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# Number System CBSE CLASS 9

Number system

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