Notes On Revisiting Rational and Irrational Numbers - CBSE Class 10 Maths
Rational number: A number which can be written in the form $\frac{\text{a}}{\text{b}}$ where a and b are integers and b ≠ 0 is called a rational number. Rational numbers are of two types depending on whether their decimal form is terminating or recurring. Irrational number: A number which cannot be written in the form $\frac{\text{a}}{\text{b}}$, where a and b are integers and b ≠ 0 is called a irrational number. Irrational numbers which have non-terminating and non-repeating decimal representation. The sum or difference of a two irrational numbers is also rational or an irrational number. The sum or difference of a rational and an irrational number is also an irrational number. Product of a rational and an irrational number is also an irrational number. Product of a two irrational numbers is also rational or an irrational number. Theorem: Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer. Theorem: If $\frac{\text{p}}{\text{q}}$ is a rational number, such that the prime factorisation of q is of the form 2a5b, where a and b are positive integers, then the decimal expansion of the rational number $\frac{\text{p}}{\text{q}}$ terminates. Theorem: If a rational number is a terminating decimal, it can be written in the form $\frac{\text{p}}{\text{q}}$ , where p and q are co prime and the prime factorisation of q is of the form 2a5b, where a and b are positive integers. Theorem: If $\frac{\text{p}}{\text{q}}$ is a rational number such that the prime factorisation of q is not of the form 2a5b where a and b are positive integers, then the decimal expansion of the rational number $\frac{\text{p}}{\text{q}}$ does not terminate and is recurring.

#### Summary

Rational number: A number which can be written in the form $\frac{\text{a}}{\text{b}}$ where a and b are integers and b ≠ 0 is called a rational number. Rational numbers are of two types depending on whether their decimal form is terminating or recurring. Irrational number: A number which cannot be written in the form $\frac{\text{a}}{\text{b}}$, where a and b are integers and b ≠ 0 is called a irrational number. Irrational numbers which have non-terminating and non-repeating decimal representation. The sum or difference of a two irrational numbers is also rational or an irrational number. The sum or difference of a rational and an irrational number is also an irrational number. Product of a rational and an irrational number is also an irrational number. Product of a two irrational numbers is also rational or an irrational number. Theorem: Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer. Theorem: If $\frac{\text{p}}{\text{q}}$ is a rational number, such that the prime factorisation of q is of the form 2a5b, where a and b are positive integers, then the decimal expansion of the rational number $\frac{\text{p}}{\text{q}}$ terminates. Theorem: If a rational number is a terminating decimal, it can be written in the form $\frac{\text{p}}{\text{q}}$ , where p and q are co prime and the prime factorisation of q is of the form 2a5b, where a and b are positive integers. Theorem: If $\frac{\text{p}}{\text{q}}$ is a rational number such that the prime factorisation of q is not of the form 2a5b where a and b are positive integers, then the decimal expansion of the rational number $\frac{\text{p}}{\text{q}}$ does not terminate and is recurring.

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