YAMT: complex numbers question

[AStar617]

All the examples I see of complex number multiplication always already have the mutliplier/multiplicand in terms of "i", with no sqrts... I'm drawing a blank on how to make the transition...

For instance, what would I do with something like (1 - sqrt[-4])(2 + sqrt[-4]) ?

Thx in advance...

 

[CheesePoofs]

(1-2i)(1+2i)= 1-4i^2 = 1+4 = 5

 

[chuckywang]

i = sqrt(-1)

Therefore sqrt(-4) = sqrt(4)*sqrt(-1) = 2i

Use that fact and see what you get.

 

[chuckywang]

Originally posted by: CheesePoofs
(1-2i)(1+2i)= 1-4i^2 = 1+4 = 5

It's (1-2i)(2+2i)

 

[AStar617]

thx.... i realize i probably should have used a slightly different example because 4 is too easily sqrt'ed. This modified version of the first example would help me better...

(1 - sqrt[-5])(2 + sqrt[-5])

 

[CheesePoofs]

Originally posted by: chuckywang

Originally posted by: CheesePoofs
(1-2i)(1+2i)= 1-4i^2 = 1+4 = 5

It's (1-2i)(2+2i)

oh, whoops. 😱 I figured it was just the complex conjugate and would multiply out nicely.

 

[chuckywang]

Originally posted by: AStar617
thx.... i realize i probably should have used a slightly different example because 4 is too easily sqrt'ed. This modified version of the first example would help me better...

(1 - sqrt[-5])(2 + sqrt[-5])

Same type of deal:

sqrt(-5) = sqrt(-1)*sqrt(5) = i*sqrt(5)

 

[CheesePoofs]

Originally posted by: AStar617
thx.... i realize i probably should have used a slightly different example because 4 is too easily sqrt'ed. This modified version of the first example would help me better...

(1 - sqrt[-5])(2 + sqrt[-5])

turn the sprt(-5) into sqrt (-1) * sqrt (5), or i * sqrt (5).

So when you multiply it out, you get 2 - i*sqrt(5) - sqrt(25)*i^2. i^2 = -1, so the problem ends up being 2 - i*sqrt(5) + 5, or 7 - i*sqrt(5) .

 

[AStar617]

Originally posted by: CheesePoofs

Originally posted by: AStar617
thx.... i realize i probably should have used a slightly different example because 4 is too easily sqrt'ed. This modified version of the first example would help me better...

(1 - sqrt[-5])(2 + sqrt[-5])

turn the sprt(-5) into sqrt (-1) * sqrt (5), or i * sqrt (5).

So when you multiply it out, you get 2 - i*sqrt(5) - sqrt(25)*i^2. i^2 = -1, so the problem ends up being 2 - i*sqrt(5) + 5, or 7 - i*sqrt(5) .

OK I think I now know where I'm getting lost... in the sign changes. 😕 Would you mind showing the four products without skipping right to 2 - i*sqrt(5) - sqrt(25)*i^2? Something in the middle (the "b" term of the resulting standard quadratic) is getting simplified there.

EDIT: NM, I think I got it. Thanks alot guys.