CH 1 REAL NUMBERS – REVISITING RATIONAL AND IRRATIONAL NUMBERS

Rational number:
A number which can be written in the form a/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 a/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 p/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 p/q terminates.

Theorem: 
If a rational number is a terminating decimal, it can be written in the form p/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 p/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 p/q does not terminate and is recurring.

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