Finding My Prime: The Inspiration Behind My Teaching Career - By Himani Kumar
My journey to becoming a math teacher began with two pivotal influences. I was fortunate to have maths teachers who built a respect for the subject in the way that they taught and presented topics, alongside an older brother who would find enjoyment in teaching 12-year-old me calculus years before I would see them in any formal educational setting.
I often say those two things were the start of my own curiosity for mathematics, as it was the reason I began watching Numberphile where they would ask thought-provoking questions such as ‘is zero even?’ and presented a mathematical approach to questions such as: ‘Will your name ever become extinct?’. This made me view maths in a completely new light, because it was no longer just about learning methods and completing a set of questions, but it made me realise that maths is universal and ubiquitous.
As a teenager, I learnt about the fundamental theorem of arithmetic which states that every integer greater than 1 is either prime or can be represented by a unique prime factorisation (prime numbers that multiply to make the original number). I found this to be truly remarkable and I quickly grabbed a pen and paper and began checking if it was true. When the realisation sank in, I had even more questions! Why are these numbers essentially the building blocks of all other numbers? What property of these numbers give them this ability?
Number theory really sparked my interest and inspired me to want to take the leap and teach maths – I hope this will be an enjoyable read as it just so happens that the largest prime number was recently discovered (and it was a Mersenne Prime).
The fundamental theorem of arithmetic was just the tip of the iceberg of what I was about to learn. I came across Goldbach’s Conjecture which states that every even number > 2 can be written as the sum of 2 primes. For example, 12 = 5 + 7, where both 5 and 7 are prime numbers. Now this has been checked to very large numbers (up to 4x10^18) but has yet to be proven for every possible number! Restricting this conjecture to twin primes is also fascinating. Twin primes are prime numbers that are 2 steps away from each other, such as 3 and 5 or 7 and 11 (which there are also infinitely many of). Goldbach’s Conjecture for twin primes says that every even number can be written as the sum of 2 twin primes with 35 exceptions (checked up to 2x10^10).
Although yet unproven, this deeply intrigued me and left me pondering even more. How could this be proven? Why is this working for all numbers up to a certain number? Are those 35 numbers part of a bigger infinite or finite group?
Things got more exciting when I heard about the Prime Number Theorem (PNT), which tells us how many prime numbers we can expect under a number n. This theorem shows that as numbers increase, prime numbers become less frequent. It says that the proportion of primes less than n is 1/ln(n). The natural log of 1 million is ~ 13.8, so that means that the proportion of primes is 1/13.8 which is ~ 7.25%. Meaning around 7.25% of numbers under 1 million are prime, so around 72,500 prime numbers. This is close to the true value as there are exactly 78,489 prime numbers below 1 million. This allows us to estimate the gaps between primes and approximate values for specific nth primes, such as the 234th prime number.
Looking further into the gaps between primes, there is Bertrand’s Postulate which states that there is always a prime between n and 2n. This has been proven in several ways and an interesting rhyme to remember is ‘Chebyshev said it, but I’ll say it again; There’s always a prime between n and 2n’.
I realised that there are almost endless patterns the more I read about prime numbers, and each one astonished me and made me love maths even more. The essence of my motivation to teach mathematics lies in my desire to cultivate a deeper reverence for the subject in others, just as my teachers did for me.
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