Number Systems

Recently, we have learned that many animals have a sort of numeracy and there is a rising suspicion in the scientific community that a very basic sort of numeracy is present in animals as an innate structure. Even some insects like bees and ants can count!

Counting, despite seeming quite simple, is an abstract task. When we count chairs, for example, we are appealing to the idea there is an abstract notion of a chair and each chair I see is an instance of that notion. If I saw, for instance, a room with 2 arm chairs, 10 desk chairs, and 20 folding chairs, I would likely report the room has 32 chairs even though there is quite a bit of variability in the quality and class of those chairs.

Humans seem to be somewhat unusual in the arena of counting in the animal kingdom. We count enormous numbers and facilitate things like inventories, wealth, and taxes with these numbers and probably have since time immemorial. Counting gives us the counting or natural numbers: $1, 2, 3, ….$ which are often denoted $\mathbb{N}$. Almost every human culture we find throughout history generates some version of the natural numbers (there are some unusual potential counterexamples).

Next, most cultures develop some notion of fractions. How exactly they notate and understand fractions is highly dependent on how sophisticated their insight into mathematics is. Egyptians developed a fractional methodology that, to the outsider, appears quite onerous and difficult to use. However, what has come to be known as Egyptian Fractions in modern number (which is quite related to the methodology of the Egyptians) has a sort of optimal convergence. A measurement using these fractions conveys the most precision with the fewest “terms” (I use that word somewhat loosely- but we will eventually see what I mean precisely when we study number theory). Fractions are ratios of whole numbers to one another. For most cultures we see a development of what we might today call $\{\frac{p}{q} \ni : p, q \in \mathbb{N}\} = \mathbb{Q}^+$ or the strictly positive set of fractions.

Next, we see a development of negative numbers. These are essentially used to indicate a direction. For instance, I might write +\$7 if I have seven dollars in my wallet and -$20 if I owe you twenty dollars. Similarly, I might say going up is moving in the positive direction and going down is moving in the negative direction. Still, we find almost all early cultures conspicuously lack the notion of zero as we might understand it today.