Number System Class 9 Notes Maths - Chapter 1

Chapter: 1

What are the Number System ?

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,....... โ€“1, โ€“2, โ€“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 โ‰  0.
  • A number which cannot be expressed in the form (p/q) (where p and q are integers and q โ‰  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 โ€“ 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) โˆšab = โˆša . โˆšb

(ii) โˆš(a/b)=(โˆša/โˆšb),provided b โ‰ 0

(iii) ( โˆša + โˆšb) ( โˆša โ€“ โˆšb) = a โ€“ b

(iv) (a + โˆšb) (a โ€“ โˆšb) = a2 - b and ( โˆša + b) ( โˆša โ€“ b) = a โ€“ b2

(v) ( โˆša + โˆšb)2 = a + 2 โˆš(ab) + b and ( โˆša โ€“ โˆšb)2 = a โ€“ 2 โˆš(ab) + b

(vi) ( โˆša + โˆšb) ( โˆšc + โˆšd) = โˆš(ac) + โˆš(ad) + โˆš(bc) + โˆš(bd)

Basics Of Number Systems

  • To rationalise the denominator of the form a + โˆšb, multiply the numerator and denominator by its conjugate, i.e. by a โ€“ โˆšb.
  • To rationalise the denominator of the form โˆša + โˆšb, multiply the numerator and denominator by its conjugate, i.e. โˆša โ€“ โˆšb.
  • Conjugate of a โ€“ โˆšb is a + โˆšb and the conjugate of โˆša โ€“ โˆšb is โˆša + โˆšb.

Laws Of Exponents

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

(i) ap ร— 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