# Irrational Numbers

An irrational number is a real number that cannot be expressed in the form $\frac{a}{b}$ , when $a$ and $b$ are integers ( $b\ne 0$ ). In decimal form, it never terminates (ends) or repeats.

The ancient Greeks discovered that not all numbers are rational ; there are equations that cannot be solved using ratios of integers.

The first such equation to be studied was $2={x}^{2}$ . What number times itself equals $2$ ?

$\sqrt{2}$ is about $1.414$ , because ${1.414}^{2}=1.999396$ , which is close to $2$ . But you'll never hit exactly by squaring a fraction (or terminating decimal ). The square root of $2$ is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:

$\sqrt{2}=\mathrm{1.41421356237309...}$

**
Historical Note:
**

According to legend, the ancient Greek mathematician who proved that $\sqrt{2}$ could NOT be written as a ratio of integers $\frac{p}{q}$ made his colleagues so angry that they threw him off a boat and drowned him!

Other famous irrational numbers are
**
the golden ratio
**
, a number with great importance to biology:

$\frac{-1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{5}}{2}=\mathrm{0.61803398874989...}$

*
$\pi $
*
(pi)
, the ratio of the circumference of a circle to its diameter:

$\pi =\mathrm{3.14159265358979...}$

and
*
$e$
*
, the
most important number in calculus
:

$e=\mathrm{2.71828182845904...}$

Irrational numbers can be further subdivided into
**
algebraic
**
numbers, which are the solutions of some polynomial equation (like
$\sqrt{2}$
and the golden ratio), and
**
transcendental
**
numbers, which are not the solutions of any polynomial equation.
$\pi $
and
$e$
are both transcendental.

The Venn diagram below shows the relationships of the various sets of numbers.