XXV Irrational Numbers and Surds


Irrational Numbers

Any number that can not be expressed as an integer or a fraction of two integers, e.g. 22\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. 13=0.33ˉ\frac 1 3 = 0.3\text{={3}}


Roots of rational numbers that are irrational, e.g. 32,sqrt[2]11,sqrt[2]19,sqrt[3]4\sqrt[2]{3},sqrt[2]{11},sqrt[2]{19},sqrt[3]{4}


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<22<1.4.1431.4142 < \sqrt[2]{2} < 1.4.143

Geometrical Construction of Surds

Considering Pythagoras' Theorem: 12+122=22\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.

22=212\sqrt[2]{2} = 2^{\frac 1 2}
(a+b)12a12+b12(a+b)^{\frac 1 2} \ne a^{\frac 1 2} + b^{\frac 1 2 }
a+ba+b\sqrt{a+b} \ne \sqrt{a} + \sqrt{b}

Multiplication, Rationalization and Simplification

See XVIII Square Root Stuff