Number Systems - Revision Notes

 CBSE Class 09 Mathematics

Revision Notes
CHAPTER – 1
NUMBER SYSTEMS


1 Rational Numbers

2 Irrational Numbers

3 Real Numbers and their Decimal Expansions

4 Operations on Real Numbers

5 Laws of Exponents for Real Numbers

  • Natural numbers are : 1, 2, 3, …………….. denoted by N.
  • Whole numbers are : 0, 1, 2, 3, ……………… denoted by W.
  • Integers : ……. -3, -2, -1, 0, 1, 2, 3, ……………… denoted by Z.
  • Rational numbers - All the numbers which can be written in the form pq are called rational numbers where p and q are integers and q0. Every integer p is also a rational number, can be written as p1.
  • Irrational numbers - A number is called irrational, if it cannot be written in the form pq  where p and q are integers and q0.
  • The decimal expansion of a rational number is either terminating or non terminating recurring. Thus we say that a number whose decimal expansion is either terminating or non terminating recurring is a rational number.
  • Terminating decimals: The rational numbers with a finite decimal part or for which the long division terminates after a finite number of steps are known as finite or terminating decimals.
  • Non-Terminating decimals: The rational numbers with an infinite decimal part or for which the long division does not terminate even after an infinite number of steps are known as infinite or non-terminating decimals
  • The decimal expansion of a irrational number is non terminating non recurring.
  • All the rational numbers and irrational numbers taken together make a collection of real numbers.
  • A real number is either rational or irrational.
  • If r is rational and s is irrational then r+s, r–s, r.s are always irrational numbers but rs may be rational or irrational.
  • If n is a natural number other than a perfect square, then n is a irrational number.
  • Negative of an irrational number is an irrational number.
  • There is a real number corresponding to every point on the number line. Also, corresponding to every real number there is a point on the number line.
  • Every irrational number can be represented on a number line using Pythagoras theorem.
  • For every positive real number x, x can be represented by a point on the number line by using the following steps:
  1. Obtain all positive real numbers x (say).
  2. Draw a line and mark a point P on it.
  3. Make a point Q on the line such that PQ = x units.
  4. From point Q marka distance of 1 unit and mark the new point as R.
  5. Find the mid-point of PR and mark the point as O.
  6. Draw a circle with centre O and radius OR.
  7. Draw a line perpendicular to PR passing through Q and intersecting the semi-circle at S. Length QS is equal to x.
  • Rationalization means to remove square root from the denominator.

3+52 to remove we will multiply both numerator & denominator by 2

  1a±b  its rationalization factor