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.

Kalid Azad loves those Aha! moments when an idea finally clicks. BetterExplained is dedicated to learning with intuition, not blind memorization, and is honored to serve 250k readers each month.

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24 Comments

  1. Multiplication in Roman numerals actually is quite easy, once you’re used to it. Take your example: IX times XXXIV:
    IX * XXXIV = (X – I) * XXXIV = X * XXXIV – XXXIV =
    CCCXL – XXXIV = CCC + XXXX – XXXIV = CCC + X – VI
    = CCC + V + IIIII – IIII = CCCVI

    All you need to know is the symbols, how to add and subtract, and the “10s” times table:
    X * I = X
    X * V = L
    X * C = M
    etc.

    This nicely iterative process is actually easier to learn than base-N multiplication in Arabic numerals, for which you need to learn the times tables, (N^2 + N) / 2 arbitrary products. Also, addition is trickier, in that you need to “carry” when a column adds to more than 10 (a step that often trips up early learners of arithmetic).

  2. Hi, thanks for the info! I’m still not convinced it’s easier though… I think Arabic Numerals were a nice upgrade :). But cool to know it’s “possible”.

  3. The real problem of Roman Numerals is that, when trying to write larger and larger numbers, the roman numeral system has to write rather large strings. Decimal allows for easier expansion (logarithmic growth as opposed to linear). Also, Division in Roman Numerals is pretty tough.

    As for “division by 0″ being a limit of the system, I think that may be a little misleading. Division by 0 is not a limitation, we understand it fairly well, indeterminant forms are fairly easy to resolve. X/0 is undefined, and theres really no good way to define it. however, we can still do math with it as a whole concept. That is, if I ask you to evaluate the product X/0 * 0/X, you can in fact do it, because you can convert it to a known indeterminant form. of 0X/0X, if we evaluate that using the concept of a limit from calculus, we can find that lim (s -> 0) of sX/sX = lim (s->0) 1 = 1

    division by zero isn’t _really_ a problem, maybe I’m being pedantic, but the way to say it makes it sound like it’s an outstanding problem in math. Unfortunately, there are some people who think that this is the case, and come up with silly, broken ideas like “nullity” to “solve” this non-problem.

    otherwise, pretty good article, keep it up

    ~~Joe

  4. joe, a little knowledge is dangerous. you can certainly slap a limit statement on a division by zero problem, invoke bernoulli’s rule, and arrive at a real number. however the question you just found an answer for isnt the question you just started with.

    in any field:
    X=X+0
    XY=(X+0)Y
    XY=XY+0Y the additive inverse of XY exists so
    0=0Y for every Y. multiplication by zero is not one-to-one so it has no inverse.

    to emphasize, zero has no multiplicative inverse, and this is a result of the natures of addition and multiplication, not of the way we “express” zero.

  5. @Greg: Thanks for the info.

    @Joe & ego: Thanks for the comments. The math that existed prior to calculus could not handle division by zero (hence the “bug”), but we do have ways to think about it. One resolution is to consider division as a limit of ratios as the denominator tends towards its “true” value, but this raises the question of whether it is in fact the same thing.

    Technically, we can label 1/0 as “undefined” and ignore the problem, but that may be similar to saying “3-5″ is undefined (prior to negatives) since subtraction is the inverse of multiplication. There is no natural number which, when 5 is added, equals 3. We had to invent “unnatural” numbers (negative integers) to address this. But yes, nullity isn’t the right way — I think one of the other number systems (http://en.wikipedia.org/wiki/Division_by_zero#Other_number_systems) may be a better approach.

    @Sharun: Thanks for the link, that site has some great articles!

  6. Kalid I must congratulate you on creating such a wonderful blog Its proably the best website I have come across in a long time. I just recentlyfinishedmy Bachelors degree here in Pakistan and wish that you had come up with this site many years earier. Its starting to get me interested in Math once again, an interest that university pofessors had managed to kill by making math so very not interesting :)…..keep up the good work and write a bit more on mathematical concepts……Thanks

  7. Hi Mohammad, thanks for the wonderful comment! I’m really glad you’ve enjoyed the site so much, and even better is getting interested in Math :).

    Yes, the great irony is that traditional education can often kill the desire to learn, if taught in the wrong manner. I’m glad you’ve overcome that, and I’ll be writing more math in the future. Thanks again.

  8. Technically, we can label 1/0 as “undefined” and ignore the problem, but that may be similar to saying “3-5″ is undefined (prior to negatives) since subtraction is the inverse of multiplication.

    That’s not quite the idea, though. 3-5 is well defined because there was an “obvious” convention once you look for consistent properties (3-5+5=3). There are times when 3-5 is still undefined (If you have three apples and you eat 5, then you’d better see a doctor… there’s no way to “owe” apples in this context), but if you look at it in a very abstract sense it always makes sense to have negatives there when you want them.

    1/0 is not so simple. It’s well defined, but in a number of different contexts, none of which being the “obvious” choice. Any time a mathematician, scientist, or engineer ends up running up against a division by zero, they should know the context well enough to make the right choice for how to get over the hump. They may even decide it’s not worth it.

    The point is that 1/0 isn’t all that crazy, it just takes more information than you get in the plug’n'chug mindset taught in high schools and their ilk. Nullity is nothing more than a default answer to a question that wasn’t properly asked.

  9. Division by 0 being meaningless is not problematic. x divides by y really means x times the multiplicative inverse of y, the value y^(-1) such that y*y^(-1)=1. Now, 0 can not possibly have an multiplicative inverse under any circumstance, since 0*anything=0. So it’s is indeed against the rules of mathematics to divide by zero!

    (in the extended complex plane z/0=ComplexInfinity only by convention)

    And we can actually count infinity! The set of integers Z and the set of quotients Q have the same cardinality (i.e. size) that is denoted by the transfinite number aleph-null. The set of real numbers R have a higher cardinality aleph-one. The set of real numbers in (0,1) have the same cardinality as the whole real number line!

  10. Heh, in less than a week I found a history manual for both Astronomy and Physics. It’d be nice though, to attribute the mathematicians who changed each system properly. I’m not sure who made Roman Numerals, and I’m not sure who came up with the concept of 0 from your article. There’s obviously historical elements to mathematics that aren’t shown here. Having them helps explain to people why they developed at the speed they did.

  11. The nature of 1/0 is inherent from the axiomatic system which we follow. It’s not an “unsolved” problem but rather a facet of the definitions we set down for the real numbers and their arithmetic operations.

  12. I learn by memorizing. I’ve always had a great memory. This allowed me to learn to read early as I remembered the sound of the words. Eventually I could discern what word should sound like based on what I already know…. I will be taking the GMAT for applications to business school that in January and I’m trying to find a way to solve the problem that irecognize I have which is that my mind finds it difficult to keep the relationships of mathmaticvariables straight in my mind. I constantly jumble it in my mind. I find it difficult to solve problems that are worded slightly differently fromthe ones I’ve done before. Data sufficiency questions involving inequalities with variables baffle me cause there are too many factors to consider. I have to think about whether x and y could be positive or negative, fraction or integer and where one relationship would yield one result and another relationship would yield another result, I find it hard to keep all that information straight. I start getting confused and losing track of the relationships. I’m great when it’s memorization but ask me to think and I get confused. I assume it’s because I don’t fully understand relationships and patterns but I don’t know how to start to see these patterns of which you speak on your post.

  13. @Mack: Thanks for writing — for these types of problems, what helps me is to have a mental model of what’s going on (some type of analogy or intuitive understanding). Once I have that, I can start playing with that model to figure out what will happen in the problem.

    It’s difficult to describe because it’s different for everyone, but for the inequality / data sufficiency examples, I might think about a number line (going left to right) and as each new equation comes along, it “shades out” different parts of the line. I think everyone has some type of model happening in their head, but it can be hard to recognize/talk about because it’s so innate.

  14. First, I must give κῦδος(kudos) to Arithmeticus Simplex for pointing out the practicality of operations with roman numerals (it’s important to acknowledge that a Roman provided with the argument of comparison would conclude that her system was vastly superior…and a Kyalian well versed in balanced ternary would correctly conclude balanced ternary to be superior to both Roman and decimal notation).

    On division by zero. I very much favor Abraham Walker, his non-standard analysis.*

    *(H. Jerome Keisler provided open access to his book, Elementary Calculus: An Infinitesimal Approach, under a Creative Commons by-nc-sa license.
    http://www.math.wisc.edu/~keisler/calc.html
    The book provides a highly approachable explanation of non-standard analysis.)

    Non-standard analysis (will be a funny name, if at some point in the future, it becomes standard) defines a range of positive numbers that are greater than zero and less than any positive real number. The additive inverse gives a negative range. And the inverses give infinite results.

    I prefer this method for dealing with infinity because, at least to me, it does not seem reasonable to think of 1 divided by zero as positive or negative or even non zero.

    To me, defining division by zero as undefined is not a bug. I think the bug is believing that we know nothing well enough assume that dividing by it should be defined (of course, if we do some day come do know nothing, then we may feel free to divide by it).

    So for all practical intents and purposes, I like to use the inverse of positive and negative infinitesimal numbers to represent infinity because I know the sign of their inverses. (I don’t like taking the integral between negative and positive infinity because I don’t think it’s reasonable to say that infinity has a sign or non zero value).

    On the other hand, I do think it may be reasonable to define zero divided by zero as 1, provided that the “1″ thus generated is given a universe of sets not compared with other divisions of zero by zero.

    (Note: I am not familiar with set theory, so please correct and forgive any errors in my use of terminology. Specifically, I’m the concept of Universe that I learned from Lewis Carrol [Dodgson], his Symbolic Logic).

    That is 0/0 of set universe A = 1 of set universe B, but 1_a does not equal 1_b. And 0_b/0_b = 1_c which does not equal 1_a or 1_b, or rather that 1_a = 1_b is one possibility out of an infinite set of roughly equivalent possibilities and therefore infinitely improbable.

  15. I was taught maths in the old-fashioned way. Mental arithmetic tests were handed out on a Thursday afternoon. There were 35 questions, and we marked our neighbour’s paper. Then we stood on our chairs.

    The headteacher came round.

    ‘Sit down if you got 35.’ Some children sat.

    ‘Well done’, she said. Sit down if you got 34.’

    You get the idea.

    When only a few of us remained standing on our chairs, say at 6 out of 35, the headteacher smacked us on the backs of our bare legs until we cried.

    I still hate Thursdays.

    The problem came for me when I thought I’d understood.

    On the bus home from school when I was about 7, I had a conversation a bit like this:

    Keith: When you times a number by nought, why does it equal nought?
    Me: Because timesing two by nought is the same as timesing nought by two. Two nothings are nothing.
    Keith: But if you times two by nothing you’re not timesing it by anything, so it’s still two.
    Me: Look, if you’ve got one nothing, or two nothings, or a squidrillion nothings, you’ve still got nothing.
    Keith: But if you times a number by nothing, you’re not timesing it at all, so it stays the same.
    Me: You smell.
    Keith: You smell of wee.
    Me: You smell of poo.

    I never really did get maths, and as I had consistently high marks in English, languages and music, I was able to avoid maths-based subjects.

    I still don’t understand why -x * -y is positive. 2 minuses make a plus because (minus sign) plus (minus sign on its side) = + was all the explanation I had.

    Your site gives me hope that I may eventually understand the subject.

    I got better at arguing, and my vocabulary increased considerably over the next 53 years.

    But Keith really did smell.

  16. Not sure if anyone has pointed this out, but I think the sentence “Luckily, irrationals are at least algebraic” should be “luckily, *some* irrationals are algebraic” since the set of algebraic numbers is countable and thus has lebesgue measure zero in the reals, meaning “almost all” real numbers are *not* algebraic. Instead they are called trancendental. This is yet another bizarre property of our familiar real number system, I just figured the author would want to note.

  17. @Jon: Thank you for the clarification! I’ll amend the post. As you mention, our number system can be really, really bizarre :).

  18. I believe the Romans used a simpler method to multiply two numbers a and b: halve a and double b until a reaches 1; then add the values of b where a was odd. Using your example (a=IX, b=XXXIV):
    odd IX XXXIV
    even IV LXVIII
    even II CXXXVI
    odd I CCLXXII
    then add XXXIV and CCLXXII to get CCCVI. Simples!
    If you examine the details, it is a binary-shift method used by digital computers, and works in any base including decimal.

  19. I like that 1/3 paradox in base 10 but in base 3 it’s fine. But again you could use a similar argument against base three saying 0.1111111… in base three is 0.5 in base 10.
    Loved the points you brought up. It’s really got me thinking about something I kinda always thought was set in stone.

    Great post and website!!

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