Number System CBSE CLASS 9

CONTENT LIST

What Are Numbers?

Numbers arithmetical value representing a particular quantity. Diffrent kinds of numbers are natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, etc.

Different kind of numbers are:

Natural Numbers

Counting numbser are called natural numbers(N).

They are positive numbers i.e. 1, 2, 3,.... and so on.

Natural numbers are also called positive integers.

Whole Numbers

When 0 added in family of natural number,It becomes whole number.

Whole numbers (W) are 0, 1, 2,..... and so on.

Whole numbers do not include any fractions, negative numbers or decimals.

Integers

Integers(Z) are numbers that include whole numbers along with negative of Natural numbers.

An integer is a number from a set of negative and positive numbers, including zero without decimal or fractional value.

-3,-1, 2, 0,4, 15, 500, etc., are examples of integers.

Rational Numbers

All numbers in the form p/q, where p and q are integers and q ≠ 0 are called Rational number.

Every whole number is a rational number because they can be written as p/q form.

Set of Rational number is representataed by Q.

Irrational Numbers

All numbers that are not be expressed as a ratio of integers are called Irrestional number; for example, √2 is an irrational number.

Set of Rational number is representataed by P.

Real Numbers

Set of rational and irrational number is called as real number.

R is the symbol used for set of real numbers.

Finding rational number between two numbers

With same denominator

CASE 1

If the value of numerator differ by a large number then simply increments of one for the numerator without altering the value of the denominator.

Example: The 3 rational numbers between 4/9 and 8/9 are 5/9,6/9,7/9

CASE 2

If the values of the numerator of given number is less than the number of rational numbers to be found

the numerators and denominators of both the rational numbers are multiplied by multiples of 10,100,1000 and so on...

Exmaple: Find 10 rational numbers are to be found between 2/7and 5/7, both the rational numbers are to be multiplied with 10/10.

2/7 x 10/10=20/70

5/7 x 10/10=50/70

The 10 rational numbers between 2/7 and 5/7 can be written as the rational numbers between 20/70 and 50/70. The 10 rational numbers are 21/70, 22/70,23/70, 24/70, 25/70, 26/70, 27/70, 28/70, 29/70 and 30/70.

With different denominator

To find the rational numbers between two rational numbers with different denominators, the denominators should be equated.

Equating the denominators can be done either by finding their LCM or by multiplying the denominators of one to both the numerator and denominator of the other.

Example

If the rational numbers between 2/3 and 3/4 are to be found.

LCM of 3 and 4 is 12. When the denominators are equated by the LCM method, the equivalent rational numbers are 8/12 and 9/12.

The same rational numbers will be obtained when the denominator of one rational number is multiplied to the numerator and denominator of the other.

2/3 x 4/4 = 8/12 and 3/4 x 3/3 = 9/12.

Once the denominators are equated, the same rules of finding the rational numbers between two rational numbers having the same denominator is used.

Irrational Numbers

• Irrational number is a number that cannot be written in the form p/q where p and q are integers and p ≠ 0 .
• Decimal expansion of the irrational number is non-terminating and non-recurring .
• √2, π are example of irretional number.

Real Numbers and their Decimal Expansions

• A set of combination of rational and irrational numbers is called real numbers.
• All the real numbers can be expressed on the number line.
• Decimal expansion of rational number are : Terminating and Non-terminating Repeating.

Operations on Real Numbers

• Addition or subtraction operation of a rational and irrational number, always result is an irrational number.
• Multiplication or division operation of a rational number with an irrational number, always result an irrational number.
• When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational number or an irrational number.

Laws of Exponents for Real Numbers

Related Topics

Polynomials: Chapter Note

Coordinate Geometry: Chapter Note

Linear equation in two variables: Chapter Note

Euclids Geometry: Chapter Note

Lines and angles: Chapter Note