Relearning Math - The Difference of Squares

While working on my undergraduate degree, I had to declare a specific topic within computing as my emphasis, and support that emphasis with relevant coursework. I chose cryptology, the study of the construction and analysis of ciphers. To this end, I took quite a few classes from the math department, specifically in abstract algebra and number theory. After completing these, another class or two earned me a minor in mathematics - an excellent pairing with my computer science major.

Despite that coursework, my professional work touches most mathematical concepts at a very surface level, and is focused on the direct applications rather than the theoretical underpinnings. In the years since university I haven't had to work through many proofs.

Lately, I've had reason to revisit my mathematical fundamentals. My children are working their way through the American elementary school mathematics curriculum - arithmetic, basic geometry, and hints of algebra.

Helping them through their examples has given me an opportunity to review my own mathematical foundations. After all, how does long division work, exactly?

Re-learning the mathematics I thought I knew from an elementary level has been immensely enjoyable.

For example: when demonstrating how squaring grows numbers faster and faster, I had my child write down the sequence of squares.

$$\begin{aligned} 0^2 &= 0 \\ 1^2 &= 1 \\ 2^2 &= 4 \\ 3^2 &= 9 \\ \end{aligned}$$

Next, we wrote down the differences between each square and the previous square. This was to observe how this difference increases with each successive square.

$$\begin{aligned} 1^2 - 0^2 &= 1 - 0 &= 1 \\ 2^2 - 1^2 &= 4 - 1 &= 3 \\ 3^2 - 2^2 &= 9 - 4 &= 5 \\ 4^2 - 3^2 &= 16 - 9 &= 7 \end{aligned}$$

I am embarrassed to admit I had never noticed how the difference of squares follows the odd integers.

This can be derived quickly using algebra:

$$\begin{aligned} (n + 1)^2 - n^2 &= (n^2 + 2n + 1) - n^2 \\ &= 2n + 1 \end{aligned}$$

And of course, the sequence of $2n + 1$ generates the odd integers.

The rich world of mathematics remains accessible and approachable, requiring nothing but patience, curiosity, and humility.