Number System Class 9 Notes - Chapter 1

CBSE Class 9 Maths Number System Notes:-Download PDF Here

Introduction to Number Systems

Numbers

Number: Arithmetical value representing a particular quantity. The various types of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers etc.

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Natural Numbers

Natural numbers(N) are positive numbers i.e. 1, 2, 3 ..and so on.

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Whole Numbers

Whole numbers (W) are 0, 1, 2,..and so on. Whole numbers are all Natural Numbers including ‘0’. Whole numbers do not include any fractions, negative numbers or decimals.
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Integers

Integers are the numbers that includes whole numbers along with the negative numbers.

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Rational Numbers

A number ‘r’ is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.

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Irrational Numbers

Any number that cannot be expressed in the form of p/q, where p and q are integers and q≠0, is an irrational number. Examples: √2, 1.010024563…, e, π

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Real Numbers

Any number which can be represented on the number line is a Real Number(R). It includes both rational and irrational numbers. Every point on the number line represents a unique real number.

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Irrational Numbers

Representation of Irrational numbers on the Number line

Let √x be an irrational number. To represent it on the number line we will follow the following steps:

  • Take any point A. Draw a line AB = x units.
  • Extend AB to point C such that BC = 1 unit.
  • Find out the mid-point of AC and name it ‘O’. With ‘O’ as the centre draw a semi-circle with radius OC.
  • Draw a straight line from B which is perpendicular to AC, such that it intersects the semi-circle at point D.

Length of BD=√x.

CBSE Notes Class 9 Maths Chapter 1-4

Constructions to Find the root of x.
  • With BD as the radius and origin as the centre, cut the positive side of the number line to get √x.

Identities for Irrational Numbers

Arithmetic operations between:

  • rational and irrational will give an irrational number.
  • irrational and irrational will give a rational or irrational number.

Example : 2 × √3 = 2√3 i.e. irrational. √3 × √3 = 3 which is rational.

Identities for irrational numbers

If a and b are real numbers then:

    • √ab = √a√b
    • √ab = √a√b
    • (√a+√b) (√a-√b) = a – b
    • (a+√b)(a−√b) = a²−b
    • (√a+√b)(√c+√d) = √ac+√ad+√bc+√bd
    • (√a+√b)(√c−√d) = √ac−√ad+√bc−√bd
    • (√a+√b)2 = a+2√(ab)+b

    Rationalisation

    Rationalisation is converting an irrational number into a rational number. Suppose if we have to rationalise 1/√a.
    1/√a × 1/√a = 1/a

    Rationalisation of 1/√a+b:

    (1/√a+b) × (1/√a−b) = (1/a−b²)

    Laws of Exponents for Real Numbers

    If a, b, m and n are real numbers then:

    • a× an= am+n
    • (am) = amn
    • am/a= am−n
    • ambm=(ab)m

    Here, a and b are the bases and m and n are exponents.

    Exponential representation of irrational numbers

    If a > 0 and n is a positive integer, then: n√a=a1n Let a > 0 be a real number and p and q be rational numbers, then:

    • ap × aq = ap + q
    • (ap)= apq
    • ap/ aq= ap−q
    • apb= (ab)p

    Decimal Representation of Rational Numbers

    Decimal expansion of Rational and Irrational Numbers

    The decimal expansion of a rational number is either terminating or non- terminating and recurring.

    Example: 1/2 = 0.5 , 1/3 = 3.33…….
    The decimal expansion of an irrational number is non terminating and non-recurring.
    Examples: √2 = 1.41421356..

    Expressing Decimals as rational numbers

    Case 1 – Terminating Decimals

    Example – 0.625

    Let x=0.625

    If the number of digits after the decimal point is y, then multiply and divide the number by 10y.

    So, x = 0.625 × 1000/1000 = 625/1000 Then, reduce the obtained fraction to its simplest form.

    Hence, x = 5/8

    Case 2: Recurring Decimals

    If the number is non-terminating and recurring, then we will follow the following steps to convert it into a rational number:

    Example –CBSE Class 9 Maths notes Chapter 1 - 1

    Step 1. Let x = CBSE Class 9 Maths notes Chapter 1 - 2  (1)

    Step 2. Multiply the first equation with 10y, where y is the number of digits that are recurring.

  • Thus, 100x = CBSE Class 9 Maths notes Chapter 1 - 3(2)Steps 3. Subtract equation 1 from equation 2.On subtracting equation 1 from 2, we get99x = 103.2x=103.2/99 = 1032/990

    Which is the required rational number.

    Reduce the obtained rational number to its simplest form Thus,

    x=172/165

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