Transcript:

So my friend asked me this question: “Is this true, (a + b)² is equal to a² + b²?” And instinctively, I said, “No, this is wrong because it is missing the 2ab.” And then I thought, “Okay, if either a or b is 0, then yes, (a + b)² will be a² + b²." And then he said, “Yes, that is the trivial answer. But there’s another structure in mathematics called rings.”

Now rings are a set in mathematics where any two elements if you multiply, add or subtract, they fall into the ring. For example, the biggest ring is the ring of integers, minus infinity all the way to 0, all the way to plus infinity. And here any two you can subtract, add or multiply, they will fall in the ring. 

Like this, he said you can define a ring where the ring is defined by the remainder of a number divided by another number. For example, this is a mod 6 ring. So this is just remainder of number divided by 6. So it has got 0, 1, 2, 3, 4, 5. And in this ring, 2² + 3² is equal to (2 + 3)².