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In mathematics, irrational numbers are all real numbers of irrational numbers, which are numbers composed of the ratio (or fraction) of integers. When the length of two line segments is irrational, the line segments are also described as incomparable, that is, they cannot be "measured", that is, they have no length ("measured").
Common irrational numbers are: the ratio of circumference to diameter, Euler number e, golden ratio φ and so on.
It can be seen that the representation of irrational numbers in the positional number system (for example, in decimal numbers or any other natural basis) will not be terminated or repeated, that is, it does not contain subsequences of numbers. For example, the decimal representation of the number π starts from 3.141592653589793, but there is no finite number that can accurately represent π, so it is not repeated. The evidence that the decimal extension of rational numbers must be terminated or repeated is different from the evidence that rational numbers must be terminated or repeated. Although this is basic and not lengthy, both proofs need some work. Mathematicians usually don't define "termination or repetition" as the concept of rational numbers.