Question

calculate (-1+i)/(1+i) in two different ways. first, make the denominator into a real number. second, convert both numbers to polar form and use the formula for division of numbers in polar form.

Answer 1

So, do you know about complex conjugates?

Answer 2

If you don't have any notes about converting to polar coordinates, this is a good start:
http://www.chemistrylearning.com/geometrical-to-polar-conversion/
it's pretty easy

Answer 3

since it's so easy why don't I set up the first part:\[\frac{-1+i}{\ \ 1+i}\ \ \frac{1-i}{\ 1-i} = \]

Answer 4

i think i know about the basics but i really can't understand what the question wants...

Answer 5

I'll walk you through it. Ready?

Answer 6

yeah...thx

Answer 7

Can you calculate the numerator and denominator above?

Answer 8

what do you mean by calculate?!calculate what?!

Answer 9

(1 + i) (1 - i) = ?

Answer 10

is it 2?

Answer 11

Yes! (1)(1) - i^2 = 2
How about the numerator?

Answer 12

(-1)(1) - i^2 = ?

Answer 13

is it 2i?

Answer 14

what is - i^2?

Answer 15

1

Answer 16

So -1 + 1 = 0. Now we have 0/1 = 0. Okay?

Answer 17

oh ... my mistake ... right

Answer 18

For a number a + bi, the complex conjugate is a - bi, and the product is a^2 + b^2. So multiplying by the complex conjugate of the denominator makes the denominator a real number, not complex.

Answer 19

To convert to polar form, you need to get r and theta. Do you know how to do that?

Answer 20

oh...got it...yeah i know about them...

Answer 21

We have -1 + i and 1 + i. What is r? It's the same for both.

Answer 22

square root of 2?

Answer 23

Yes.

Answer 24

Do you know the formula for theta?

Answer 25

tan^-1 (b/a)

Answer 26

So what is theta for 1 + i?

Answer 27

pi/4

Answer 28

and for -1+i is 3pi/4?

Answer 29

Yes.
Last step.

Answer 30

To divide two complex numbers we divide r for each and subtract theta

Answer 31

oh thanksss...got it all...thx...:)

Answer 32

I get r = 1, theta = pi,
@TuringTest have I done something wrong? Please help.

Answer 33

my theta is pi/2...3pi/4 - pi/4 = pi/2

Answer 34

theta 1 is pi/4
theta 2 is -3pi/4
I get pi

Answer 35

So I get x =1, y = 0
But we calculated the quotient is equal to 0 above.

Answer 36

Oh I switched thetas.

Answer 37

What do you do when converting back to x,y coord and theta is pi/2? tan is undefined.

Answer 38

@eliassaab Help?

Answer 39

Well, come back in a while and maybe they will have seen the pings. I'll see if I can figure it out too.

Answer 40

sure ... thx ... :)

Answer 41

Ah.
z=r cos(theta) +i r sin(theta)

Answer 42

We have 0 for the first term.

Answer 43

And i for the second.

Answer 44

But it still doesn't match.

Answer 45

I know what I did. In multiplying the numerator
(-1 + i)(1 - i)
I did it wrong and forgot an imaginary part.

Answer 46

am going through it again...mayb i've calculated something wrong!!!

Answer 47

We should get 2i/2 = 1.

Answer 48

Whew!

Answer 49

I mean, equal i.

Answer 50

Sound right?

Answer 51

yeah...totally...got it...

Answer 52

Thanks for hanging in there.

Answer 53

thanks...u helped me alot...:)

Answer 54

yw

Answer 55

\[
\frac{-1+i}{1+i}=\frac{(-1+i) (-1+i)}{(1+i)
(-1+i)}=\frac{i^2+1-2 i}{-1+i^2}=\frac{-2
i}{-2}=i\\
\frac{-1+i}{1+i}=\frac{\sqrt{2}\, e^{\frac{i 3 \pi
}{4}}}{\sqrt{2}\, e^{\frac{i \pi
}{4}}}=e^{\frac{i \pi }{2}}=i

\]

Answer 56

Thank you.

Answer 57

yw