# Complex number calculation

1. Jul 4, 2007

### Trail_Builder

not sure if this is in the right section as i havn't got to calculus yet lol. anyways, was wondering whether you could check i have calculated this right as I'm learning complex numbers at the moment and have no way of checking my working. thnx

1. The problem statement, all variables and given/known data

Calculate $$(\surd5 - 2i)^{11}$$

2. Relevant equations

3. The attempt at a solution

Find polar form of $$(\surd5 - 2i)$$

$$r = complex modulus$$

$$r = \sqrt{5 + 4}$$
$$r = 3$$

$$Arg(z) = sin^{-1}(\frac{2}{3})$$

sub all that into $$r(cosArg(z) + i*sinArg(z))$$

but before actual calculate take the power of 11 into consideration so...

$$(\surd5 - 2i)^{11} = r^{11}(11*cosArg(z) + 11i*sinArg(z))$$

I then calculate all that as

$$177147(8.19..+7.33..i)$$

$$1452413.3..+1299078i$$

hope that's right :D

thnx

2. Jul 4, 2007

### neutrino

The argument of a complex number (x+iy) = ????

3. Jul 4, 2007

### Trail_Builder

ermmm, did i just work out the angle of the triangle instead of working out the angle counterclockwise from the x-axis? (on the argand diagram i think its called/complex plane)

think i might of

would the arguement be instead $$360 - sin^{-1}(\frac{2}{3})$$?

4. Jul 4, 2007

### HallsofIvy

Staff Emeritus
Why not just sin-1(-2/3)? (Admittedly, that is the same as 360- sin-1(2/3).)