CBSE Class 9 Maths Number System Notes:-Download PDF Here
Introduction to Number Systems
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(N) are positive numbers i.e. 1, 2, 3 ..and so on.
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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 are the numbers that includes whole numbers along with the negative numbers.
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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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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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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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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.
- 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 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:
- am × an= am+n
- (am) n = amn
- am/an = am−n
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)q = apq
- ap/ aq= ap−q
- apbp = (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
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:
Step 1. Let x = (1)
Step 2. Multiply the first equation with 10y, where y is the number of digits that are recurring.
- Thus, 100x = (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,
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