Absolute value of imaginary numbers?
Here's something that stumped my pre-calculus teacher.
Supposedly, the absolute value of an imaginary number such as 4 + 3i is found by throwing away the i and using the pythagorean theorem on it. i.e. sqrt( 4^2 + 3^2 ) = 5
So according to that formula, if you have 3 numbers:
x = 4 + 3i
y = 3 + 4i
z = 0 + 5i
| x | = | y | = | z | = 5
But if that is true, then the following should also be true, but it isn't:
Part of the reason why I think that the formula is invalid is because they ignore the i which is part of the number.
x^2 = y^2 = z^2
(4 + 3i)^2 = (3 + 4i)^2 = (5i)^2
(16 + 24i - 9) = (9 + 24i - 16) = (-25)
7 + 24i != -7 + 24i != -25
The only "proof" that I've seen is graphical, but since they are dealing with imaginary numbers, how can they know that the imaginary number line is perpendicular to the real number line, or that the points on the imaginary number line have the same spacing as real numbers or even if it has a linear progression? If we knew the properties of imaginary numbers, they wouldn't be very imaginary. ;)