## Definitions

### Irrational Numbers

Any number that can not be expressed as an integer or a fraction of two integers, e.g. $\sqrt{2}$ can only be expressed in decimals as approximation to a certain number of significant figures.

However numbers with recurring decimal places may still be rational numbers when they can be expressed as fractions of two integers, e.g. $\frac 1 3 = 0.3\text{={3}}$

### Surds

Roots of rational numbers that are irrational, e.g. $\sqrt{3},sqrt{11},sqrt{19},sqrt{4}$

### Incommensurables

Irrational numbers that are not roots of rational numbers.

## Irrational Numbers on the Number Line

While irrational numbers aren't representable by decimals with a finite number of figures their limits can still be stated with any required degree of accuracry.

$1.4142 < \sqrt{2} < 1.4.143$

## Geometrical Construction of Surds

Considering Pythagoras' Theorem: $\sqrt{1^2+1^2} = \sqrt{2}$

## Operations with Surds

Any root of a number is considered a surd in algebra, regardless of whether it may be a rational or irrational number. They follow the laws of algebra as formulated for rational numbers.

$\sqrt{2} = 2^{\frac 1 2}$
$(a+b)^{\frac 1 2} \ne a^{\frac 1 2} + b^{\frac 1 2 }$
$\sqrt{a+b} \ne \sqrt{a} + \sqrt{b}$