Type of Numbers

Natural Numbers (N)

1,2,3,4,5,….. are called Natural Numbers.
It is denoted by N.

Whole Numbers (W)

0,1,2,3,4,5,….. are called Whole Numbers.
It is denoted by W.

Integers (Z)

….-3,-2,-1,0,1,2,3,….. are called Integers.
It is denoted by Z.

Integers and be further classified as Positive (\(Z^+\)) and Negative Integers(\(Z^-\)).

Rational Number (Q)

Any number which can be expresses in \(\frac{p}{q}\) form where \(q\ne 0\) is a Rational Number.
It is denoted by Q.

Irrational Number (I)

Any number which cannot be expresses in \(\frac{p}{q}\) form where \(q\ne 0\) is an irrational Number.
It is denoted by I.

Real Number (R)

All rational numbers and all irrational numbers together make the collection of real numbers.
It is denoted by R.

Important Properties -1

I + R = I
I - R = I
R - I = I
R + R = R
R - R = R
\(I\times R = I\)
\(I\times I = I or R\)
\(\cfrac{R}{I} = I\)
\(\cfrac{I}{R} = I\)

Important Properties - 2

For positive real numbers a and b,

\(\sqrt{a}\sqrt{b} = \sqrt{ab}\)
\(\sqrt{\cfrac{a}{b}} = \cfrac{\sqrt{a}}{\sqrt{b}}\)
\((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = (a-b)\)
\((a+\sqrt{b})(a-\sqrt{b}) = (a^2-b)\)
\((\sqrt{a}+\sqrt{b})^2 = (a + 2\sqrt{ab} + b)\)

Important Properties - 3

If m and n are rational numbers and a,b are positive real numbers. Then,

\(a^ma^n = a^{m+n}\)
\((a^m)^n = a^{mn}\)
\(\cfrac{a^m}{a^n} = a^{m-n}\)
\(a^mb^m = (ab)^m\)

NCERT EXERCISE - 1.1

Question - 1

Is zero a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Solution :

0 can be expressed as \(\cfrac{0}{1}\) which is of the form \(\cfrac{p}{q}\).
Hence, 0 is a rational number.

Question - 2

Find six rational numbers between 3 and 4.

Solution :
To find 6 rational numbers between 3 and 4, we multiply and divide the numbers by 6+1=7.
\(3\times \cfrac{7}{7} =\cfrac{21}{7}\)
\(4\times \cfrac{7}{7} =\cfrac{28}{7}\)

Hence, \(\cfrac{22}{7}, \cfrac{23}{7},\cfrac{24}{7},\cfrac{25}{7},\cfrac{26}{7}\) and \(\cfrac{27}{7}\) are the six rational numbers.

Question - 3

Find five rational numbers between \(\cfrac{3}{5}\) and \(\cfrac{4}{5}\).

Solution :
To find 5 rational numbers between \(\cfrac{3}{5}\) and \(\cfrac{4}{5}\), we multiply and divide the numbers by 5+1=6.
\(\cfrac{3}{5}\times \cfrac{6}{6} =\cfrac{18}{30}\)
\(\cfrac{4}{5}\times \cfrac{6}{6} =\cfrac{24}{30}\)

Hence, \(\cfrac{19}{30}, \cfrac{20}{30},\cfrac{21}{30},\cfrac{22}{30}\) and \(\cfrac{23}{30}\) are the five rational numbers between \(\cfrac{3}{5}\) and \(\cfrac{4}{5}\).

Question - 4

State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

(ii)Every integer is a whole number.

(iii)Every rational number is a whole number.

Solution :
i. TRUE
Natural Number are 1,2,3,4,….
Whole Numbers are 0,1,2,3,4,…

We can say that whole numbers have all Natural numbers and Zero.

Hence, every natural number is a whole number but the revrse is not true.

ii. FALSE
Whole Numbers are 0,1,2,3,4,…
Integers are ….,-2,-1,0,1,2,….

We can clearly see that the engative integers are not whole numbers.

Hence, every integer is not a whole number but every whole number is an integer.

iii. FALSE
Whole Numbers are 0,1,2,3,4,…
Rational numbers are the one which can be expressed in the form of \(\cfrac{p}{q}\), where \(q\ne 0\).

We can clearly see that whole numbers does not include the fractions and the negative integers.

Hence, every rational number is not a whole number however, every whole number is a rational number.

NCERT EXERCISE - 1.2

Question - 1

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

i. Every irrational number is a real number.

ii. Every point on the number line is of the form √𝑚 , where m is a natural number.

iii. Every real number is an irrational number.

Solution :
i. TRUE
Irrational Number : Numbers which cannot be represented in \(\cfrac{p}{q}\) form are called irrational numbers.
Real Numbers : Collection of rational and irrational numbers is called Real numbers.

Hence, Every irrational number is a real number however, every real number is not irrational.

ii. FALSE
The negative numbers cannot be expressed as square roots.

Hence, Every point on the number line is not of the form \(\sqrt{m}\), where m is a natural number.

iii. FALSE
Irrational Number : Numbers which cannot be represented in \(\cfrac{p}{q}\) form are called irrational numbers.
Real Numbers : Collection of rational and irrational numbers is called Real numbers.

Hence, every real number is not irrational.

Question - 2

Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Solution :
Square root of all positive integers are not irrational.
Examples:
\(\sqrt{4} = 2\), which is not a irrational.
\(\sqrt{9} = 3\), which is not a irrational.

We can see that, \(\sqrt{4}\) and \(\sqrt{9}\) is a rational number.

Question - 3

Show how √5 can be represented on the number line.

Solution :
We know that, \(\sqrt5 = \sqrt{2^2+1^2}\)

  1. Draw a line OA = 2 units
  2. Draw a line AB perpendicular to OA of 1 unit.
  3. Join point OB which is equal to \(\sqrt5\).
  4. Construct an ARC equal to OB by center at O and interescting the number line at D.
  5. OD here represents \(\sqrt5\) on number line.

Question - 4

Classroom activity (Constructing the ‘square root spiral’) :

Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion.

Start with a point \(O\) and draw a line segment O\(P_1\) of unit length. Draw a line segment \(P_1P_2\) perpendicular to \(OP_1\) of unit length.

Now draw a line segment \(P_2P_3\) perpendicular to \(OP_2\).

Then draw a line segment \(P_3P_4\) perpendicular to \(OP_3\).

Continuing in this manner, you can get the line segment \(P_{n-1}P_n\) by drawing a line segment of unit length perpendicular to \(OP_{n-1}\).

In this manner, you will have created the points \(P_2\) , \(P_3\) ,….,\(P_n\) ,… ., and joined them to create a beautiful spiral depicting 2, 3, 4, …

Solution:

NCERT EXERCISE - 1.3

Question - 1

Write the following in decimal form and say what kind of decimal expansion each has :

i. \(\cfrac{36}{100}\)

ii. \(\cfrac{1}{11}\)

iii. \(4\cfrac{1}{8}\)

iv. \(\cfrac{3}{13}\)

v. \(\cfrac{2}{11}\)

vi. \(\cfrac{329}{400}\)

Solution i. :
\(\cfrac{36}{100} = 0.36\) which is a terminating decimal expansion.

Solution ii. :

\(\cfrac{1}{11} = 0.0909.... = 0.\overline{09}\) which is a non terminating decimal expansion.

Solution iii. :

\(4\frac{1}{8} = \cfrac{33}{8}= 4.125\) which is a terminating decimal expansion.

Solution iv. :

\(\frac{3}{13} = 0.\overline{230769}\) which is a terminating decimal expansion.

Solution v. :

\(\frac{2}{11} = 0.\overline{18}\) which is a terminating decimal expansion.

Solution vi. :

\(\frac{329}{400} = 0.8225\) which is a terminating decimal expansion.

Question - 2

You know that \(\cfrac{1}{7} = 0.\overline{142857}\). Can you predict what the decimal expansions of \(\cfrac{2}{7} , \cfrac{3}{7} , \cfrac{4}{7} , \cfrac{5}{7} , \cfrac{6}{7}\) are, without actually doing the long division? If so, how?

Solution :

Given : \(\cfrac{1}{7} = 0.\overline{142857}\)
\(\cfrac{2}{7} = 2\times \cfrac{1}{7} = 2 \times0.\overline{142857} = 0.\overline{285714}\)
\(\cfrac{3}{7} = 3\times \cfrac{1}{7} = 3 \times0.\overline{142857} = 0.\overline{428571}\)
\(\cfrac{4}{7} = 4\times \cfrac{1}{7} = 4 \times0.\overline{142857} = 0.\overline{571428}\)
\(\cfrac{5}{7} = 5\times \cfrac{1}{7} = 5 \times0.\overline{142857} = 0.\overline{714285}\)
\(\cfrac{6}{7} = 6\times \cfrac{1}{7} = 6 \times0.\overline{142857} = 0.\overline{857142}\)

Question - 3

Express the following in the form \(\cfrac{p}{q}\), where p and q are integers and \(q\ne0\).

i. \(0.\overline{6}\)

ii. \(0.4\overline{7}\)

iii. \(0.\overline{001}\)

Solution i. :
\(0.\overline{6} = 0.666...\)
Assume, \(x = 0.666..\) ………………(1)
Multiply both sides of Eq(1) by 10, we get,
\(10x = 6.666...\)
\(10x = 6 + 0.666..\)
\(10x = 6 +x\)
\(9x = 6\)
\(x = \cfrac{2}{3}\)
\(0.\overline{6} =\cfrac{2}{3}\)

Solution ii. :
\(0.4\overline{7} = 0.4777...\)
As there is one repeating digit, multiplying (1) by 10 on both sides, we get,
Assume, \(x = 0.477..\) ………………(1)
Multiply both sides of Eq(1) by 10, we get,
\(10x = 4.77..\)
\(10x - x = 4.77... - 0.477...\)
\(9x = 4.3\)
\(x = \cfrac{43}{90}\)
\(0.4\overline{7}=\cfrac{43}{90}\)

Solution iii. :
\(0.\overline{001} = 0.001001...\)
As there is one repeating digit, multiplying (1) by 10 on both sides, we get,
Assume, \(x = 0.001..\) ………………(1)
Multiply both sides of Eq(1) by 1000, we get,
\(1000x = 1.001..\)
\(1000x - x = 1\)
\(999x = 1\)
\(x = \cfrac{1}{999}\)
\(0.\overline{001} = \cfrac{1}{999}\)

Question - 4

Express 0.99999 …. in the form \(\cfrac{p}{q}\) . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution :
Assume, \(x = 0.9999...\) ………………..(1)
As there is one repeating digit, multiplying (1) by 10 on both sides, we get,
Multiply both sides of Eq(1) by 1000, we get,
\(10x = 9.999....\) …………….(2)
Eq(2) - Eq(1), we get,
\(10x - x = 9.9999... - 0.9999....\)
\(9x = 9\)
\(x = 1\)
\(0.9999... = 1\)

Question - 5

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\cfrac{1}{17}\) ? Perform the division to check your answer.

Question - 6

Look at several examples of rational numbers in the form \(\cfrac{p}{q}\) (\(q \ne 0\)), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Question - 7

Write three numbers whose decimal expansions are non-terminating non-recurring.

Question - 8

Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Question - 9

Classify the following numbers as rational or irrational :

i. √23

ii. √225

iii. 0.3796

iv. 7.478478…

v. 1.101001000100001…

NCERT EXERCISE - 1.4

Question - 1

Visualise 3.765 on the number line, using successive magnification.

Question - 2

Visualise \(4.\overline{26}\) on the number line, up to 4 decimal places.

NCERT EXERCISE - 1.5

Question - 1

Classify the following numbers as rational or irrational:

1. 2 − √5

2. (3+ √23)−√23

3. \(\cfrac{2√7}{7√7}\)

4. \(\cfrac{1}{√2}\)

5. 2π

Question - 2

Simplify each of the following expressions:

1. (3+√3)(2+√2)

2. (3+√3)(3−√3)

3. \((√5+√2)^2\)

4. (√5−√2)(√5+√2)

Question - 3

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = \(\cfrac{c}{d}\)⋅ This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

Question - 4

Represent \(\sqrt{9.3}\) on the number line.

Question - 5

Rationalise the denominators of the following:

1. \(\cfrac{1}{\sqrt{7}}\)

2. \(\cfrac{1}{\sqrt{7} - \sqrt{6}}\)

3. \(\cfrac{1}{\sqrt{5} - \sqrt{2}}\)

4. \(\cfrac{1}{\sqrt{7} - 2}\)

NCERT EXERCISE - 1.6

Question - 1

Find :

1. \(64^{\frac{1}{2}}\)

2. \(32^{\frac{1}{5}}\)

3. \(125^{\frac{1}{3}}\)

Question - 2

Find :

1. \(9^{\frac{3}{2}}\)

2. \(32^{\frac{2}{5}}\)

3. \(16^{\frac{3}{4}}\)

4. \(125^{\frac{-1}{3}}\)

Question - 3

Find :

1. $2^{} .2^{} $

2. \((\cfrac{1}{3^3})^7\)

3. \({\cfrac{11\frac{1}{2}}{\cfrac{11\frac{1}{4}}}\)

4. $7^{} .8^{} $

Practice Set - 1

Question - 1

Is 21 a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Question - 2

Is -12 a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Question - 3

Is \(\cfrac{-1}{2}\) a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Question - 4

Is \(\cfrac{-\sqrt{2}}{3}\) a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Question - 5

Is \(4.\overline{27}\) a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Question - 6

Is \(4.\overline{27}\) a rational number? Can you write it in the form \(\cfrac{p}{q}\), where p and q are integers and \(q \ne 0\) ?

Practice Set - 2

Question - 1

Find six rational numbers between 2 and 3.

Question - 2

Find nine rational numbers between 6 and 2.

Question - 3

Find three rational numbers between \(\cfrac{2}{5}\) and \(\cfrac{3}{5}\).

Question - 4

Find seven rational numbers between \(\cfrac{9}{3}\) and \(\cfrac{10}{3}\).

Question - 5

Find nine rational numbers between \(\cfrac{-10}{3}\) and \(\cfrac{-9}{3}\).

Practice Set - 3

Question - 1

Show how √2 can be represented on the number line.

Question - 2

Show how √3 can be represented on the number line.

Question - 3

Show how √5 can be represented on the number line.

Question - 4

Show how √6 can be represented on the number line.

Question - 5

Show how √8 can be represented on the number line.

Question - 6

Show how √9 can be represented on the number line.

Question - 7

Show how √10 can be represented on the number line.

Question - 8

Show how √11 can be represented on the number line.

Question - 9

Show how √12 can be represented on the number line.

Question - 10

Show how √13 can be represented on the number line.

Practice Set - 4

Question - 1

Write the following in decimal form and say what kind of decimal expansion each has :

i. \(\cfrac{33}{100}\)

ii. \(\cfrac{11}{111}\)

iii. \(4\cfrac{3}{8}\)

iv. \(\cfrac{7}{13}\)

v. \(\cfrac{3}{11}\)

Question - 2

Express the following in the form \(\cfrac{p}{q}\), where p and q are integers and \(q\ne0\).

i. \(0.\overline{7}\)

ii. \(0.3\overline{7}\)

iii. \(0.\overline{01}\)

Question - 3

Express the following in the form \(\cfrac{p}{q}\), where p and q are integers and \(q\ne0\).

i. \(7.\overline{7}\)

ii. \(1.3\overline{7}\)

iii. \(2.\overline{01}\)

Question - 4

Express the following in the form \(\cfrac{p}{q}\), where p and q are integers and \(q\ne0\).

i. \(2.\overline{05}\)

ii. \(9.3\overline{07}\)

iii. \(2.\overline{101}\)

Question - 5

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\cfrac{2}{17}\) ? Perform the division to check your answer.

Question - 6

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\cfrac{3}{17}\) ? Perform the division to check your answer.

Question - 7

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\cfrac{5}{17}\) ? Perform the division to check your answer.

Question - 9

Find three different irrational numbers between the rational numbers 7/11 and 9/11.

Question - 10

Find three different irrational numbers between the rational numbers 2/5 and 3/5.

Practice Set - 5

Question - 1

Visualise 1.975 on the number line, using successive magnification.

Question - 2

Visualise 9.23467 on the number line, using successive magnification.

Question - 3

Visualise 2.3546 on the number line, using successive magnification.

Question - 4

Visualise -1.2236 on the number line, using successive magnification.

Question - 5

Visualise -10.123456 on the number line, using successive magnification.

Practice Set - 6

Question - 1

Classify the following numbers as rational or irrational:

1. 2 + √5

2. (3 - √23) + √23

3. \(\cfrac{2√5}{5√5}\)

4. \(\cfrac{1}{√3}\)

5. 3π

Question - 2

Simplify each of the following expressions:

1. (3-√3)(2-√2)

2. (5+√5)(5−√5)

3. \((√5-√2)^2\)

4. (√3−√2)(√3+√2)

Question - 3

Simplify each of the following expressions:

1. (5-√3)(5-√2)

2. (7+√5)(7−√5)

3. \((2√2-3√3)^2\)

4. (√5−√3)(√5+√3)

Question - 4

Represent \(\sqrt{5.3}\) on the number line.

Question - 5

Represent \(\sqrt{8.2}\) on the number line.

Question - 6

Represent \(\sqrt{5.1}\) on the number line.

Question - 7

Rationalise the denominators of the following:

1. \(\cfrac{1}{\sqrt{2}}\)

2. \(\cfrac{1}{\sqrt{2} - \sqrt{3}}\)

3. \(\cfrac{1}{\sqrt{7} - \sqrt{3}}\)

4. \(\cfrac{1}{\sqrt{5} - 2}\)

Question - 8

Rationalise the denominators of the following:

1. \(\cfrac{1}{\sqrt{11}}\)

2. \(\cfrac{1}{\sqrt{11} - \sqrt{7}}\)

3. \(\cfrac{1}{\sqrt{11} - \sqrt{5}}\)

4. \(\cfrac{1}{\sqrt{11} - 4}\)



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