XXV Irrational Numbers and Surds

Definitions

Irrational Numbers

Any number that can not be expressed as an integer or a fraction of two integers, e.g. $\sqrt[2]{2}$ 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. $\sqrt[2]{3},sqrt[2]{11},sqrt[2]{19},sqrt[3]{4}$

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.

$1.4142 < \sqrt[2]{2} < 1.4.143$

Geometrical Construction of Surds

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

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.

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

Multiplication, Rationalization and Simplification

See XVIII Square Root Stuff