Complex numbers - Argument question (1 Viewer)

[Ragerunner]

The question says to find the modulus, argument and polar form for z = 4, -4, i and -2i

I'm not sure how to find the argument of it.

For z = 4

the answers says Arg(4) = tan^-1 0 (i.e. inverse tan of zero)

what is the tan bit for? And how do I find the argument of the other values of z?

thanks

 

[Affinity]

hmm.. draw the point 3+4i in the argand diagram and find the argument... naturally it involves arctan.

for other values for z hmmm.. same drill

for general z=x+iy

|z| = sqrt(x^2 + y^2)

arg(z) = arctan(y/x) if x is positive
and = arctan(y/x) +Pi if x is negative

 

[...]

but how come its 3+4i?

 

[CM_Tutor]

Ragerunner, it might to remember what an argument is. For a complex number like 3 + 4i (which I'm sure Affinity has chosen as an example), the argument is the direction that 3 + 4i lies from the origin.

So, since 4 lies on the positive real axis, it is 0 radians in direction from O, and hence arg 4 = 0.
Similarly, arg i = pi / 2.

Thus, for z = 4, |z| = 4, arg z = 0, and in polar form, z = 4(cos 0 + i sin 0)

For z = i, |z| = 1, and arg z = pi / 2, and thus in polar form z = 1[cos(pi / 2) + isin(pi / 2)]

I'll leave the others for you to thikn about / try.