By Kate Ertmann

The following blog first appeared on Kate Loves Math*.  

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.

*When I say satisfaction I mean solace – as in the truest of all comforts – is what comes over me when I can just focus on the task-at-hand of reconciling a bank register. This is why I was the bookkeeper for my company the entire 15years that I was also the President of it. I actually looked forward to Friday afternoons at 2pm because that meant that I was done with interacting with humans for the week, and I could get up-to-date with all Accounts Payable and Accounts Receivable for the business, and then go into my weekend with a feeling of true balance and assurance cuz Knowledge is Power.

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*.

*A zero is described to be an “additive identity” — which means that adding zero to any number leaves that number unchanged.

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.

*;) I wink because in order to talk about things like the weather & climate change as well as talk about deposits and debits in a bank account, you need negative numbers in your mathematical vernacular.

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 1/8th or 1/16th 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 irrational and they might just go on and on and on and here is how they’re described in The Princeton Companion to Mathematics: 

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.

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.

Kate Ertmann is a math researcher, a recovering entrepreneur, and a public speaker. She helps math make sense in the real world while celebrating and honoring the work of the academics. To learn more about Kate, read more of her essays, or to ask her to speak at your event, join her at www.katelovesmath.com,