Numbers are funny. Even with all the order and the systematicness, you will always find numbers cropping up in the most unexpected places. To look at some of these remarkable instances is the purpose of this series.

Let’s start off with something simple. Everyone knows how to reduce fractions to their simplest form, right? But what if I was annoyed at the conventional simplification, and started doing this?

16 / 64 = 1 / 4

Not good enough? You want to see more?

26 / 65 = 2 / 5
19 / 95 = 1 / 5
49 / 98 = 4 / 8

Why only two digits, you ask? Okay then. (from Wolfram Mathworld)

364 / 637 = 4 / 7

Still not satisfied? Maybe four digits is what you’re looking for. (note, got this one from here)

4277 / 2730 = 47 / 30

Not yet? Alright, you asked for it.

143185 / 1701856 = 1435 / 17056

But wait. This is only base 10. Why not try and find this in other bases?

15₆ / 53₆ = 1₆ / 3₆
2C₁₅ / C9₁₅ = 2₁₅ / 9₁₅

But how? Well, that’s something to think about. This phenomenon is called anomalous cancellation. Perhaps you can find a way to generate these strange cancelling fractions?