Number System Class 9 Notes Maths - Chapter 1

Chapter: 1

What are the Number System ?

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    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