A Quirky Introduction To Number Systems

Everyone’s got quirks. Me, I like finding new ways to think about problems, and I’ve started seeing numbers in a new way. Today I’m bringing you along for the ride.

And why do you care?

You’ll see math as one of several approaches. We’re still improving our number system.
You’ll start noticing core relationships and apply them to diverse areas
You’ll have a way to approach “weird” concepts like division by zero and imaginary numbers.

Math is Software For Your Brain

Though our favorite encyclopedia describes math as “the body of knowledge centered on such concepts as quantity, structure, space, and change” I think there’s more to it than that.

Math is software for your brain:

Your brain is a raw computer.
You learn new math (install “Counting 1.0” and “Algebra XY Pro”) and suddenly you can solve new types of problems.
Sometimes math has bugs. “Roman Numerals I” was ok, but Decimals 2.0 was a much-needed upgrade. But we still have a few issues, like dividing by zero.

It’s a strange analogy, but I’m a bit strange, so I think it works out.

But Math Has Its Limits

Quick quiz: Can you multiply two Roman numerals? No cheating, no converting to decimal: I’m talking about “IX times XXXIV”. Ready, set, multiply!

…

Having fun yet? It’s CCCVI.

Does this horrendous experience mean multiplication is “hard”? Or are we thinking about multiplication in the wrong way, using the wrong mental software?

If you upgrade your brain from “Roman Numerals I” to “Decimals 2.0”, you’ll find that 9 times 34 is a much easier question: after some work you’d get 306. Same problem + different mental software = drastic difference.

Painting With Notepad

Yes, you could squeeze multiplication into Roman Numerals. But it’s neither fun nor easy, and don’t get me started on long division.

Our difficulties are often due to our approach, not the concept.

Think of it like trying to draw in Notepad. It’s a nice tool, and you can even “type” the Mona Lisa, but the software just wasn’t built with images in mind.

Similarly, Roman Numerals were built when we were still learning to count — zero wasn’t even invented yet! Math is a software system that gets better over time, and Roman Numerals were due for an upgrade.

But before we get too high-and-mighty, realize our current number system is a patchwork of new features and bug fixes, used to improve our understanding of the universe.

And when we hit difficulties (What’s 1/0? The square root of -1?) we need to wonder if we’re hitting universal “laws” or walls of our own making. Like the Romans trying to multiply, let alone do fractions, my money’s on the latter.

From Ug to Infinity

Our number system developed over time. We started counting on our fingers, moved to unary (lines in the sand), Roman Numerals (shortcuts for large numbers) and Arabic Numerals (the decimal system) with the invention of zero.

Along the way we found “bugs” in our number system and had to invent new ways around it. Again, the bug was in our thinking (our mental software).

Ugware

Ugware is the counting system devised by Ug the caveman: counting on your fingers and toes. Ug’s bug was that he was limited to 20 items!

The fix was to abstract the need for physical objects: you don’t need 20 cows to count 20 cows. You can make 20 lines in the sand. Or take shortcuts like C for 100.

Unary and Roman Numerals

Having numbers represented abstractly let us do cool things like add and subtract, even fairly large numbers. I + II = III. X + XX = XXX. Not bad.

But there was still a few “bugs” — what is III – III?

Zero

What a fantastic, beautiful invention: using the symbol 0 to represent nothingness! It’s a mind-bending and useful idea: we can keep track of “no” cows at all!

This development led to our familiar positional number system: 204 means two “hundreds”, zero “tens” and four ones.

Integer division and multiplication became possible in ways the Romans (and Ug) had never imagined. You could work out 1234 × 5678 if given enough time. What a great feature!

Negatives

But zero didn’t solve everything; subtraction still had problems. What happens when we take 5 from 3? One solution is to throw up our hands and say “it’s a bug and it’s undefined”, but we’ll do better.

We can think about the problem differently, and entertain the possibility that a number can be “negative” — a number that is less than nothing! (Pretty mind-bending, no?).

There are many interpretations (a lack of cows, a debt of cows) and negatives were invented to handle this “bug” in subtraction. Of course, it took a few thousand years to accept this new feature — negative numbers were still controversial in the 1700s!

Rational Numbers

Division introduced bugs as well. 8/4 is fine, but what is 3/4? It’s a bug!

The fix is to find a way to represent “numbers between numbers”. 3/4 is really 75/100, or “0.75”.

We invented the decimal point to handle the crazy idea of a number more than zero but less than one. Wow! Pretty wild, but we included these crazy types of numbers to make our mental software better. Lo and behold, fractions have their uses. The average family can have 2.3 kids and we know what it means.

Irrationals Make Greeks Angry

Here we are, minding our own business when we see a right triangle:

The sides are 1 and 1. And there, staring us square in the face, is the square root of 2. It taunts us, asking to be written down. We can’t — it’s an infinite, non-repeating decimal number that can’t be expressed as a fraction! And yet it’s right there on paper.

It’s more than a conundrum — it’s madness! The guy who discovered irrationals got thrown off a boat.

Luckily, irrationals are at least “algebraic” in that they are the solution to some algebra equation. We can consider $\sqrt{2}$ as shorthand for “the solution to the equation $x^2 = 2$”. We often forget $\sqrt{9}$ is really both 3 and -3, don’t we? Convention implies the positive root.

Complex Numbers

Now some a smart aleck asks, “Ok bub, what number is the solution to the equation x^2 = -1?”.

What to do? Declare this to be impossible and non-sensical, just like zero, fractions, rationals and irrationals were once “impossible and non-sensical”? Or do we accept that maybe, just maybe, our human understanding of the universe is not complete and we have more to learn. You know where my money lies.

Imaginary numbers are just as “realistic” as other numbers (or equally contrived, depending on your viewpoint). But, we don’t have an intuition for them because they’re often “explained”: Oh, you don’t have an Electrical Engineering degree? Didn’t learn about complex impedance? No intuitive imaginary numbers for you!

I’ve been thinking about these numbers and plan to address this issue. But not yet — have patience.

Why .9999… = 1, and why you should care

Our number system is a way of thinking, but it still has a few gaps. We’re not quite sure how to deal with infinity and infinitely small numbers.

Here’s a brain-buster for you:

You: What’s 1/3?

Me: Um, .33 repeating.

You: Ok. What’s 1/3 + 1/3?

Me: 2/3 You: Sure, but do it in decimals.

Me: Um, .33 repeating plus .33 would be… .666 repeating.

You: Great. Now what’s 1/3 + 1/3 + 1/3, in decimal?

Me: Uh… .33 repeating plus .33 repeating plus .33 repeating… is .99 repeating.

You: But doesn’t 1/3 + 1/3 + 1/3 = 1? So .99 repeating = 1.

Me: What manner of trickery is this? (Pushes you off blog).

Try that argument on a kid (or adult) — it’s fun to see people’s reactions. Clearly, 1/3 + 1/3 + 1/3 = 1, but somehow when we try to “add it in decimal” the result seems a bit strange. Again, is the strangeness due to the concept, or our thinking?

What is .33 repeating, anyway? Is it a monkey writing 3’s until the end of time? Is it a number beyond our notation that we’re hopelessly trying to approximate, like the square root of 2? If we simply switch to base 3, the problem goes away: 1/3 = .1 in base 3, so .1 + .1 + .1 = 1 (again, in base 3).

And why do you care? It may be time for a number system upgrade. Discussing infinity with our current numbers is like drawing in notepad. It’s crude and feels “tacked on” (like saying 1/0 = infinity. What about 2/0 or 0/0?).

Mathematicians are working on new number systems where infinity is built-in, but there’s still unsolved problems about how to “count” infinity.

Let’s seek the “a ha!” insights that made zero, fractions and negative numbers understandable, not just the results of manipulating equations. We’ve been able to overcome every previous mathematical roadblock.

Going Forward

The goals of this article were simple:

Show how math is like mental software that improves over time
Explain that “nonsense” like zero or negative numbers can start as a paradox and become intuitive as we adapt our approach.
Today, we still have trouble with ideas like infinity (or at least I do). It’s ok to admit it.

This is a way to think about math; combine it with your own understanding. Don’t stand in a daze, unable to add because you’re unsure what 1/3 really means.

Insights deepen our understanding, but sometimes only emerge with use. Newton didn’t have a “formal” understanding of infinitesimals when he invented calculus, but it seemed to work fine for him (equations got solved). I don’t advocate plug-and-chug, but for certain ideas you need to hammer away before the insights come.

But enough philosophy. Upcoming articles will show real, concrete ways to think about arithmetic and complex numbers, which can aid the “mechanical” understanding we have today. Happy math.

PS. If you’re curious, there’s more on .999… and division by zero on wikipedia.