### 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 are called rational numbers where p and q are integers and Every integer p is also a rational number, can be written as
- Irrational numbers - A number is called irrational, if it cannot be written in the form where p and q are integers and
- 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 may be rational or irrational.
- If n is a natural number other than a perfect square, then 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 can be represented by a point on the number line by using the following steps:

- Obtain all positive real numbers (say).
- Draw a line and mark a point P on it.
- Make a point Q on the line such that PQ = units.
- From point Q marka distance of 1 unit and mark the new point as R.
- Find the mid-point of PR and mark the point as O.
- Draw a circle with centre O and radius OR.
- Draw a line perpendicular to PR passing through Q and intersecting the semi-circle at S. Length QS is equal to .

- Rationalization means to remove square root from the denominator.

to remove we will multiply both numerator & denominator by

its rationalization factor