In this post I intend to tell a concise and perfected story of how and why numbers came to be. Please don't expect this to be a factual account of all the trial and tribulations that accompanied the process. So, for instance, I won't dwell in all the misconceptions that plagued negative numbers for a long time, nor will I tell in a detailed way how complex numbers came to be (even though I'm saving this bit for a later post).
Here the story will have no false starts, no dead-ends and everything will flow perfectly and rationally.
— 1. Operations —
Let us first introduce the concept of an operation. For us an operation will be a process that will allows to act on a set of numbers to obtain another number:
- Addition, which is represented by the symbol $ {+} $, as an operation that picks up two numbers and obtains a third one. In symbols: $ {a+b=c} $.
- Subtraction is defined as the inverse operation to addition and is represented by the symbol $ {-} $. So, we'll say that $ {c-b=a} $ if it is $ {a+b=c} $.
- Multiplication is represented by the symbol $ {\times} $ or $ {\cdot} $ and also associates two numbers to a third one: $ {a \times b = c} $.
- Division is the inverse operation to multiplication and is represented by the symbol $ {/} $. So we'll say that $ {c/b=a} $ if it is $ { c = a \times b} $. Another way of representing $ {c/b} $ is by the symbol $ {\dfrac{c}{b}} $.
Fortunately for us as human beings these are not the only operations available but for now they are all that we need.
Some readers may be taken aback with the level of abstraction that I used and with the fact that we still don't know what these numbers really are. To those readers I say that in Mathematics this is the normal way to present things. Yes, I'll do things in a more digestible way from now on (cringe worthy to mathematicians I'd assume) but nevertheless I wanted to show how things work out in the real world.
— 2. Numbers —
Well, on to numbers we go. The first numbers that we need are what called the natural numbers. The symbol to denote them is $ { \mathbb{N} } $ and they appeared and evolved from the need to count how many objects of a given class we possess:
$ \displaystyle 0,1,2,3,4,5,6,\ldots $
Where the symbols $ { \ldots } $ mean that that list doesn't have an end. Note that I included the number $ {0} $ in this list. The history of the number $ {0} $ is a fascinating one and I urge the reader to find out a little bit about it.
— 2.1. Operations with Natural numbers —
In order to break down the level of abstraction that we had while introducing mathematical operations we'll now see what they look like for natural numbers.
- Addition. The first thing I want to say about addition in the special case of natural numbers is that we can see it like this:
$ \displaystyle m+n=p $
$ {m} $ is the number from where we start and $ {n} $ is the number of steps we'll walk till we stop in point $ {p} $
$ {0} $ is giving no steps at all, $ {1} $ is what we define as giving one step, and the natural numbers that follow are defined in relationship to $ {1} $. For instance: we define $ {2=1+1} $ and so we interpret $ {n+2} $ as giving two steps starting from point $ {n} $. All the natural numbers that follow have the definitions and interpretations that you expect them to have.
- Subtraction is the inverse operation to addition. We can see addition, $ {m+n} $ as starting from $ {m} $ and giving $ {n} $ steps to the right, thus $ {m-n} $ can be seen as starting from $ {m} $ and giving $ {n} $ steps to the left.
- Multiplication is defined to be consecutive additions. By that we mean that we interpret $ { 2 \times 3} $ as being $ {2+2+2} $ and $ {3 \times 2} $ as being $ {3+3} $. As we can see it is $ {2 \times 3 = 3 \times 2 = 6} $. In general we can say that for every natural number it is $ {m \times n = n \times m = p} $
- Division is the inverse operation to multiplication and thus can be seen as consecutive subtractions.
For example let's compute $ {6/2} $.
It is $ {6-2=4} $. Now we take the result and go subtracting $ {2} $ until we can.
$ {4-2=2} $, and $ {2-2=0} $. Since we were allowed to make three subtractions and the end result was $ {0} $ we say that $ {6/2=3} $ with remainder $ {0} $.
Another example is $ {7/3} $. This time it is $ {7-3=4} $, $ {4-3=1} $ and here we have to stop. This time we made two subtractions and ended up with $ {1} $. So what we say is seven divided by three is two with remainder one.
The beautiful thing about numbers and these four operations is that now a whole new realm of nifty and useful things pop out almost for free. All that we need to do is to think and to be ambitious.
Let us analyze with a more watchful eye to the inverse operations and let's see what we can do with them.
— 3. Augmenting the available numbers —
— 3.1. Negative numbers —
First off let s take a look at subtraction and the numbers we already have. The numbers are $ { \mathbb{N} = \left\lbrace 0,1,2,3,4,\ldots \right\rbrace } $. For instance $ {7-2=5} $ and $ {10000-1000=9000} $. This notion of inverse operation to addition seems to work really well. But we don't need to go very far to encounter some possible problems $ {3-7} $. In our current number system we can't go seven steps to the left starting in three. But naturally we'd like for subtraction to work all the time and so we'll take matter into our own hands.
We'll just increase the numbers available to us. We define $ {-1=0-1} $ and $ {-2=-1-1} $, $ {-3=-1-1-1} $, and so on... Now subtraction is always possible. For instance the once problematic $ {3-7} $ just equals $ {-4} $ and the new number system that we end up with is $ { \mathbb{Z}=\ldots,-3,-2,-1,0,1,2,3,\ldots } $.
Besides providing for a way to subtraction being always possible negative numbers can also represent debts, heights below a reference level, etc.
— 3.2. Rational numbers —
After being provided with a whole new set of numbers via subtraction we will now construct a new subset of numbers in order to division always have a meaningful result.
Now the problematic cases are $ { \dfrac{m}{n} } $ with $ {n>m} $. Imagine that you had seven breads to divide for 10 people. How would you do that? If you limited yourself to numbers in $ {\mathbb{Z}} $ you couldn't share the bread. Hence to make division always possible (and remember that this problem certainly doesn't lack practical motivation) we have to further increase the scope of the numbers that are available to us. These new numbers will be called fractional numbers. To the set that is formed by the reunion of the set of fractional numbers and the set of integer numbers ($ { \mathbb{Z} } $) we'll give the name of rational numbers $ { \mathbb{Q} } $.
So for instance imagine that you have one bread and you want to share it between you and two friends. You just break the bread in three equal parts and there you go. The number that represents this process is $ {\frac{1}{3}} $. One of the disconcerting things about fractional numbers is that they allow an infinite number of representations:
$ \displaystyle \frac{1}{3} = \frac{2}{6}= \frac{3}{9} = \ldots $
This problem is circumvented with the notion of equivalent classes, but in a more pedestrian way just think about sharing breads and you'll see why those fractions do have the same value.
— 4. Catching up —
At this point we can be very happy about ourselves. We started off with just the ordinary natural numbers, the addition operation, the notion of inverse operation, some brain power and a lot of ambition.
After deciding what addition meant in the context of the natural numbers we introduced the notion of subtraction as being the inverse operation to addition. Multiplication was defined as being a form of shorthand to repeated additions. Imagine that you had to add fifty times the number twenty. Instead of writing $ {20+20+20+20+\ldots+20} $ one just writes $ {20 \times 50} $ and everything is said.
After defining multiplication in this way once again we wanted to know what the inverse operation would look like. We defined division as being consecutive subtractions and so once again an operation appeared as a short hand for an already known operation.
After having arrived to these four operations (starting from just one) we allowed ourselves to apply them to the natural numbers without any reservations. In that way we came to the conclusion that either the numbers we were working with weren't enough or that we had to put limits in to the applications of the inverse operations. The second option wasn't compatible with our principle of ambition so we had to extend the amount of numbers in our disposal:
$ \displaystyle \mathbb{N} \rightarrow \mathbb{Z} \rightarrow \mathbb{Q} $
— Glimpses of a near future —
In the next post we'll continue in our search of coherence and wider scope of application of inverse operations. Powers and radicals will be introduced and we'll see that the operation of extracting radicals forces us to once again widen the scope of our numbers, and this time not once, but twice!