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Number System CBSE CLASS 9

Number system


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
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