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First steps in the wonderful world of Numbers
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Someone recently asked to me why rational numbers are called "rational" and whether there are some numbers that are not rational.


The short answers to those questions are:
1. Because they can be written as ratio (quotient) of two integers.
2. Yes.

... but while I am at it, why not say a bit more about (real and complex) numbers.

The most straightforward numbers in the universe are the natural integers. Natural integers are number such as 0, 1, 2, 3 etc... They are rather easy to understand, and they are naturally used to count things. For instance there are currently 3 people living in my (shared) house. Something very important about natural integers (probably one of the most important thing ever in mathematics) is that the set of natural integers is infinite. In other words, the sequence 0, 1, 2, 3, etc.... never ends. I said that they were straightforward, actually that is not entirely true. All of them but a particular one are. The number 0 has waited long before being considered (being discovered) as a proper number during the history of mankind. It represents "nothing", and in the past it wasn't obvious for people that "nothing" could be represented by "something".

It is not that people in the past were stupid, it was just that some concepts were not very obvious to them, that's all. I am sure that in our future, some people will find funny that in the 20th/21st centuries, people still believed in God and that Evolution wasn't obvious for everyone.

After the natural integers, you have the integers. Integers are like the natural integer, but you can use them together with a sign. For instance -4 is an integer, integers are the numbers ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, etc...
As you can see it goes for ever in both directions.

The world would be a perfect place if we only needed integers, but sometimes we need something more. Let us say for instance that I have to equally share 5 cakes between 3 people. I need a number to represent how much of cake I will give to everyone of them. Note that I wrote "how much of cake" and not "how many cakes", the difference between those two formulations is a crucial point here.
Most people will figure out that I have to give to each of them 5/3 of a cake. 5/3 is the answer to the problem because 5/3 + 5/3 +5/3 = 5. (If you are not sure about that, don't worry). The problem here is that "5/3" doesn't look like an integer. Actually it is not. 5/3 is a new kind of number. A number between 1 and 2 (closer to 2 than to 1 by the way). 5/3 is what is called a rational number. It is a number that can be written as a ratio (quotient) of two integers.

Before moving ahead, we can notice that integers are rational numbers. Let us take 7 for instance. We have 7=7/1.
7 can be written as a ratio of two integers, so 7 is also a rational number.

5/3 is a number which can also be written 1.66666666... This way of writing it is not very good actually. Why ? Because most numbers, when we write them using theirdecimal development (that's the name of writing a number with all those digits after the dot) in order to write them correctly we would have to write down all the digits. We obviously can't. In the case of 5/3, I wrote "1.66666...". The last 3 dots "..." mean "and you continue for ever writing 6". If you try with you pocket calculator (or even a powerful computer), you will have "1.66666666666667". As you can see, machines are not very good at representing numbers using their decimal development.

In our everyday life, we are nevertheless used to numbers written down with their decimal development. For some reasons we understand them better, even if we never write them with all their digits. For instance, if I say to someone "Do you want 2 cakes or 5/3 of cake ?", this person will wonder "hum.....". But if I say "Do you want 2 cakes or 1.66666... cakes ?", this person will answer very quickly "2 cakes, please."; because one sees quicker that 2 is better than 1.666666666

We have seen that 5/3=1.666666666666..., one might be tempted to think that, after all, all numbers can be written as rational numbers. Unfortunately, this is not true. We can state this more correctly

Result: Some (real) numbers are not rational.

Now I know that you are thrilled to see one of them. That's easy. Haven't you ever seen the number 1.41421356... Yes, you have, you just don't remember. This number is sqrt(2). This is the number such that when you multiply it by itself you get 2. In other words sqrt(2)*sqrt(2)=2

There are at least 2 interesting things about this number
1. The proof of the fact that it is not a rational number is not difficult (but I won't do it as it is a bit subtle and I don't want to lose those of you who accepted to read me as far as here).
2. More than 2000 years ago, some people close to a guy called PYTHAGORAS discovered that sqrt(2) is not a rational number. This discovery has been kept secret for a long time, because the view of the world that they had at that time didn't allow non rational numbers to exist. It was as if tomorrow someone comes up with a perfect proof of the fact that Evolution did happen and that Earth is not 6000 years old (as more or less stated in the Bible). Some deep beliefs of some people will be destroy, and most of them are not prepared to this.

Coming back to numbers, when a number is not rational, we usually say that it is irrational.

One of my favorite mathematical proof is the fact that the number \pi is not rational. Even though the fact that sqrt(2) is not rational can be understood by anyone, the fact that \pi is not rational is a bit more subtle and requires at least a university first year in maths.

Irrational numbers are traditionally cut in two parts: algebraic numbers and transcendent numbers.

Algebraic numbers are the numbers that are solution of polynomial equations. Transcendent numbers are the others. For instance, even if sqrt(2) is not a rational number, sqrt(2) is nevertheless solution of the equation x^2 - 2 = 0. So sqrt(2) is an algebraic number. \pi is not a solution of a polynomial equation, so \pi is not an algebraic numbers, ie \pi is a transcendent number. The proof that \pi is a transcendent numbers is very difficult. I don't know the proof. But I guess that it would take me about two weeks to understand it and learn it by heart.

I remember that when I was a kid I once asked to the teacher why he used to always say "real numbers" instead of "numbers". I claimed that he could say "numbers" because they were all the numbers anyway. He then calmly replied that I was wrong and that there are some numbers that are not real.

And I was like "what!?!?". Real numbers are all the numbers that we can write (of think of) a decimal development for. How could a number not have a decimal development, even an infinite one....

In High School we've been told about the complex numbers. I then understood the answer of the teacher and the fact that a world of concepts was ahead, but we I started my math studies at the university I discovered that what I knew so far was just a very single page (the first page) of a very long and complex story... In particular I met the quaternions (and their friend the hypercomplex numbers). And then the Octonion (the kind of number which exists in an algebra of dimension 8).

After the Octonion, well, nothing. We know that 8 is the maximum number of dimensions that an algebra over the field or real numbers may ever have.

This said, (lots of) other number systems exist ...
... but they are built on different logics than the ones we are customized to in our everyday lifes. Most of them are easy to understand though.

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