In The Big Bang of Numbers, I take the reader through a thought experiment, where together, we build all the different kinds of numbers, and then, from these, go on to trying to build geometry, algebra, patterns, physics, and everything in the universe. In Chapter 4, we just learned how to construct irrational numbers like pi and the square root of 2, whose decimal expansions go on and on, without repeating. In Chapter 5, we start discussing how this can help us in the universe we are building.
So how do the irrationals you’ve just created enhance your universe?
On an aesthetic level, they provide a sense of completeness. You can rest secure in the knowledge that you’ve created each and every decimal expansion possible, without any omissions. If you felt a touch of mathematical appreciation producing the rationals, that’s now grown to full-fledged intellectual satisfaction. However, this benefit is somewhat abstract. What do irrationals contribute in practical terms to your universe?
Most mathematicians would answer precision. With irrationals at your disposal, you can identify the exact square root of 2, the exact ratio of circumference to diameter. But there’s a gap here, since you can never actually write down the exact value of or pi. Each has an infinite…
