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.

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/a-quirky-introduction-to-number-systems/

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 =

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

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).

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”.

Roman numerals have a symbiotic relationship with the abacus .

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, 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:
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.

There is a very nioe write up on why Euler needed to invent “i” here - http://resonanceswavesandfields.blogspot.com/2007/08/eulers-equation-and-complex-numbers.html

@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!

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

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.

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.

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 yy^(-1)=1. Now, 0 can not possibly have an multiplicative inverse under any circumstance, since 0anything=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!

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.

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.

[…] Numbers: number systems, visual arithmetic, different bases, Prime numbers […]

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.

@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.

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.
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.

What is math

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.