On Sun, 8 Jun 1997, William Chops Westfield wrote:

> I'm not convinced.

>                           .. anyone who is actually a mathematician
> would look at you funny for even suggesting that there was anything
> wrong with "i = sqrt(-1)".

Oh but they will look funny, these mathematicians. Almost as funny as when
you ask them if they can prove that they can prove anything :-)

> And I assume that we are talking about real-world situations.

Nope, we're talking about mathematics.

> real world formulas ..

Here's another example of where imaginary numbers are really "useful":
  cos(3x) + i*sin(3x) =
  ( cos(x) + i*sin(x) )^3 =
  cos(x)^3 + 3*cos(x)^2*i*sin(x) + 3*cos(x)*i^2*sin(x)^2 + i*sin(x)^3 =
  cos(x)^3 - 3*cos(x)*sin(x)^2 + i*( 3*cos(x)^2*sin(x) - sin(x)^3 )
which horrible formula yields both:
  cos(3x) = cos(x)^3 - 3*cos(x)*sin(x)^2
  sin(3x) = 3*cos(x)^2*sin(x) - sin(x)^3
Interestingly, the same method applies to sin(4x), sin(5x), cos(999x), etc.
Without complex numbers, these deductions would be really horrible to do.

Let's continue this thread as an imagainary one.


Joost