Proof by contradiction

Proofs are amazing things. In mathematics, a proof allows us to establish the truth of a statement. Math is a unique subject because it is the only subject where you can be 100% certain about something. For example, you can prove that an even number plus an even number will always be an even number. That is pretty remarkable.

Proofs come in all shapes and sizes. Some are long and some are short. Some require extensive background knowledge and some only use the simplest of concepts. Most proofs can be grouped into some common categories. The category I want to focus on today is “proof by contradiction.” This technique is one of the hardest to grasp but the most rewarding. Let’s start with an example.

I have a question. What is the biggest whole number? Is there such a thing? If you play around a little bit it should be clear that 1,000 is pretty big. But 4,000 is even bigger. 40,000 is bigger still. But how can we be sure that there is no biggest whole number? Here is where we use the technique. We try to prove the opposite.

Let us suppose that there is a biggest whole number, which we will call “x”. What would happen if we add 1 to this number? Clearly, this new number would be bigger. But we assumed that there is only one biggest whole number. We now have a contradiction. Here is the rub. Since we assumed that there was a biggest whole number, and this assumption led us into logical nonsense, the only other logical option is that there is no biggest whole number. Done.

I want to do another quick example with you. Is there a smallest positive fraction? Well 1/2 is pretty small. 1/10 is smaller still. And 1/100 is getting pretty tiny. You probably see where this is headed.

Suppose we have a smallest fraction, call it “x”. What would happen, if we divided this number by 2? In other words, what if we cut it in half? Well clearly this number would still be positive. Think of an analogy. If you have a really tiny piece of pizza, and you cut it in half, you still have pizza.* But you have less pizza. Hence, we have found a smaller positive fraction. This is a contradiction. We arrived at this contradiction by assuming that there was a smallest positive fraction. Thus, we can logically conclude, that there is no smallest positive fraction. Neat.

Back to the pizza analogy. This proof means that you could keep cutting a pizza into smaller and smaller pieces forever. You could always construct a smaller piece. That is kind of interesting. Next post, I will investigate one of the most famous proofs by contradiction.

*Like all analogies, the pizza analogy ultimately breaks down. At the level of atoms, the pizza can no longer be cut into smaller pieces. This raises interesting questions about the nature of pizza, reality, continuity, metric space, and the universe itself. These concepts may be explored in future posts. Grab a slice and stay tuned J


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