Math Is Fun Forum

pban92

Member

Registered: 2006-10-12

Posts: 3

Multiplication of imaginary number with euler formula (?)

Hi,

I got this following relationship:

1) (a+ib)(cosx+isinx) = acosx - bsinx
2) (a-ib)(cosx+isinx) = acosx + bsinx
3) i(cosx+isinx) = -sinx

I wonder if there exists any formula/property to justify/prove the validity of above relationship. Any comment/idea would be very much appreciated!

Thanks

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: Multiplication of imaginary number with euler formula (?)

Hi pban,

Hate to say it, but those formulas aren't right. To take number 3 as an example, you have

[align=center]

[/align]

Simplify the left hand side:

[align=center]

[/align]

And this is true if and only if cos(x) = 0. This equation puts a constraint on x - namely, that it must be one of the values {...,-3pi/2,-pi/2,pi/2,3pi/2,...}, but this relationship is surely not true for all x.  You mentioned the euler formula, which allows you to write

[align=center]

[/align]

So if you wanted to you could write your three conditions in terms of this, but that does not make them any more or less true.

Wait .... now I see what's probably confusing you here. Your answers on the right side are all the "real parts" of the equations on the left. Real parts just mean this - any complex number can be written as F = a + ib, where a is the "real part" and b is the "imaginary part" because it is being multiplied by i. Sometimes you don't care what the imaginary part is so you can throw it away ... it depends on the context of the problem, but those equations as written aren't really true.