# Real Numbers

Chapters

- Chapter-1-Real-Numbers
- Chapter-2-Polynomials-and-Factorisation
- Chapter-3-The-Elements-of-Geometry
- Chapter-4-Lines-and-Angles
- Chapter-5-Co-Ordinate-Geometry
- Chapter-6-Linear-Equation-in-Two-Variables
- Chapter-7-Triangles
- Chapter-8-Quadrilaterals
- Chapter-9-Statistics
- Chapter-10-Surface-Areas-and-Volumes
- Chapter-11-Areas
- Chapter-12-Circles
- Chapter-13-Geometrical-Constructions
- Chapter-14-Probability
- Chapter-15-Proofs-in-Mathematics

- Chapter-1-Real-Numbers
- Chapter-2-Polynomials-and-Factorisation
- Chapter-3-The-Elements-of-Geometry
- Chapter-4-Lines-and-Angles
- Chapter-5-Co-Ordinate-Geometry
- Chapter-6-Linear-Equation-in-Two-Variables
- Chapter-7-Triangles
- Chapter-8-Quadrilaterals
- Chapter-9-Statistics
- Chapter-10-Surface-Areas-and-Volumes
- Chapter-11-Areas
- Chapter-12-Circles
- Chapter-13-Geometrical-Constructions
- Chapter-14-Probability
- Chapter-15-Proofs-in-Mathematics

- Solutions
- PDF Download
- Question Papers
- Videos

Solutions

a) Write any three rational numbers.

Solution:

3/4,5/9,2/7

b) Explain rational number is in your own words.

Solution:

A number which can be expressed in algebraic form i.e., in p/q form is called a rational number.

E.g.: 3/5, ?4/9 etc.

Give one example each to the following statements.

i) A number which is rational but not an integer.

:

7/11

ii) A whole number which is not a natural number.

Solution:

‘0’ (Zero)

iii) An integer which is not a whole number.

Solution:

-8

iv) A number which is natural number, whole number, integer and rational number.

Solution:

5

v) A number which is an integer but not a natural number.

Solution:

-4

Find five rational numbers between 1 and 2.

Solution:

Five rational numbers between 2/3 and 3/5

Solution:

Represent 8/5 and -8/5 on a number line.

Solution:

Step 1 : Draw a number line.

Step 2 : Divide each unit into 5 equal parts.

Step 3 : Take 8 -equal parts from ‘0’ on its right side and mark it as 8/5 (similarly) on left side -8/5.

Express the following rational numbers as decimal numbers.

Solution:

II. i) 2/3

Solution:

ii) ?25/36

Solution:

iii) 22/7

Solution:

iv) 11/9

Solution:

Express each of the following decimals in p/q form where q ≠ 0 and p, q are integers.

i) 0.36

Solution:

0.36 =36/100=925

ii) 15.4

Solution:

15.4 =154/10=77/5

iii) 10.25

Solution:

10.25 =1025/100=41/4

iv) 3.25

Solution:

3.25 =325/100=13/4

Question 8.

Express each of the following decimal number in the p/q form.

x = 0.12777

Multiplying by 10 on both sides

Without actually dividing find which of the following are terminating

decimals.

i) 325

Solution:

Check the denominator, if it consists of 2’s or 5’s or combination of both then only it reduces to a terminating decimal.

25 = 5 x 5

Hence 3/25 is a terminating decimal.

ii) 11/18

Solution:

Denominator 18 = 2 × 3 × 3,

hence 11/18 is a non-terminating decimal 13

iii)13/20

Denominator 20 = 2 × 2 × 5,

hence 13/20 is a terminating decimal.

iv)41/42

Solution:

Denominator 42 = 2 × 3 × 7,

hence 41/42 is a non-terminating decimal.

Classify the following numbers as rational or irrational.

- i)√27
- ii)√441
- iii) 30.232342345
- iv) 7.484848
- v) 11.2132435465
- vi) 0.3030030003

Solution:

- i)√27 irrational number
- ii)√441 = 21 - rational
- iii) 30.232342345 - irrational number
- iv) 7.484848 - rational number
- v) 11.2132435465 - irrational number
- vi) 0.3030030003 - irrational number

Explain with an example how irrational numbers differ from rational numbers ?

Solution:

Irrational numbers can’t be expressed in p/q form where p and q are integers and q ≠ 0.

E.g.√2,√3;√5,√7 etc.

Where as a rational can be expressed in pq form

E.g. :- -3 = ?3/1 and 5/4 etc.

Find two irrational numbers between 0.7 and 0.77.

Solution:

Two irrational numbers between 0.7 and 0.77 can take the form

0.70101100111000111…………. and 0.70200200022……………

Find the value of √5 uPto 3 decimal places.

Solution:

[√5 is not exactly equal to 2.2350679………….. as shown ¡n calculators]

Find the value of √7 upto six decimal places by long division method.

Solution:

Locate √10 on number line.

Step 1 : Draw a number line.

Step 2 : Draw a rectangle OABC at zero with measures 3 x 1. i.e., length 3 units and breadth 1 unit.

Step 3 : Draw the diagonal OB.

Step 4 : Draw an arc with centre ‘O’ and radius OB which cuts the number line at D.

Step 5 : ‘D’ represents (sqrt{mathrm{10}}[latex] on the number line.

Find atleast two irrational numbers between 2 and 3.

Solution:

An irrational number between a and b is Tab [latex]sqrt{mathrm{ab}}) unless ab is a perfect square.

∴ Irrational number between 2 and 3 is √6

∴ Required irrational numbers are 6^{1/2}, 24^{1/4}

Method – II:

Irrational numbers between 2 and 3 are of the form 2.12111231234………….. and 3.13113111311113…….

State whether the following statements are true or false. Justify your answers.

Solution:

- Every irrational number is a real number – True (since real numbers consist of rational numbers and irrational numbers)
- Every rational number is a real number – True (same as above)
- Every rational number need not be a rational number – False (since all rational numbers are real numbers).
- √n is not irrational if n is a perfect square – True. (since by definition of an irrational number).
- √n is irrational if n is not a perfect square – True. (same as above)
- All real numbers are irrational – False (since real numbers consist of rational

Visualise 2.874 on the number line, using successive magnification.

Solution:

Visualise5.28 on the number line, upto 3 decimal places.

Solution:

Simple the following expressions.

i) (5 + √7) (2 + √5)

Solution:

(5 + √7) (2 + √5)

= 10 + 5√5 + 2√7 + √35

ii) (5 + √5) (5 - √5)

Solution:

(5 + √5) (5 - √5)

= 5^{2} + (√5)^{2}

= 25 - 5 = 20

(iii) (√3 + √7)^{2}

Solution:

(√3 + √7)^{2}

= (√3)^{2} + (√7)^{2} + 2(√3)(√7)

= 3 + 7 + 2√21

= 10 + 2√21

iv) (√11 - √7) (√11 + √7)

= (√11)^{2} - (√7)^{2<.sup>
= 11 - 7 = 4}

Classify the following numbers as rational or irrational.

i) 5 - √3

ii) √3 + √2

iii) (√2 - 2)^{2}

iv) 2√7/7√7

v) 2π

vi) 1/√3

vii) (2 +√2) (2 - √2)

Solution

i) 5 - √3 - irrational

ii) √3 + √2 - irrational

iii) (√2 - 2)^{2} - irrational

iv) 2√7/7√7 - irrational

v) 2π - Transcendental number. (not irrational)

vi) 1/√3 - irrational

vii) (2 +√2) (2 - √2) - rational

In the following equations, find whether variables x, y, z etc., represents rational or irrational numbers.

- i) x
^{2}= 7 - ii) y
^{2}= 16 - iii) z
^{2}= 0.02 - iv) u
^{2}=17/4 - v) w
^{2}= 27 - vi) t
^{4}= 256

Solution:

i) x2 = 7
⇒ x = √7 is an irrational number.

ii) y2 = 16 ⇒ y = 4 is a rational number.

iii) z2 = 0.02 ⇒ z = √0.02 is an irrational number.

iv) u2 = 174 ⇒ x = √17/2 is an irrational number.

v) w2 = 27 ⇒ w = 3√3 an irrational number.

vi) t4 = 256 ⇒ t2 = √256 = 16

⇒ t = √16 = 4 is a rational number

The ratio of circumference to the diameter of a circle c/d is represented by π. But we say that π is an irrational number. Why?

Rationalise the denominators of the following.

Solution:

ii) 1/√7?√6

Solution:

iii) 1/√7

Solution:

iv) √6/√3?√2

Solution:

Simplify each of the following by rationalising the denominator.

i) 6?4√2/6+4√2

Solution:

ii) √7?√5/√7+√5

Solution:

iii) 1/3√2?2√3

Solution:

iv) 3√5?√7/3√3+√2

Solution:

Find the value of √10?√5/2√2 upto three decimal places. (take √2 = 1.414, √3 = 1.732 and √5 = 2.236).

Solution:

Find

i) 64^{1/6}

Solution:

= (2^{6})^{1/6}

= 6

ii) 32^{1/5}

Solution:

32^{1/5}

= (2^{5})^{1/5}

= 2

iii) 625^{1/4}

625^{1/5}

= (5^{4})^{1/4}

= 5

iv) 16^{3/2}

Solution:

16^{3/2}

= (4^{2})^{3/2}

v) 243^{2/5}

Solution:

243^{2/5}

= (3^{5})^{2/5}

vi) (46656)^{-1/6}

Solution:

Simplify ∜81 ? 8 ∛343+ 15 32 + √225

Solution:

If ‘a’ and ‘b’ are rational numbers, find the values of a and b in each of the following equations.

i)√3+ √2/√3 - √2=a+b√6

Solution:

Given that√3+ √2/√3 - √2=a+b√6

Rationalising the denominator we get

Comparing 5 + 2√6 with a + b√6

We have a = 5 and b = 2

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