## Where we learn that all numbers

are *not* created equally

I want to go back to the basics, to one of the fundamental participants in all mathematical calculations. When I say I love math, most folx presume that means I love arithmetic, which is not quite true. I mean, I definitely get some satisfaction when reconciling a bank register** ^{}** but I’ve never considered working in accounting, or with data or statistics, or following some unrealized true calling of becoming a world-renowned card-counting poker player.

Nope, when I say that I love math, I’m primarily talking about theoretical math and the application of mathematical concepts to the world we live in, and that (sometimes intricate) sojourn through logic and reasoning, in order to explain …everything. That’s the math pocket I like to hang out in.

That being said, I’m gonna take y’all back to elementary school because I want to give some respect to what I think is the most straight-up what-you-see-is-what-you-get of all the numbers, your friend and mine, the integer.

According to the fine folx at Oxford Languages, the first definition they list for the word integer is:

Ah, numbers. Everyone’s first conscious intro to mathematics. Counting. Then adding, and subtracting. Multiplication! Division! Oh the things you can do with numbers!

Now there are different classifications (and systems) of numbers and… this is one of those times I’m *not* going to take you down that whole particular path of discovery.** ^{}** But here’s the basics of the basics:

First, there are natural numbers like 1,2,3,4,…

And then there are whole numbers – which are *the same thing* *as *natural numbers, but they also include the number zero (0) in their pack.^{}

And then you have integers – which *include *that previous number-pack known as whole numbers as well as their negative counterparts like -1,-2, -3, -4,..

**Attention readers:** For the remainder of this post, I'm going to be verbally anthropomorphizing my descriptions of the classification of numbers.^{} Welcome, everyone, to how my brain works.

This now brings me to Oxford’s second definition of the word integer, which is:

Oooooooh. Didja just get a lil shiver from that tiny sentence? I sure did. Considering that definition, it’s as if the word integer is the ultimate destination for oneself. Like, honestly, my ultimate goal in life should just be becoming an integer. Hey kid, what do you want to be when you grow up? Well, gosh, I want to be an integer, of course!

The description of an integer is, IMO, the same as the characteristics of a really solid human being, the kind of person I wanna hang out with and have a beer. They are whole. They are complete. Sure, sometimes they can be negative but that’s ok because – more often than not, unless they’re always talking about something weather-related or how they have overdrawn on their checking account** ^{}** – they are positive.

But I can tell ya one thing for certain that integers are *not*, and that would be fragmented or fractured — cuz then we’re talking about rational numbers, what with their obsession with precise measurement, like, down to the 1/8^{th} or 1/16^{th} or whatever you’d call the smallest fraction of an inch. Ergo, rational numbers include numbers that are fractions and decimals and percentages.

And then there are those that are real (numbers, yes, I’m still talking about numbers), like Real numbers with a capital R because they can be totally *ir*rational and they might just go on and on and on and here is how they’re described in The Princeton Companion to Mathematics:

…irrational numbers will appear everywhere we look…They are not used directly for the purposes of measurement, but they are needed if we want to reason theoretically about the physical world by describing it mathematically.

The Princeton Companion to Mathematics

pg. 18

So, like, I totally get that real numbers are necessary. I’m down with the realness, it’s important to just get real sometimes. Real numbers, in the mathematical sense, are incredibly necessary if you are at the transition point in your mathematical education where you are going from a world of memorization and rules and are now entering the realm known as mathematical analysis and that’s a new place filled with proofs and positing and you’re discovering how there may be more than just one path in your search for an answer…

…and that’s exactly why I* don’t* want to grab a beer with a real number. It’d be like college all over again. A real number would totally stay up all night, yapping away, saying “But what if…” and “Have you ever thought about…” countless times. Bleh, I only need to experience that once in my life.

And then I’m just going to say the bare minimum about the last group in the number system, and that is that they are Complex numbers, and hoo ha, no. Not now. Did you not see that I just said the word ‘complex’? My beer time has no business being complex.

I leave you with a rather long passage from the book Mathematics for the Nonmathematician by Morris Kline. This is some heart-filled math nerdery.

Just as we are inclined to accept the sun, moon, and stars as our birthright and do not appreciate the grandeur, the mystery, and the knowledge that can be gleaned from the contemplation of the heavens, so are we inclined to accept the number system. There is, however, this difference. Many of us would not claim the latter and would gladly sell it for a mess of pottage. Because we are forced to learn about numbers and operations with numbers while we are still too young to appreciate them – a preparation for life which hardly excites our interest in the future – we grow up believing that numbers are drab and uninteresting. But the number system warrants attention not only as the basis of mathematics, but because it contains weighty and beautiful ideas which lend themselves to powerful applications.

Mathematics for the Nonmathematician

pg. 58

In conclusion — what do you think is integer’s favorite beer, and why? Because the first one is totally on me.

Thanks, math. You’re the best.

In other bases besides ten, what happens to these definitions? For example in base 1/3 do we think of 1 as an integer even though it represents the repeating base 10 number 0.33333….?

Firstly, for anyone else reading this: A number system that is base 10 is aka The Decimal System. And ok, ok, I see what you’re throwing down, Gerald. 😉 That’s a totally valid question — and I will totally take it into account when I really get into some Numerical Analysis. For now, my answer to your question is that the definitions presented herein don’t change…but they do expand as the decimal place value would need to be taken into consideration, now.

To give a direct quote (again) from _The Princeton Companion to Mathematics_ – pg 81: “…whole numbers are nicer than decimals.” 😀

All I know for sure now is that all I want in life is to be an integer, too. And even though I’m not a beer drinker, I’d love to be a fly on the wall when others are discussing what beer is an integer’s favorite.

Tho I like to think that an integer would be into IPAs, I’d bet they stick with lagers and, on those days when they just want something a little bit different, maybe a really approachable pale ale.