I saw something on YouTube today that was interesting. To relieve you of having to watch the extremely slow-moving video, I will just provide the proof right here.
First assume that
i = √(-1)
Next, assume that
√(ab) = √(a) * √(b)
Now, we will derive the proof:
1 + 1 = 1 + √(1)
= 1 + √[(-1)(-1)]
= 1 + √(-1)√(-1) = 1 + (√(-1))²
= 1 + i²
= 1 + (-1)
= 1 - 1
= 0
Does that not blow your mind? Now, obviously this is crazy. So let’s take a look at how the proof is flawed.
The assumption that √(ab) = √(a) * √(b)
is only true if at least one of a
and b
are positive numbers (including zero). If both a
and b
are negative then √(ab) = -√(a) * √(b)
. Hence,
1 + 1 = 1 + √(1)
= 1 + √[(-1)(-1)]
= 1 - √(-1)√
(-1) = 1 - (
√
(-1))²
= 1 - i²
= 1 - (-1)
= 2
And the world makes sense again. Now I feel like my feet are back on the ground!
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