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Originally Posted by OngBonga
It's fine until the 4th one.
sqrt is not a value, it's a function. You can't multiply by a function. What's ten multiplied by multiplication? But that's what you're essentially doing. You're breaking the bracket by multiplying -1 by square root.
I think. I'm not great with this sort of thing but obviously there's some hand waving nonsense going on here.
You can totes multiply functions. You can even put functions inside other functions and multiply those by other functions.
You're mixing up functions and operators, though. Multiplication is an operation. A function may not imply operations.
f(x) = 2 is a function, but no matter what input you give f, f always returns 2. No operation was performed.
As long as the output is uniquely determined by the input, it's a function.
SQRT is, as such, technically, not a function, but an operation. However, it's common in proofs and math texts to use notation that where not explicitly stated, the SQRT operator returns a positive value, and is therefor defined as a function.
Depending on how you look at the 2nd line, that's the problem or the 4th line is the problem because of the 2nd line.
The 2nd line says 1 = sqrt(1), which is, strictly speaking, not true. sqrt(1) = +/-1.
If you don't ignore the negative part of the square root, this line is simply false.
However, given the form of the proof, it is a reasonable assumption that they're using the notation in which since the negative wasn't explicitly stated, we're using the function form of the square root.
With that interpretation, we have a problem going from line 3 to line 4.
The rule applied is valid
sqrt( a*b ) = sqrt(a)*sqrt(b)
if we allow all the sqrt()'s to be + or -.
However, if we strictly interpret those sqrt()'s as positive, then that rule breaks if both numbers are negative and you don't cancel the minus signs before you break it up.
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