Complex numbers Problem

kelxx

Represent the following in polar form:

1) (√3/2) - (3/2)i


And for this part just wondering would you square it out 1st and then multiply by the conjugate or multipy by the conjugate and then square it out??
2) ((6+8i)/(4-3i))*2

Thanks!

Michael Collins

kelxx wrote: »

Represent the following in polar form:

1) (√3/2) - (3/2)i

Imagine a line segment from the origin to that complex number in the complex plane.

The polar form needs two things

1) the length of that line segment
2) the anti-clockwise angle from the positive real axis to the same line segment.

For the length, just use the formula for magnitude of a complex number.

For the angle, draw an Argand* diagram with a line from the origin to that complex number and mark the angle from the real axis. Then use your diagram and [latex] \tan^{-1} [/latex] (inverse tan) to work out what the angle is.

And for this part just wondering would you square it out 1st and then multiply by the conjugate or multipy by the conjugate and then square it out??
2) ((6+8i)/(4-3i))*2

Thanks!

Either way will work, but getting it into the standard form of [latex] a + b\cdot i [/latex] first might be a bit less work i.e. do the complex conjugate thing first.

*An Argand diagram is just a diagram in the complex plane, i.e. just a plane with the real part on the x-axis and the imaginary part on the y-axis.

If you've any questions, just ask.

bnt

Before you get to the complex conjugate stage, it can help (I think), to treat i as if it were a variable like x, and use general algebraic methods. Then, when you get an i². replace it with -1 and simplify. You can square the numerator and denominator separately:
[latex]\displaystyle
\[\frac{{\left(6 + 8\,i\right) }^{2}}{{\left( 4+3\,i\right) }^{2}} = \frac{{\left(36 + 96\,i + 64\,i^2 \right) }}{{\left( 16 + 24\,i + 9\,i^2\right)}} \]
[/latex]

kelxx

ok so i got
1) r = √3 (length)

2) tanθ = -(3/√3)
get tan inverse then θ = -60 (60°)

Then since its in the 4th quadrant do you take 60 away from 360 and then convert it to radians??
So,
360° - 60° = 300°
300° = 5pi/3 radians

So for my final answer i got √3(cos(5pi/3) + isin(5pi/3)
Does this sound correct??

Thanks

Michael Collins

Yeh kelxx that's it exactly. Those answers are correct. It's often fine to leave your answer in degrees, and in the fourth quadrant you don't have to worry about taking it away from 360˚ if you aren't asked to. You can leave it as -60˚ if you like (or [latex] -\pi/3 [/latex] radians).

kelxx

ok thanks for the help!