I Now Understand, the Binomial Theorem!

I don't just mean:  rote learnt the theorem, punched in the a,b and n values and then crunched those variables.

I actually mean that having studied M208 Unit I2 this week, which included some set theory work based around counting finite sets and elements of finite sets; followed by a brief recap and proof of the Binomial Theorem.  That I have been able to actually see, through the symbols,  and to truly understand, at the basic level; how and why the Binomial Theorem actually works!

I think this is what I love about studying University level mathematics.  It just seems to keep revealing such lovely surprises where and when you least expect them.

Take the Binomial Theorem as a perfect example.  Anyone who has studied A-Level maths at school or as part of pre-university maths modules, will have met the Binomial Theorem before.  One of its uses, is to be able to expand out polynomials such as (a+b)^2

However, M208 has taken a sprinkling of set theory, a dash of algebra and a soupcon of factorial magic, to show in very clear and simple terms, how many different ways there are, to put a set of different elements, into different combinations. i.e.

The set {1,2}, can be put in the order of either  {1,2} or {2,1}


This then leads to counting the number of ways this can be done and how many separate sets, each main set can be split into.  Following on from this, we can then begin to use this principle, to treat a polynomial as a finite set with (n) elements, which can then be simply expanded in full, using the Theorem.


This is, yet again, one of those realisations and personal moments, when the connections have fired between maths skills  learnt 30+ years ago (counting objects and putting them in different order), 2+ yrs ago (expanding polynomials using formulas), to finally understanding (I mean really understanding), how it all connects and how it all fits together.


These little revelations have been keeping me so fired up this week, that I even took a detour when I was studying some number theory yesterday (Implications and equivalences),  I actually pondered an 'I wonder?' moment and proved a simple statement, that I made up off of the top of my head, scrawling manically on my white board and then dancing around the kitchen when I proved it  (this was very uncool, according to my daughter).


I think that is my first ever bit of creative mathematics.


Beautiful.