Beautiful Equations Explained.. Binomial Theorem

Mathematical equations are undoubtedly beautiful to look at. If you’re lucky enough to study them, then you’ll quickly realise that their beauty is not limited to the elegant symbols and funky x’s… We’re passionate about sharing this beauty with the world, breaking them down one equation at a time to explain their meaning, purpose and history! First up, a handy tool for expanding brackets…

The binomial theorem is an algebraic description of the expansion of a binomial to the power of some natural number n. Or more clearly, say you have some brackets in algebra (technically a polynomial) and inside there are two terms. If this bracket is multiplied by itself a certain number of times (it must be a positive whole number) then instead of writing it out this many times and expanding each bracket one by one this formula gives the solution! In order to use this equation your first step is to change “a” and “n” for the values in your own brackets. The greek epsilon symbol in the equation means “Sum''. To use it, you take all the positive whole numbers from 1 to n, putting them in the place of k one at a time and then adding each of these terms together. And that’s it, you have your expansion!

This theorem pops up in surprising places across mathematics and is a useful tool in finding proofs and deriving other formulas in probability and calculus, for example. It also links beautifully to Pascal’s triangle in pure mathematics, as the binomial coefficient is also the (k + 1)th number in the (n+1)th row of the triangle. 

Special cases of the theorem such as choosing  n = 2 can be traced back as far as the Greek mathematician Euclid in 4th century BC and for n = 3 to 6th century AD in India. The generalised form of the theorem is credited to Isaac Newton in the mid-17th century; however, it was the lesser known Lichfield-born mathematician John Colson who first published a proof in 1736.