Complex Numbers

[adambinch]

Can you please help me with the following:

(1) How can I express, z = 7e^i(pie)/3 in the form of the compex number z = 4 + 3i

(2) How can I express....

x-1 / 2+j + y-1 / -1+2j
...In the form a +jb, where x, y, a and b are real.

Thanks

 

[tkhunny]

These are very difficult with your present notaion.

Do you mean \(\displaystyle \frac{7}{3}\;e^{i\;\pi}\) or \(\displaystyle 7e^{i\frac{\pi}{3}}\) or something else?

 

[soroban]

Hello, Adam!

\(\displaystyle \text{(2) Express: }\;\frac{x-1}{2+j} + \frac{y-1}{-1+2j}\:\text{ in the form }\,a +bj\)

\(\displaystyle \text{We have: }\;\frac{x-1}{2+j} - \frac{y-1}{1-2j}\)

Rationalize the fractions:

. . \(\displaystyle \frac{x-1}{2+j}\cdot\frac{2-j}{2-j} - \frac{y-1}{1-2j}\cdot\frac{1+2j}{1+2j}\)

. . \(\displaystyle =\;\frac{(x-1)(2-j)}{4-j^2} - \frac{(y-1)(1+2j)}{1-4j^2}\)

. . \(\displaystyle = \;\frac{2x - xj - 2 + j}{4+1} \;-\; \frac{y - 2yj - 1 - 2j}{1 + 4}\)

. . \(\displaystyle =\;\frac{(2x-xj - 2 + j) - (y - 2yj - 1 - 2j)}{5}\)

. . \(\displaystyle = \;\frac{2x - xj - 2 + j - y + 2yj + 1 - 2j}{5}\)

. . \(\displaystyle =\;\frac{2x - y - 1 - xj + 2yj + 3j}{5}\)

. . \(\displaystyle = \;\frac{(2x - y - 1) + (-x + 2y + 3)j}{5}\)

. . \(\displaystyle =\; \underbrace{\left(\frac{2x-y-1}{5}\right)}_a \;+ \;\underbrace{\left(\frac{-x+2y+3}{5}\right)}_bj\)