Dr Matrix's math problem

Factual Questions
1

Dr. Matrix put this is a thread in the Pit, about abolishing math (apparently “proving” 1=2).

Let a = b
a * b = b * b – multiply by b
a * b - a * a = b * b - a * a – subtract a * a
a(b - a) = (b + a)(b - a) – factor
a = b + a – divide by common factor (b - a)
a = a + a – substitution
a = 2a – combine
1 = 2 – divide by common factor a

The problem is I can’t figure out what’s wrong with this! Which, is very irritating. What am I missing?! Geez, I can estimate values very accurately, but I hate these problems.

2

Divison by zero. Since a = b, a - b =0, therefore dividing by (a - b) is dividing by zero, and thus undefined.

3

In slightly more detail:
From the fact that
ab = ac
we can, if a is not 0, deduce that b = c.
However, the statement
0b = 0c (1)
is true for every b and c.
What Dr Matrix did was to prove an equation of the form (1) and infer ( fallaciously) that b = c.

4

Ah. Cute.

Thanks a lot, this one was bugging me since it didn’t seem to be a bait-and-switch problem and it looked like it was following “the rules” properly.

5

And in this line, we prove that 0 = 0, since if a = b, then

ab - aa = bb - aa

=> aa - aa = aa -aa =0

Hence proving why Dr Matrix’s arguments are fallacious from the get-go.