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A high school student asked me whether learning Binomial theorem would help him with complex numbers. At first, I said no but then I realised I was too rusty in the subject to make a judgement.

What should I tell him? If there is some relevance of Binomial Theorem in complex numbers, to what extent?

Also, specific examples where Binomial Theorem is relevant to complex analysis would be great!

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    Binomial Theorem is critical in some proofs regarding power series, which is important in complex analysis. – 2017-01-03
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    Is it a must study before complex? – 2017-01-03
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    Well here's one: cos$(nx) + i$sin$(nx) = \sum_{k = 0}^n \binom{n}{k} $cos$^n(x)(i$sin$(x))^{n-k}$. I always find trigometric sums like that pretty cool, and it uses the binomial theorem together with a formula (Euler's) from basic complex numbers. – 2017-01-03
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    If you want to expand $(a+bi)^n$... – 2017-01-03
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    You say "must study" but I don't think it's the right way to look at it. Think of mathematical subjects as tools in a toolkit. As you progress, you gradually accumulate these tools. You never know when a particular tool may come in handy. Before starting on complex number theory (not even analysis), I would say being experienced with all the tools pertaining to basic algebra is important. I think most people would consider binomial expansions (integer powers) to be part of that basic algebra toolkit. So a resounding "yes" from me. Others have already brought up concrete examples of application – 2017-01-03
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    I see what you mean. I'll be sure to relay this to him :) – 2017-01-03
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    The binomial theorem is useful in proofs. If he is simply looking to understand complex numbers without writing down proofs, it's not necessary. – 2017-01-03
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    @Deepak Do you wan't to put this comment as an answer? – 2017-01-03
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    @TripleA Thank you, I have done so. Fleshed it out with a couple of examples too. – 2017-01-03

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You say "must study" but I don't think it's the right way to look at it. Think of mathematical subjects as tools in a toolkit. As you progress, you gradually accumulate these tools. You never know when a particular tool may come in handy. Before starting on complex number theory (not even analysis), I would say being experienced with all the tools pertaining to basic algebra is important. I think most people would consider binomial expansions (integer powers) to be part of that basic algebra toolkit. So a resounding "yes" from me.

Here are a couple of examples where your student may need to apply binomial theorem when dealing with complex numbers at even an elementary level.

Example 1:

Simplify $(2+3i)^4$, expressing the result in the form $a + bi$.

There is more than one way to handle this problem, and a slightly sophisticated approach would be to transform the number into exponential form before taking the power. However, for such a low exponent, a simple application of binomial theorem is probably easiest. $(2 + 3i) ^4 =2^4 + \binom 4 12^3(3i) + \binom 4 22^2(3i)^2 + \binom 4 32(3i)^3 + (3i)^4$ which can then be easily simplified by evaluating and grouping the real and complex terms together.

Example 2:

As your student delves deeper, he/she will cover de Moivre's theorem, which affords a very easy way to take powers of complex numbers. It's especially useful in the computation of the multiple angle formulae in trigonometry.

For example, suppose you wanted to compute $\cos 3\theta$ in terms of powers of $\cos\theta$.

de Moivre's theorem allows you to immediately state that $(\cos\theta + i\sin\theta)^3 = \cos 3\theta + i\sin 3\theta$

Since the real parts on both sides have to be equal, you can use binomial expansion to determine and simplify the real part of the LHS, and then derive $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ without much fuss.

The real power comes when you want to generalise this to any multiple $n$, as in this Wolfram article on Multiple Angle formulae