How can be 1 is equal to 2?

Question

It may be a silly question. But I don't know it. So I'm questioning. Recently I've got a proof that proves 1=2.

Is there any fault in the proof? If so then what is the fault??

Answer 1

The problem is here:

$a^2=b^2 \Rightarrow a=b \underbrace{\vee}_{\text{or}} a=-b$.

In this case, only the second one counts:

$(1-\frac32)^2=(2-\frac32)^2 \Rightarrow \require{cancel} \cancel{1-\frac32=2-\frac32} \vee 1-\frac32=-2+\frac32$, but the first one gives a contradiction, so it doesn't count.

Answer 2

$$\left(1-\frac{3}{2}\right)^2 =\left(2-\frac{3}{2}\right)^2 \implies $$

$$ \implies \left|1-\frac{3}{2}\right| =\left|2-\frac{3}{2}\right|$$

because

$$\sqrt{x^2} = |x| $$