# Number System Class 9 Notes Maths - Chapter 1

## Basics Of Number Systems

• Natural numbers are counting numbers, like 1, 2, 3,.......
• Whole numbers are just like natural numbers but they start from O, like 0, 1, 2, 3,.......
• If we put the whole numbers and their negative together, the new collection of numbers will look like 0, 1, 2, 3,....... –1, –2, –3....... and such numbers are called integers.
• A number is called a rational number, if it can be written in the form (p/q) , where p and q are integers and q ≠ 0.
• A number which cannot be expressed in the form (p/q) (where p and q are integers and q ≠ 0) is called an irrational number.
• All rational numbers and all irrational numbers together make the collection of real numbers.

## Decimal Expansions Of Real Numbers

• Decimal expansion of a rational number is either terminating or non-terminating recurring.
• Decimal expansion of an irrational number is non-terminating non-recurring.

## Operations On Real Numbers

• The sum, the difference and the product of two rational numbers is a rational number. Also, the quotient of two non-zero rational numbers is also a rational number.
• If u is a rational number and v is an irrational number, then u + v and u – v are irrationals. Also, if u is non-zero rational, then uv and (u/v), are irrationals.
• If u and v both are irrationals, then u + v, u - v, uv and (u/v) are either rational or irrational.

## Some Important Identities

For any positive real numbers a and b:

(i) √ab = √a . √b

(ii) √(a/b)=(√a/√b),provided b ≠0

(iii) ( √a + √b) ( √a – √b) = a – b

(iv) (a + √b) (a – √b) = a2 - b and ( √a + b) ( √a – b) = a – b2

(v) ( √a + √b)2 = a + 2 √(ab) + b and ( √a – √b)2 = a – 2 √(ab) + b

(vi) ( √a + √b) ( √c + √d) = √(ac) + √(ad) + √(bc) + √(bd)

## Basics Of Number Systems

• To rationalise the denominator of the form a + √b, multiply the numerator and denominator by its conjugate, i.e. by a – √b.
• To rationalise the denominator of the form √a + √b, multiply the numerator and denominator by its conjugate, i.e. √a – √b.
• Conjugate of a – √b is a + √b and the conjugate of √a – √b is √a + √b.

## Laws Of Exponents

If a and b are positive real numbers and p and q are rational numbers, then

(i) ap × aq = ap+q

(ii) (ap)q = (aq)p = apq

(iii) (ap/aq)=a(p-q) and (1/ap)=(a0/ap)= a0-p

(iv) (ab)p = apbp

(v) (a/b)p=(ap/bp) and (a/b)-p=(b/a)p