.

Answer 6

The second last step in incorrect. You cannot cancel 0 from both sides. a-x=0

Answer 7

Because by your terms: a = x so that when you divide by a-x, which equals 0, near the end of your work, the result become undefined and specious.

Answer 8

the mistake is when you divide by (a-x), because a-x = 0 (a=x, so a-x =0), and you can not divide by 0.