Paradox: 1 = 0

In the first part of the paradox series, I listed four categories of paradoxes. Let’s look at an old favourite today, which falls in one of the categories I haven’t given an example of yet.

We have a proof that 1 = 0:

1. Let x and y be positive numbers such that   x = y
2. Square both sides:   x2 = y2
3. Subtract y2 from both sides:   x2 - y2 = 0
4. Factor x2 - y2:   (x + y)(x - y) = 0
5. Divide both sides by x - y:   x + y = 0
6. Since x = y, substitute x for y:   2x = 0
7. Divide both sides by 2x:   1 = 0

— Quod Erat Demonstrandum.

The flaw, of course, is in step 5: if x = y then x - y = 0, and we’re dividing by zero in step 5. (But note that because I made x a positive number, there’s no division by zero in step 7.) Since division by zero is not well defined, everything after that point is invalid. That puts this into category two:

It actually is not true, because it’s based on a false premise (that seems true, or that’s been lost in the weeds).

There are many variations on this, “proving” slightly different things, but they all involve dividing by zero. When we use an invalid premise — here, the premise that division by zero is legitimate — we can claim to deduce anything from it. That’s because of how conditional statements work.

Put mathematically, if P and Q are statements, and P is false, the statement if P then Q is always true. But that doesn’t mean that Q is true — if P is false, we can say nothing at all about Q, either way — and that’s the part that people often miss.

We’ll come back to that point later in this series.