Irrational Numbers

Irrational numbers are real numbers that neither repeat nor terminate in decimal form. This includes pi and the square roots of any prime numbers.

Examples
Here is a small list of irrational numbers.

$$\pi\,$$

$$\sqrt{5}\,$$

$$0.101001000100001000001000000100000001000000001000000000100000000001...$$

$$\sqrt{13}\,$$

$$e$$

$$\sqrt{\pi\,}\,$$

$$\sqrt{19}\,$$

Why You Can't Write Irrational Numbers As Fractions In Simplest Form
We all know we can't write irrational numbers as fractions in simplest form. But how do we know?

Allow me to demonstrate.

For the time being, let's assume that I can write out the square root of 2 as a fraction in simplest form. Since I don't know the values of the numerator and denominator, it will be set to a over b.

$$\sqrt{2}\, = \frac{a}{b}\,$$

If we play around with it a little, we end up with 2b2 is equal to a2.

$$\sqrt{2}\, = \frac{a}{b}\,$$

$$2 = \frac{a^2}{b^2}\,$$

$$2b^2 = \frac{a^2}{b^2}\,b^2$$

$$2b^2 = a^2$$

With this, we can say that a2 is even. Hence, a is an even number. Since a is even, let's set a equal to 2c.

$$2b^2 = a^2$$

$$2b^2 = (2c)^2$$

$$2b^2 = 4c^2$$

$$\frac{2b^2}{2}\, = \frac{4c^2}{2}\,$$

$$b^2 = 2c^2$$

With this, we can say that b2 is even. Hence, b is an even number.

But that's the problem. I've just proven a and b are both even numbers. Any fraction with the numerator and denominator being both even numbers is not in simplest form.

$$\frac{2}{4}\, = \frac{1}{2}\,$$

$$\frac{4}{16}\, = \frac{1}{4}\,$$

$$\frac{16}{34}\, = \frac{8}{17}\,$$

$$\frac{8}{2}\, = \frac{4}{1}\, = 4$$

$$\frac{32}{6}\, = \frac{16}{3}\,$$

$$\frac{54}{80}\, = \frac{27}{40}\,$$

$$\frac{16}{48}\, = \frac{1}{3}\,$$

Note how all fractions in simplest form have at least one odd number? The fraction I tried making of the square root of 2 has no odd numbers! Therefore...

$$\sqrt{2}\, = \frac{a}{b}\,$$ is an impossible fraction!