Mathematical Induction Without Dominoes

How to actually understand Mathematical Induction

Imagine you want to show that the number 5 has a certain property. Let’s say this property is that the sum of all natural numbers up until 5 (0, 1, 2, 3, 4 and 5) is equal to

Now you could just add 0, 1, 2, 3, 4 and 5 together and show that this sum equals 15, which is also what the above fraction gives.

But you could also assume the property holds for the number 4, and work from there to show it also holds for 5.

Sounds odd, right? Bear with me. We assume the property holds for 4 — we’ll come back to this assumption later. So the sum of 0, 1, 2, 3 and 4 is

From there, can we show that the sum of 0, 1, 2, 3, 4 and 5 equals

Yes! The sum of 0, 1, 2, 3, 4 and 5 is just the sum of 0, 1, 2, 3 and 4 plus 5. We assumed the sum of 0, 1, 2, 3 and 4 equals

Which means the sum of 0, 1, 2, 3, 4 and 5 must be

4 + 1 = 5, and also, 5 equals (5 x 2)/2. So we have

Which equals

Note that 4 + 2 = 5 + 1. So we have that the sum of 0, 1, 2, 3, 4 and 5 equals

Which is what we wanted to show!

Okay, that’s fine and all, but you assumed the property holds for 4. You have to justify that assumption!

True! And I will. Again, we have two choices: we can calculate that 0 + 1 + 2 + 3 + 4…