In continuation of the philosophical and foundational nature of the book thus far, Chapter 3 opens with a discussion on kinds of numbers, our reliance on the appearance of some of them in nature and how that fueled their original derivation, and how we can rederive them without ties to our subjective, messy world experience.
The chapter begins with a demonstration of a powerful proof of contradiction demonstrating the existence of irrational numbers. Firstly, the proof (and original inspiration: Pythagorean Theorem) relies on whole numbers and rational numbers existing (rational numbers being ratios of whole numbers). Assuming that’s agreed upon, we can get on with the proof.
Assume we have a triangle with sides length 1 and, by the Pythagorean Theorem, a diagonal with a squared length of 2. We’ll then assume that the square root of 2 can be represented by a rational number (a ratio of whole numbers).
We then rearrange the above to put a on one side and b on the other.
We know that since 2 is a factor on the right hand side that a^2 must be an even number that is divisible by 2^2 and therefore a is divisible by 2. We’ll now introduce a new number c that must also be a whole number that is the other factor of a when divided by 2.
Reducing this down we’ll see that we’re right back where we started but with c instead of a .
This pattern clearly continues on ad infinitum; eventually we would be unable to divide these whole numbers any further, demonstrating that the square root of 2 must not be rational.
The chapter continues to try to confront the discussion on validity and practicality of various kinds of numbers. Clearly, in tasks such as counting sheep, natural numbers, whole values starting with 1 and increasing, are pretty clear participants in the natural world. That being said, the set of integers, whole numbers including their negative counterparts such as -1, -2, -3, …, is an interesting topic since you may believe they occur often in the natural world; however, as scalar values (representations of magnitude irrespective of direction) they do not.
It was believed for a period of time that integers, as scalar values, did not appear. It was first when we began work with Quantum Mechanics did we discover that the charge of an electron and the charge of the proton were exactly opposite of each other irrespective of time, angle, position, etc.
These values are known as additive quantum numbers which are values that both describe particles and are only added together (with sign included). It’s interesting that unlike scalars at the macroscopic scale which operate on natural or whole numbers, additive quantum numbers operate in the realm of integers.
Compared to the last chapter, this felt like a fun break. I enjoyed rounding out my knowledge on the kinds of numbers there are as, admittedly, I struggled to recall the difference between real and natural numbers. It was strange trying to understand where the usefulness of this lesson would come in except to perhaps cover proofs by contradiction, but the note on additive quantum numbers helped drive home the importance of understanding what kinds of numbers a phenomenon could accurately be described by.