Numbers were invented for a reason -- they help us solve real-world problems. Natural numbers were needed for simple counting -- the earliest form of maths. You need them to answer questions like “how many deer in the herd”, and “how many people went out on the lion hunt” and “how many people returned from it”.
Integers expand the scope of the problems you can solve. We can now answer questions like “There were seven apples in the basket. John ate three. How many remain?”
Rationals expand the problems we can find answers to. For instance, if Iannis has 3 acres of land, and 5 children, how does he divide the land equally among them? If $x$ is the number of acres each child gets, we have $5x = 3$, leading to the rational solution: $x = \frac{3}{5}$.
However, even rationals are not sufficient to solve all problems. Consider the following question: “Peter has a square field. Both sides are exactly one league long. He wants to build a fence diagonally across the field. How long must the fence be?”
You will recognize the above as an instance of the equation: $x^2 = 2$. It is easy to show that $x$ can never be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers.
What does it mean to say that $x^2 = 2$, for a rational number $x \in \rationals$? Quite literally, it means that there exists an ordered pair of integers $(p,q)$, $q \neq 0$, such that $p^2 = 2q^2$.
Proof
We will employ proof by contradiction. To do so, however, we must build up the proof in steps.
This is easy to prove by contradiction. Assume that the above statement is false, and that there exists a rational number $x$ such that it cannot be written as $\frac{p}{q}$ where $p$ and $q$ are mutually prime. Let $x = \frac{m}{n}$ be one of the representations of $x$. By our assumption, $GCG(m,n) > 1$.
Let $k = GCD(m,n)$. By our assumption, $k > 1$. So we can write $m = kr$ and $n = ks$, where $r$ and $s$ have no further common divisors other than 1, i.e. they are mutually prime. We claim that $\frac{m}{n} = \frac{r}{s}$. This is easily verified. For the equality to hold true, we need $ms = nr$. But we know that $m = kr$ and $n = ks$. Replacing $m$ and $n$ in the equation by $kr$ and $ks$, we find this to be trivially true. Thus we have found a rational representation $x = \frac{r}{s}$ such that the numerator and denominator have no common divisor, contradicting our original assumption that $x$ does not have such a representation.
Ergo, our assumption can never be true, so the above lemma holds.
Again, we will use proof by contradiction. Assume the above statement is incorrect, and that $x^2 = 2y$ for some integer $y$. This implies that $x^2$ is even.
Since $x$ is odd, it can be written as $x = 2z + 1$, where $z$ is some integer. Using this definition $x^2 = 4z^2 + 2z + 1 = 2(2z^2 + z) + 1$. Since $2z^2 + z$ is also an integer, $x^2$ is odd. This contradicts our assumption.
Hence, the above lemma is correct.
This follows directly from the earlier lemma.
Now return to our original problem of proving that $x^2 = 2$ has no rational solution.
Assume that there exists a rational number $x$ such that $x^2 = 2$. Let $x = \frac{p}{q}$ where $p$ and $q$ are mutually prime, and have no common divisors greater than 1. This leads to the following conclusion: \[ \left(\frac{p}{q}\right)^2 = 2\\ \Rightarrow \frac{p^2}{q^2} = 2 \\ \Rightarrow p^2 = 2 q^2 \]
Implying that $p^2$ is even, and hence $p$ is even.
Since $p$ is even, $p = 2r$ for some integer $r$. This means that \[ (2r)^2 = 2q^2 \\ \Rightarrow 4r^2 = 2q^2 \\ \Rightarrow 2r^2 = q^2 \]
Implying that $q$ is even and can be expressed as $2s$ for some integer $s$.
But then, this implies that both $p$ and $q$ have a common divisor, $2$, contradicting our original assuption that $p$ and $q$ have no common divisor. This in turn means that we cannot have an $x = \frac{p}{q}$ such that $x^2 = 2$.
The fact that there is no rational number $x$ such that $x^2 = 2$ was first discovered by the Pythagoreans in the 6th century BC. The discovery shocked them -- they had firmly believed until then that all numbers were rationals.
They Pythagoreans decided that the fact that such numbers -- which could not be expressed as ratios of integers -- existed was dangerous knowledge, and should not be shared with the common public.
Unfortunately Hippasus of Metapontum could not keep the secret to himself. He revealed the secret to the public.
So, to keep the secret from spreading further, the Pythagoreans murdered Hippasus by drowning him in the Aegean sea.
In general, an irrational number is a number that cannot be expressed as a ratio of integers.