Introduction to the Binomial Theorem

Introduction to the Binomial Theorem

In the following sections, we build up to the inductive steps Abu Bakr Al-Karaji takes to explain the binomial coefficients. Our survey of the proof Al-Karaji provides is of relevance to algebra and logic; The binomial theorem is an algebraic theorem, while induction is a common and powerful proof technique.

The binomial theorem is a theorem with applications to probability, statistics, calculus, complex analysis, and abstract algebra. The binomial theorem states that, given two variables x and y and an integer n greater than or equal to 0, we have 7:

n (x + y)² = [ (1) a²-kyk. k=0

For instance, in an introductory algebra class, one would learn that for n=2, we have:

Σ(3) [a²¬^y^ = (3²) a²y° + (²) a¹v¹ + ( ² )aºy² = x² + 2xy + y²! y² (x + y)² =

So, one can use the binomial theorem to calculate the expansion of (x + y)” for any natural number n.