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We spend a lot of time in school mathematics learning how to work with numbers, but what are they? What is a number and how can we construct our collection of these very abstract things that we use every day in conversation and planning?

a more formal way to define numbers

We use them along with arithmetic like `times` and `+` to help us describe and understand many properties of the world we observe around us. An astonishing variety of very different quantities we measure behave like numbers and have important properties derived using arithmetic. In physics we measure distance, time, mass and electrical charge and from them calculate properties like position, area, volume, speed, acceleration, force, pressure, temperature, density and energy to build a model of the way physical things interact.

We use numbers for momentum, angle, probability and anything we can count as well as all kinds of properties we are interested in analysing, talking about and making predictions about that we can measure, put in an order or assign meaningful values to. Not only science and technology but also design, business, planning and management rely on mathematical models to use and understand the systems they are engaged with. Essentially numbers are part of understanding, describing and predicting how many kinds of things behave.

The view of numbers described here is rather more formal than is usual in school, but it can be a helpful way to think about these things for some students. There are several different approaches to this kind of exploration of what lies behind mathematics. Here we consider numbers as a mathematical abstraction, a way of describing and working with both quantities and the order of things. We count and measure things, and we use algebra to reason about those things using numbers. Some of the words I will use here are defined and explained first. The ideas are more important than the words, but you will need to know some of the words to understand the explanations, or the questions you could be asked — and to communicate about these ideas.

some important ideas first ...

commutative means you can swap the order, that is: `qquad2times3=3times2`.
It works for `times` and for `+` (these, strictly, only have a meaning for pairs of numbers)

but `-` and `div` are not commutative:
`quad 2-3 quad`is the negative of`quad 3-2 quad`and
`quad 2div3 quad`is the reciprocal of`quad 3div2`

associative means you can group addition however you want: `qquad(2+3)+4 = 2+(3+4)`.

This applies to `times` as well, together these rules mean that you can rearrange `+` or `times` operations.

If you have `-` or `div` then it is not so simple, they are not commutative.

It does not work when operations `times` and `+` are mixed together: `quad(2+3)times4 ne 2+(3times4).quad` This means we need to make up a rule to tell us what we mean when we write something like `4+6times5^2`…

Order of operations in arithmetic:
do the brackets first;
then any powers;
then multiply and divide;
then finally plus and minus.

distributive tells how `+` and `times` work together, `quad 2times(3+4)=(2times3)+(2times4)=14`. This looks much neater (and is easier to read and understand) in our algebra notation where we do not use the `times` symbol… \[\large a(b+c)=ab+ac.\]

Saying that counting numbers are closed under addition means that whenever you add two counting numbers the result is also a counting number. If we want numbers to be “closed under division” we need to include fractions as well. Then if we want numbers to be “closed under subtraction” we need our numbers to include zero and the negative numbers. Understanding zero as a number, and the particular problems this brings up, was difficult. It was the last part of this puzzle to be solved, thousands of years after fractions, powers, roots and negatives were widely used.

Number Algebra is the symbolic language that we use to write down and work with ‘facts’ we are given, or ‘facts’ we want to know, about numbers and things we can measure or count that behave like numbers.

We use letters to represent the values we are interested in but do not know yet, the ‘unknowns’ in our problem, the values we are trying to find.

We write algebra a little differently compared to arithmetic. We do not usually write the `times` or `div` symbols. We always use a fraction-bar instead of `div` and we write numbers and letters next to each other to mean multiplication, so that: \begin{align*} \phantom{\frac12}42ab\quad\text{means} \quad&42 \times a \times b,\\ \text{and: } \quad&\\ \frac{4x}{yz}\quad\text{means} \quad&(4 \times x)\div(y \times z). \end{align*} We call these groups of symbols a term.

→

The collection of numbers found by starting with one, then adding, then dividing, then subtracting we call the Rational Numbers. The Hindu mathematicians put these ideas together in a formal, mathematical way during the time Europeans call the Dark Ages, from about 650AD. Their analysis was soon translated into Arabic. In the 1500s the Persian and Islamic mathematics that followed was translated and published in Europe, including much of our basic algebra and the algorithms we use for arithmetic. Those words come from the Arabic title and Persian author of one of those books. The modern symbols for the ten digits are originally from the Hindu written script of that time.

This is a rather important idea, we only just touch on it a little in high school — but mathematicians keep creating new kinds of entities like these to talk about new kinds of models of things that we observe in the world around us. We develop the algebras that work with these new collections … what we are learning now is Number Algebra, and we look at Set Algebra also, manipulating sets and their elements using operations including union and intersection with relations like in. At school we also explore Vector Algebra.

What is a number?

We can start by deciding that “one is a number”

… now let’s make some more!

First we learn how to count.

We give the name two to the number after one, three to the number after two and make up a way of giving names to every next number. For example after ninety nine we call the next number one hundred. So we end up with an endless list of number-names, in a fixed order.

I will call this first collection of named numbers the Counting Numbers. Sometimes we call them Natural Numbers.

I am going to use some words that I explained above.

  • we could find all the counting numbers by starting with one, and adding.
    • addition of counting numbers can be seen as repeated 'next’ number operations
    • that is: adding them is done by counting along the sequence of number-names
      • the first thing we learn (or define, or teach) about numbers is their names, in order
      • then we learn how to count a collection of things by matching each one to a number-name
        • How many?” is answered by the name of the last counting number matched
      • then we learn to add two numbers this way, perhaps using our fingers as a tally
      • that is how we started our journey into mathematics!
  • addition is commutative, associative and closed
    • this means that we can rearrange any sequence of additions
    • and that the addition of numbers always gives a number
  • we can multiply two numbers, multiplication by a counting number is repeated addition
    • for example we learn to “count by threes”, using our fingers to multiply three by a number, keeping a tally of how many times we add `3`.
    • multiplication is also commutative, associative and closed
      • that means that you can rearrange products (reorder the number and letters within a term)
      • multiplication of numbers always gives a number, we say numbers are “closed under multiplication
    • multiplying any number by one leaves it unchanged, so we call one the “multiplicative identity
      • hence we can leave out the `1` in any term — it is always implied
  • multiplication is distributive over addition
    • certainly not the other way round!
      • this is the motivation for the conventional order of operations in arithmetic
        • in arithmetic we use brackets to change (or emphasise) the order of operations
      • and why we write equations with `+` and `-` signs between terms (rather than using `times` symbols)
    • hence we ‘collect like terms’ by adding their number-parts
    • hence we multiply an expression by multiplying every (top level) term of the expression.
  • the reciprocal, or multiplicative inverse, of a number is the number you multiply it by to get one
    • or, the same thing if you already can divide: one divided by a number is its reciprocal
    • we use this to do division, we multiply by the reciprocal to divide.
    • we need fractions to make a set of numbers (excluding zero) closed under division

      Fractions (that is: division and ratio) transform our number system from counting distinct things into one able to measure continuous properties like length, time, weight or probability.

      This is a very big shift, we start to use this new and very different kind of number in primary school, but it takes a few years to start to really understand that difference.

    • we can find all the fractions by dividing pairs of counting numbers (then including their negatives)
    • the fraction bar in algebra combines brackets and dividing
  • we decide that zero is a number (this might be very familiar, but it is not at all trivial or obvious!)
    • adding zero (the additive identity) to any number leaves it unchanged
  • the negative, or additive inverse, of a number is the number you add to it to get zero
    • or, put the other way round: a number subtracted from zero is its negative
    • thus we define subtraction: we add the negative to subtract.
    • put another way: zero is any number subtracted from itself
    • we need zero and negative numbers to make our set of numbers closed under subtraction
  • now we have our new set of numbers — zero, the Counting Numbers, fractions and negatives — we call them the Rational Numbers. We can add, subtract, multiply and (almost always) divide these numbers to get another as the result.
  • multiplication of any number by zero gives zero
    • which means that zero does not have a reciprocal
    • so we cannot divide by zero!
    • in algebra we can leave out any term with `0` as its number-part, or imagine one is there.

These properties, these ‘facts’ about numbers, are in the right hand column in the more formal, abstract representation here.

We use these rules to build our Number Algebra.

more numbers

Mathematicians extend the idea of number, and apply it in all sorts of practical or abstract models and explorations.

One very important example is described here.

there are important non-numbers

When describing numbers above we saw that dividing by zero was not possible, it did not make sense. Yet in other cases like that we pressed on, we made new numbers to answer the questions the old ones could not and so made our collections of numbers more complete. We cannot find any solution to \(\ x^2=-1\ \) in the Real Numbers, so we name a new number that is the solution and give it a name: \(\rm i\). It is not a Real Number of course but we find that a whole new kind of number emerges. We call them Complex Numbers and find they do behave very nicely as numbers — following all the rules we talked about here. We also discover many areas where they are very useful, we find many systems where values are naturally expressed as Complex Numbers.

Perhaps, then, we should simply define a new number, call it \(\large\infty\) and extend our number system even further. The problem is that it does not play by the number rules. If we include it as a number we lose much of what makes numbers so useful and powerful, we cannot use it as a value within our algebra without extra care and work.

This is an ancient problem, discussed here. The ways we deal with it are very important and we get started with this mathematics in school. Beyond school we dig much more deeply (and into many other kinds of abstractions and generalisations as well). The mathematics we call “Advanced” touches on this Infinitessimal Calculus in the last year of school, in our “Extension” courses we get quite thoroughly into basic, 1-variable calculus — especially as it applies to rates of change in continuous quantities. This was the huge advance in mathematical thinking, physics and analysis that was developed over the 1600s by a few generations of mathematicians including many very famous names — Keppler, Descarte, Pascal, Galilleo, Leibniz and Newton are probably the best known. This application of calculus to understanding how physical objects move is called dynamics.

teaching/topics/number/axioms.txt · Last modified: 2024/05/02 13:04 by simon