Samacheer Class 9 Maths - Real Numbers

First convert the rational number into mixed form.

So, the number lies:

• between (3) and (4)
• closer to (4)

Therefore, the correct arrow is the one pointing at:

on the number line.

Solution

We need rational numbers greater than

and less than

Possible rational numbers are:

Answer

Any three rational numbers are:

# (i) Between (\frac14) and (\frac15)

Step 1: Take LCM denominator

Now choose fractions between:

Answer

Five rational numbers are:

(Any valid five rational numbers are acceptable.)

# (ii) Between (0.1) and (0.11)

Step 1: Write with same decimal places

Now choose numbers between them.

Answer

# (iii) Between (-1) and (-2)

We need numbers greater than (-2) and less than (-1).

Answer

All these lie between (-2) and (-1).

# (i) (\frac{2}{7})

Solution

The digits repeat continuously.

Answer

Type of decimal expansion:

Non-terminating recurring decimal

# (ii) (-5\frac{3}{11})

Step 1: Convert mixed fraction into improper fraction

Step 2: Convert into decimal

Therefore,

Answer

Type of decimal expansion:

Non-terminating recurring decimal

# (iii) (\frac{22}{3})

Solution

Answer

Type of decimal expansion:

Non-terminating recurring decimal

# (iv) (\frac{327}{200})

Solution

Answer

Type of decimal expansion:

Terminating decimal

Solution

The repeating block is:

This block contains 6 digits.

Answer

Length of the period:

Step 1: Use (\frac{1}{11})

We know:

Step 2: Find (\frac{1}{33})

Therefore,

Step 3: Find (\frac{71}{33})

Now,

Therefore,

Answer

The decimal expressions are not visible in the question provided.
Please share the decimal numbers clearly to convert them into rational numbers.

A rational number has a terminating decimal expansion if the denominator in simplest form contains only the prime factors:

# (i) (\frac{7}{128})

Only factor (2) is present.

Answer

has a terminating decimal expansion.

# (ii) (\frac{21}{15})

Simplify:

Denominator:

Only factor (5).

Answer

has a terminating decimal expansion.

# (iii) (4\frac{9}{35})

Convert fractional part:

Denominator:

Factor (7) is present.

Answer

has a non-terminating recurring decimal expansion.

# (iv) (\frac{219}{2200})

Simplify:

Factorize denominator:

Factor (11) is present.

Answer

has a non-terminating recurring decimal expansion.

# (i) (\sqrt3)

So, on the number line:

• Mark (0), (1), and (2)
• Locate the point approximately at (1.732)

Answer

lies between (1) and (2).

# (ii) (\sqrt{4.7})

So, on the number line:

• Mark (2) and (3)
• Locate the point approximately at (2.168)

Answer

lies between (2) and (3).

# (iii) (\sqrt{6.5})

So, on the number line:

• Mark (2) and (3)
• Locate the point approximately at (2.549)

Answer

lies between (2) and (3).

# (i)

and

Solution

Any non-terminating non-recurring decimals between them are irrational.

Possible answers:

Both lie between the given numbers.

Answer

Two irrational numbers are:

and

# (ii) Between (\frac67) and (\frac{12}{13})

Step 1: Convert into decimals

Now choose irrational numbers between them.

Answer

Two irrational numbers are:

and

Approximate values:

Both lie between (0.857) and (0.923).

# (iii) Between (\sqrt2) and (\sqrt3)

Approximate values:

Choose irrational numbers between them.

Answer

Two irrational numbers are:

and

Approximate values:

Both lie between (\sqrt2) and (\sqrt3).

and

Solution

Any terminating decimals between the given numbers are rational numbers.

Possible answers:

Both lie between the given numbers.

Answer

Two rational numbers are:

and

# (i) (5.348)

Solution

Step 1

First note that:

lies between (5) and (6).

Step 2

Divide the portion between (5) and (6) into 10 equal parts using a magnifying glass.

This gives:

Now (5.348) lies between:

Step 3

Divide the portion between (5.3) and (5.4) into 10 equal parts.

This gives:

Now (5.348) lies between:

Step 4

Divide the portion between (5.34) and (5.35) into 10 equal parts.

This gives:

Now (5.348) lies between:

Step 5

Mark the point corresponding to:

on the number line.

We observe that (5.348) is closer to (5.349) than to (5.347).

# (ii) (6.\overline{4}) up to 3 decimal places

Step 1

Up to 3 decimal places:

The number lies between (6) and (7).

Step 2

Divide the portion between (6) and (7) into 10 equal parts.

Now (6.444) lies between:

Step 3

Divide the portion between (6.4) and (6.5) into 10 equal parts.

Now (6.444) lies between:

Step 4

Divide the portion between (6.44) and (6.45) into 10 equal parts.

Now (6.444) lies between:

Mark the point corresponding to:

on the number line.

We observe that (6.444) is closer to (6.445) than to (6.443).

# (iii) (4.\overline{73}) up to 4 decimal places

Step 1

Up to 4 decimal places:

The number lies between (4) and (5).

Step 2

Divide the portion between (4) and (5) into 10 equal parts.

Now (4.7373) lies between:

Step 3

Divide the portion between (4.7) and (4.8) into 10 equal parts.

Now (4.7373) lies between:

Step 4

Divide the portion between (4.73) and (4.74) into 10 equal parts.

Now (4.7373) lies between:

Step 5

Divide the portion between (4.737) and (4.738) into 10 equal parts.

Now (4.7373) lies between:

Mark the point corresponding to:

on the number line.

We observe that (4.7373) is closer to (4.7373) than to (4.7374).

# (i) (625)

Solution

Answer

# (ii) (\frac15)

Solution

Answer

# (iii) (\sqrt5)

Using:

Solution

Answer

# (iv) (\sqrt[3]{125})

Since:

Solution

Answer

# (i) (16)

Solution

Answer

# (ii) (8)

Solution

Also,

Therefore,

Answer

# (iii) (32)

Solution

Since,

Answer

The expressions are not visible in the uploaded question.
Please share the full expressions clearly (or upload the image again) so I can provide the exact validated solutions.

# (i) (\sqrt5)

Solution

Answer

# (ii) (\sqrt[2]{7})

Solution

Answer

# (iii) ((\sqrt[3]{49})^5)

Solution

Using:

Answer

# (iv)

The expression is missing in the question provided.
Please share the complete expression.

# (i) (32)

Solution

Therefore,

Answer

# (ii) (243)

Solution

Therefore,

Answer

# (iii) (100000)

Solution

Therefore,

Answer

# (iv) (\frac{1024}{3125})

Solution

Therefore,

Answer

# (i) (5\sqrt3+18\sqrt3-2\sqrt3)

Since all are like surds, combine coefficients.

Solution

Answer

# (ii) (4\sqrt[3]{5}+2\sqrt[3]{5}-3\sqrt[3]{5})

Solution

Answer

# (iii) (3\sqrt{75}+5\sqrt{48}-\sqrt{243})

Step 1: Simplify each surd

Step 2: Substitute

Answer

# (iv) (5\sqrt[3]{40}+2\sqrt[3]{625}-3\sqrt[3]{320})

Step 1: Simplify cube roots

Step 2: Substitute

Answer

# (i) (\sqrt3\times\sqrt5\times\sqrt2)

Solution

Answer

# (ii) (\sqrt{35}\div\sqrt7)

Solution

Answer

# (iii) (\sqrt[3]{27}\times\sqrt[3]{8}\times\sqrt[3]{125})

Solution

Answer

# (iv) ((7\sqrt a-5\sqrt b)(7\sqrt a+5\sqrt b))

Use:

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Solution

Answer

# (v)

Step 1: Simplify each surd

Step 2: Perform division

Step 3: Subtract

LCM:

Answer

Given:

# (i) (\sqrt{40}-\sqrt{20})

Step 1: Simplify

Step 2: Substitute values

Answer

# (ii) (\sqrt{300}+\sqrt{90}-\sqrt8)

Step 1: Simplify

Step 2: Substitute values

Answer

# (i) (3\sqrt5,\ 9\sqrt4,\ 6\sqrt3)

Step 1: Find approximate values

Descending order

# (ii) (2\sqrt3\sqrt5,\ 3\sqrt4\sqrt7,\ \sqrt{\sqrt3})

Step 1: Simplify

Descending order

# (i) Sum of two surds

Yes.

Example:

is a pure surd.

# (ii) Difference of two surds

Yes.

Example:

is a pure surd.

# (iii) Product of two surds

Yes.

Example:

which is a pure surd.

# (iv) Quotient of two surds

Yes.

Example:

which is a pure surd.

# (i) Sum of two surds

Yes.

Example:

which is rational.

# (ii) Difference of two surds

Yes.

Example:

which is rational.

# (iii) Product of two surds

Yes.

Example:

which is rational.

# (iv) Quotient of two surds

Yes.

Example:

which is rational.

# (i) (\frac1{\sqrt{50}})

Solution

# [

\frac{\sqrt{50}}{50}
]

# [

\frac{\sqrt{25\times2}}{50}
]

# [

\frac{5\sqrt2}{50}
]

# [

\frac{\sqrt2}{10}
]

Answer

# (ii) (\frac5{3\sqrt5})

Solution

# [

\frac{5\sqrt5}{15}
]

# [

\frac{\sqrt5}{3}
]

Answer

# (iii) (\frac{\sqrt{75}}{\sqrt{18}})

Solution

# [

\frac{\sqrt{3\times5\times5}}{\sqrt{3\times2\times3}}
]

# [

\frac{5\sqrt3}{3\sqrt2}
]

Rationalising:

# [

\frac{5\sqrt6}{6}
]

Answer

# (iv) (\frac{3\sqrt5}{\sqrt6})

Solution

# [

\frac{3\sqrt{30}}{6}
]

# [

\frac{\sqrt{30}}{2}
]

Answer

# (i)

Step 1: Simplify surds

Thus,

# [

\frac{4\sqrt3+4\sqrt2}{3\sqrt3-3\sqrt2}
]

# [

\frac{4(\sqrt3+\sqrt2)}{3(\sqrt3-\sqrt2)}
]

Step 2: Rationalise

Multiply by conjugate:

Denominator

Numerator

Final Answer

# (ii)

Step 1: Multiply by conjugate

Denominator

Numerator

Answer

# (iii)

Step 1: Multiply by conjugate

Denominator

Numerator

Answer

# (iv)

Step 1: Take LCM

# [

\frac{\sqrt5(\sqrt6-2)-\sqrt5(\sqrt6+2)}
{(\sqrt6+2)(\sqrt6-2)}
]

Denominator

Numerator

Final Answer

# Exercise 2.7

Solution

Rationalise the denominator.

Using:

genui{"math_block_widget_always_prefetch_v2":{"content":"(a+b)(a-b)=a^2-b^2"}}

Numerator

Denominator

Therefore

# [

-\frac43\sqrt7+\frac{11}{3}
]

Comparing with:

we get:

Answer

find the value of

Step 1: Find (x^2)

Step 2: Find (\frac1x)

Rationalising:

Step 3: Find (\frac1{x^2})

Step 4: Add

Answer

find the value of

correct to 3 decimal places.

Step 1: Rationalise the denominator

Denominator

Numerator

Step 2: Substitute value

Answer

# (i) (569430000000)

Move decimal point after first digit.

Answer

# (ii) (2000.57)

Answer

# (iii) (0.0000006000)

Move decimal 7 places right.

Answer

# (iv) (0.0009000002)

Answer

# (i) (3.459\times10^6)

Move decimal 6 places right.

Answer

# (ii) (5.678\times10^4)

Answer

# (iii) (1.00005\times10^{-5})

Move decimal 5 places left.

Answer

# (iv) (2.530009\times10^{-7})

Answer

# (i)

Step 1: Write in scientific notation

Step 2: Apply powers

Answer

# (ii)

Step 1: Convert to scientific notation

Step 2: Simplify

Answer

# (iii)

Step 1: Scientific notation

Step 2: Apply powers

Numerator:

Denominator:

Step 3: Divide

Answer

# (i) World population

Answer

# (ii) One light year

Answer

# (iii) Mass of an electron

Answer

# (i)

Step 1: Equalise powers

Step 2: Add

Answer

# (ii)

Step 1: Equalise powers

Step 2: Subtract

Answer

# (iii)

Multiply coefficients

Add powers

Answer

# (iv)

Step 1: Divide coefficients

Step 2: Subtract powers

Answer

# Activity – 3

Complete the table and arrange the planets in order of magnitude

# Completed Table

| Planet | Decimal Form (in km) | Scientific Notation (in km) |
| ------- | -------------------- | --------------------------- |
| Jupiter | 778000000 | (7.78\times10^8) |
| Mercury | 58000000 | (5.8\times10^7) |
| Mars | 228000000 | (2.28\times10^8) |
| Uranus | 2870000000 | (2.87\times10^9) |
| Venus | 108000000 | (1.08\times10^8) |
| Neptune | 4500000000 | (4.5\times10^9) |
| Earth | 150000000 | (1.5\times10^8) |
| Saturn | 1430000000 | (1.43\times10^9) |

# Arrangement in order of magnitude

(Closest to the Sun to farthest)

1. Mercury

2. Venus

3. Earth

4. Mars

5. Jupiter

6. Saturn

7. Uranus

8. Neptune

# Final Order

1. always a natural number
2. always an irrational number
3. always a rational number
4. may be rational or irrational

Solution

If (n) is a perfect square,

Example:

which is rational.

If (n) is not a perfect square,

Example:

which is irrational.

Answer

1. Every rational number is a real number.
2. Every integer is a rational number.
3. Every real number is an irrational number.
4. Every natural number is a whole number.

Solution

Real numbers include both rational and irrational numbers.

Hence statement (3) is false.

Answer

1. always an irrational number
2. may be a rational or irrational number
3. always a rational number
4. always an integer

Solution

Example:

is irrational.

But,

is rational.

Answer

1. (\frac5{64})
2. (\frac89)
3. (\frac{14}{15})
4. (\frac1{12})

Solution

A rational number has a terminating decimal if the denominator contains only factors (2) and/or (5).

Hence,

has a terminating decimal.

Answer

1. (\sqrt{25})
2. (\sqrt{\frac94})
3. (\frac7{11})
4. (\pi)

Solution

are rational numbers.

is irrational.

Answer

1. (\sqrt{11})
2. (\sqrt5)
3. (\sqrt{2.5})
4. (\sqrt8)

Solution

We need a number between (4) and (6.25).

lies between them.

Hence,

lies between (2) and (2.5).

Answer

1. (\frac1{10})
2. (\frac3{10})
3. (3)
4. (30)

Solution

which has terminating decimal:

Answer

then the value of

is

Solution

# [

5\times0.\overline{142857}
]

# [

0.\overline{714285}
]

Answer

1. (\sqrt{32}\times\sqrt2)
2. (\sqrt{27}\div\sqrt3)
3. (\sqrt{72}\times\sqrt8)
4. (\sqrt{54}\div\sqrt{18})

Solution

(1)

(2)

(3)

(4)

which is irrational.

Hence it is different from others.

Answer

The complete question is not visible.
Only the solution fragment:

is visible.

Please share the full question/options for exact validation.

1. The square root of (25) is (5) or (-5)
2. (-\sqrt{25}=-5)
3. (\sqrt{25}=5)
4. (\sqrt{25}=\pm5)

Solution

The symbol:

represents only the principal positive square root.

Hence,

not (\pm5).

Answer

1. (\sqrt{\frac8{18}})
2. (\frac73)
3. (\sqrt{0.01})
4. (\sqrt{13})

Solution

rational.

rational.

rational.

irrational.

Answer

1. (\sqrt{39})
2. (5\sqrt6)
3. (5\sqrt3)
4. (3\sqrt5)

Solution

Answer

then (k=)

Solution

Hence,

Answer

Solution

Answer

Solution

# [

\frac{2\sqrt6}{6}
]

# [

\frac{\sqrt6}{3}
]

Answer

Solution

Using:

genui{"math_block_widget_always_prefetch_v2":{"content":"(a-b)^2=a^2-2ab+b^2"}}

Answer

Solution

Thus,

# [

\left(\frac3{10}\right)^{-6}
]

# [

\left(\frac{10}{3}\right)^6
]

Answer

then (x=)

Solution

Squaring and simplifying gives:

Answer

and

metres respectively. Find the area.

Solution

Area:

Answer