# How To Evaluate Complex Numbers

1. Dec 2, 2009

### urduworld

Hi PFs

i want to know how to evaluate the complex number, and what are the meaning of the evaluating a complex number

Let Z be the complex number, Z = (3-4i)^5 what i have to do, just give me hint

2. Dec 2, 2009

### sylas

Put the complex number into its polar representation: r.e.

3. Dec 2, 2009

### urduworld

its polar representation is Z = r[Cos(thita) + iSin(thita)]
is it is Evaluation
i want to know what actually i have to do in evaluation
Thanks for your help :)

4. Dec 2, 2009

### sylas

You need to go in the reverse direction, using x and y as your variables, not r and θ. Can you complete the following with substitutions for r and θ in terms of x and y?

$$r^{i\theta} = x + iy$$​

Alternatively, can you rewrite 3-4i into polar form, to, say, four decimal places?

Cheers -- sylas

5. Dec 2, 2009

### urduworld

i was thinking that evaluation is like polar form i don't know how to resolve this, i know nothing about
$$r^{i\theta} = x + iy$$
please tell me what i have to do,

6. Dec 2, 2009

### HallsofIvy

To find powers you need "DeMoivre's formula":
$$(r[cos(\theta)+ i sin(\theta)])^n= r^n (cos(n\theta)+ i sin(n\theta))$$.

If DeMoivre's formula is not in your text, did you consider just sitting down and multiplying 3- 4i by itself 5 times? That would probably have been faster.

7. Dec 2, 2009

### urduworld

oOps this is like that yes Demorvies theorem is in mine book, i have to apply this, and is n will the power like 0,1,2....n

8. Dec 2, 2009

### sylas

I fool so feelish....

HallsofIvy is quite right; simply multiplying it is probably the easiest thing to do when you have an integer exponent.

For example:
$$(3-4i)(3-4i) = 9 - 24i + 16i^2 = 9 - 16 - 24i = -7 -24i$$​
Keep going from there. Multiply by (3-4i), and again, and once more. You don't need the polar form.

However, since I mentioned it, I'd better go ahead with the answer:
$$\sqrt{x^2+y^2} \times e^{\text{atan2}(y, x) i} = x + iy$$​
You don't need to worry about this if you take HallsofIvy's hint about repeated multiplications.

Cheers -- sylas