Introduction Irrational Numbers

In mathematics we have different names for different types of collections of numbers. There are reasons as to why we have these, a big factor is historical considerations. In the beginning, people thought that the numbers 1, 2, 3, … all the way to infinity were all the numbers we had. But soon enough we discovered many exotic types of numbers, such as negative ones or even irrational numbers. Let’s see what these are all about. We actually need to know all of them before we are able to define irrational numbers.

The Natural Numbers

Let us start with the easiest example, and this is called the natural numbers. This collection is just the numbers 1, 2, 3, … al the way up to infinity. So 10.000.000 is an example of a natural number, but 4/3 is not, and so all other fractions and so on. some authors include 0 in this set of natural numbers, and some do not. It is just based on convention. The way we denote this set of numbers is with the symbol \mathbb{N} which is called the blackboard N.

Integers and Rationals

Now, but of course you know that we have more numbers than just these whole numbers. What about all the negative numbers for example? Well we can include them by expanding this set of numbers, by adding all the numbers on the left side. Then we get the numbers … -3, -2, -1, 0, 1, 2, 3… . This set of number is denoted by a \mathbb{Z} . But then there are also numbers in between these whole numbers. For example the numbers 1/2, or -3/4 or 0,125. These are called the rational numbers.

So, a rational number is any number that can be written as a fraction. More formally, it can be written as a \frac { p }{ q }, where p is called the numerator and q the denominator. Also note that q cannot be zero, because division by zero is not possible. Check out an upcoming post and YouTube video of why we can’t do that! The set of all rational numbers is denoted by a \mathbb{Q}.

Note that the denominator can be 1. So if we for example have a the number 2/1, we simply get the number 2, which is a natural number, or an integer. So we can get all the natural numbers by taking these fractions that divide by one. We can also get all the integers by dividing by one but adding negative numbers on the top as well. So we can clearly see that the set of rational numbers is bigger than the set of integers which is also bigger than the set of natural numbers.

In set-theoretical terms, we say that \mathbb{Q} contains \mathbb{Z} and that \mathbb{Z} contains \mathbb{N}. If this is the case, then \mathbb{Q} also contains \mathbb{N}. In other words, you contain your brain, and your brain contains braincells, so you contain braincells. But it is not the other way around. Your braincells do not contain you.


Before we define what an irrational number is, I want to skip this for one second and define on other thing: real numbers. This set \mathbb{R} contains basically all the numbers you can think of. All the integers and fractions are included, but also all other numbers with infinite options behind the decimal point. Numbers such as 0.999999999… or 3.1415…, or 3.12076547328 and so on.

Irrational Numbers

Okay, now we are ready to define what an irraitonal number is. An irrational number, is a real number which is not a rational number. In other words, it is a comma number which cannot be written as a fraction. That is pretty crazy right! No matter what we do, some numbers are just so weird that they cannot be written as a fraction. We can write most numbers as a fraction. For example, 1 is just 1/1 and -1 is -1/1. Also 0.5 is just 1/2, and 1,5416666…. Is 37/24 and 0.07142857142857… = 3/42. Since we have infinite numbers we can put as the numerator, and infinite numbers we can put as the denominator, we should be able to approach basically any comma number we would like. But there are certain numbers that just won’t allow this.

One famous example of a number that cannot be written as a fraction is \sqrt { 2 }. See the proof of this, and a bit of history about this special number in this post:

Or, in this YouTube Video.

Further reading:


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