Any number that can not be expressed as an integer or a fraction of two integers, e.g. 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.
Roots of rational numbers that are irrational, e.g.
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.
Geometrical Construction of Surds
Considering Pythagoras' Theorem:
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.