# Complex Numbers

• Jan 16th 2008, 10:35 PM
OzzMan
Complex Numbers
I need to find the rectangular form of i ^ (-3) + i ^ (-2)
i = imaginary number
The answer in my book says -1 + i but i do not know how they got it
• Jan 16th 2008, 10:40 PM
mr fantastic
Quote:

Originally Posted by OzzMan
I need to find the rectangular form of i ^ (-3) + i ^ (-2)
i = imaginary number
The answer in my book says -1 + i but i do not know how they got it

$i^{-3} = \frac{1}{i^3} = \frac{1}{i \times i \times i} = \frac{1}{-i}$

Multiply top and bottom by i:

$= \frac{i}{-i \times i} = \frac{i}{1} = i$.

$i^{-2} = \frac{1}{i^2} = \frac{1}{i \times i} = \frac{1}{-1} = -1$.

Therefore ......
• Jan 16th 2008, 10:43 PM
earboth
Quote:

Originally Posted by OzzMan
I need to find the rectangular form of i ^ (-3) + i ^ (-2)
i = imaginary number
The answer in my book says -1 + i but i do not know how they got it

Hello,

$i^{-3}+i^{-2}= \frac{1}{i^3} + \frac{1}{i^2}= \frac{1}{(-1) \cdot i} + \frac{1}{-1}= \frac{1}{(-1) \cdot i} \cdot \frac{i}{i} - 1 = \frac{i}{(-1) \cdot (-1)} - 1 = i-1$
• Jan 16th 2008, 11:37 PM
OzzMan
why do you multiply top and bottom by i ?
• Jan 17th 2008, 12:03 AM
OzzMan
oh is it because the conjugate of -i is i?
• Jan 17th 2008, 12:14 AM
mr fantastic
Quote:

Originally Posted by OzzMan
I need to find the rectangular form of i ^ (-3) + i ^ (-2)
[snip]

Quote:

Originally Posted by mr fantastic
[snip]
Multiply top and bottom by i:

$= \frac{i}{-i \times i} = \frac{i}{1} = i$.
[snip]

Quote:

Originally Posted by OzzMan
why do you multiply top and bottom by i ?

Quote:

Originally Posted by OzzMan
oh is it because the conjugate of -i is i?

Well, if you have to give an answer in rectangular form, then $\frac{1}{-i}$ is not really in the form you want, is it? So what would you do .....?

Returning to

Quote:

Originally Posted by OzzMan
why do you multiply top and bottom by i ?

Quote:

Originally Posted by OzzMan
oh is it because the conjugate of -i is i?

Well, yes it is. But it also happens to be the easiest way to express $\frac{1}{-i}$ in rectangular form .....