I’ve been asked a few times for more practice questions on complex numbers. This is where Wolfram Alpha can be your friend (like it’s not already!).

I’ll just give a few examples of questions from the tut on complex numbers which you could have solved using Wolfram Alpha, and from this you will be able to set up your own questions.

For instance, question 48 c) Find the roots of $z^5=1+\sqrt(3) I$ can be solved in Wolfram Alpha with the command:

Solve[z^5==1+sqrt[3]I,z]

Moreover it will solve this for you, give you the five roots and plot them in the complex plane. So now you can come up with any root question you can possibly think of. There’s an infinite number of questions to start you off. You can thank me later!

If you want to convert between the trigonometric form and the exponential form, you can use the two commands:

TrigToExp[Sin[x]+2 I Cos[x]]

ExpToTrig[Exp[I z+3]]

Though remember the definition of the hypergeometric trig functions from a previous tut.

If you want to get an expression in the normal cartesian form, then you have to be a bit more careful. For instance, $(1+\sqrt(3) I)(1+\sqrt(3)I)$: Here you can’t just plug it in because it doesn’t know the form you’re looking for it in. Here you have to put:

Re((1+sqrt[3] I)(1+sqrt[3]I))+I Im((1+sqrt[3] I)(1+sqrt[3]I))

ie. ask for the real part and the imaginary parts separately.

If you want to find the solutions to: $e^{3z}=1+I$, you have to do this in two steps. First you have to put in:

Solve[Exp[3z]==1+I]

This will return for you:

$z=\frac{1}{3}(2\pi n i+\log(1+I)$

but we know that we don’t want the log of a complex number. So you have to take $\log(1+I)$ and calculate that in terms of the cartesian form:

Re(log(1+I))+I Im(log(1+I))

Now you have all the information you need to solve this question.

To find the argument of a complex number you use the command Arg, and to find the magnitude, you use the command Abs. If you want to write $(\sqrt(3)+I)^{17}$ in modulus argument form, you can calculate:

Abs[(sqrt(3)+I)^17]  and  Arg[(sqrt(3)+I)^17]

separately and then write it yourself together in whatever form you want.

Note also that the conjugate of a complex number is Conjugate[z].

Hopefully this should be enough information for you to be able to go through the tut questions and write some of your own. Let me know if there are questions which you can’t answer using Wolfram Alpha or your own complex numbers intuition…

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