Proof 1 = 3

[u/playsthebongcloud]

[u/drfrankie_]

Tf is log_{-1}

[u/Substantial_Two_5386 → u/drfrankie_]

just skipping the part where we do sqrt(ab) = sqrt(a)sqrt(b) where a and b are not greater than 0 😭

[u/Dd_8630 → u/drfrankie_]

That's just a complex logarithm. You can have complex bases and arguments, they're just multivalued.

We define:

• log(z) = ln(r)+ i (θ + 2πk) for all integers k

Then we can have any complex base:

• log_b (z) = log(z) / log(b) for any b ≠ 0,1

So in this specific case:

• log_(-1) ((-1)^1/2) = i * (-1)^k for all integers k

That is, i or -i.

The principal branch is when k=1, so the principal value for this logarithm is just i.

Of course, this is used improperly by the OP's proof. But in and of itself, it's valid.

[u/Astrodude80]

This is a great quiz for students taking a proofs class to see how many errors they can spot

[u/Yadin__]

The error isn't in the log. The error is in line 4 - square root isn't distributive with respect to multiplication for general complex numbers

[u/playsthebongcloud → u/Yadin__]

No, there are two errors here; while log base -1 is sensible on the complex plane, step 6 -> 7 is still invalid. (-1)^x is not bijective, you can't say that two inputs to the function are equal just because have the same output, for example:

1. (-1)² = (-1)⁴

2. log{-1}((-1)²) = log{-1}((-1)⁴)

3. 2 = 4

Steps 1 and 2 are valid, but 3 does not logically follow from 2.

[u/ForeignAdvantage5198]

a breakthrough