ABSTRACT
In this chapter we show how ideas from two mostly unrelated areas of mathematics—complex numbers and linear algebra—can be used to prove binomial identities. Complex numbers turn out to be a useful tool for proving a variety of identities involving aerated, alternating binomial sums. (For example, https://www.w3.org/1998/Math/MathML"> ∑ k ≥ 0 ( n 2 k ) ( − 1 ) k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351215824/cd461a07-713f-408e-a472-f82f6bbf751f/content/eq1596.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is such a sum because it alternates and because it features every other binomial coefficient from row n of Pascal’s triangle.) Proving an identity involving such a sum is often accomplished by substituting i or, more generally, a complex root of unity into the binomial theorem. We review basic properties of complex numbers before diving into the details of this process.
