Did you know that 1 = 2?

Mar 2, 2021 at 4:57am
This will blow your mind, well, at least it blew my mind. Some of you other math nerds will probably be able to blow down my card house, but until someone does...

Anyway, one does equal two, and I can prove it.

So, without any further ado:

         a = b
        a2 = ab
   a2 + a2 = a2 + ab
 2a2 - 2ab = a2 + ab -2ab
 2a2 - 2ab = a2 - ab
2(a2 - ab) = 1(a2 - ab)
         2 = 1


Well, there's a catch in it, if you can find it, you're doing better than I did (I had to ask my pre-calculus professor)!

Luck,
max
Last edited on Mar 2, 2021 at 5:00am
Mar 2, 2021 at 5:53am
if a = b, then a^2 - ab = 0

Sorry to steal your excitement but this is yet another example of the common division by zero puzzler.
Mar 2, 2021 at 2:46pm
a^2 + a^2 = a^2 + ab
is not necessarily equivalent to
2a^2 - 2ab = a^2 + ab - 2ab
You added a^2 - 2ab on the left side but added only -2ab on the right side. The above two expressions are only equivalent if a^2 - 2ab = -2ab
a^2 - 2ab = -2ab
a^2 = 0
a = 0

Since a = b = 0, the last step in the argument, where you divided both sides by a^2 - ab, is invalid, since a^2 - ab = 0 and it would be a division by zero. More correctly,
2(a^2 - ab) = a^2 - ab
2 * 0 = 0
0 = 0
Mar 2, 2021 at 5:54pm
Darn, you guys got it already. I was kind of hoping someone would post and ask what the heck I meant by 1 = 2. But oh well.

Yes, @helios is right, it's a division by zero, so 1 does not actually equal 2. (If it did, we would probably still be living in the Stone Age).
Mar 3, 2021 at 10:15am
a^2 + a^2 = a^2 + ab
is not necessarily equivalent to
2a^2 - 2ab = a^2 + ab - 2ab

Really?
Mar 3, 2021 at 12:48pm
Oops. My bad, I must have misread. The step where you add -2ab on both sides is not erroneous.
The more proper correction would be

2(a^2 - ab)/(a^2 - ab) = 1(a^2 - ab)/(a^2 - ab)
implies that a^2 - ab != 0
a^2 - ab != 0
a^2 != ab
aa != aa
0 != 0
Since 0 != 0 is not true, it's not true that 2 = 1.
Mar 3, 2021 at 1:12pm
All good but it's even simpler than that because the result of dividing any number by 0 has no meaning, it's undefined.
https://en.wikipedia.org/wiki/Division_by_zero

eg
n = 1234
let n/0 = p => n = p.0 = 0
so n = 0 which contradicts n = 1234
Mar 3, 2021 at 5:39pm
Wow. My math prof only said that it was division by zero, he didn't explain why. Thanks, guys!
Mar 3, 2021 at 6:26pm
Any division is only valid if the divisor is non-zero. Every time you cancel out some multiplication in an equation you're implying that something is not zero. Forgetting to follow up on those implications is how you arrive at contractions such as the one above, or how you miss solutions to a system of equations. It's quite common to do something like
x^2 = x
x = 1
And forget that x = 0 is also a solution. I'm simplifying, of course; not every example is this obvious.
Mar 3, 2021 at 10:14pm
x = x^2
x - x^2 = 0 ( subtract overcomes possibility that x could be 0 )
x(x - 1) = 0
So, x = 0 or x -1 = 0 => x = 1
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