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# NCERT solutions Mathematics Class 9 chapter 1 Number Systems

## Chapter 1 - Number Systems

#### Page 5

Is zero a rational number? Can you write it in the form p/q, where p and q are integersand q ≠ 0?

Q 1 | Page 5

Find six rational numbers between 3 and 4.

Q 2 | Page 5

Find five rational numbers between 3/5 and 4/5.

Q 3 | Page 5

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.

Q 4 | Page 5

#### Page 8

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 sqrt m, where m is a natural number.

(iii) Every real number is an irrational number.

Q 1 | Page 8

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.

Q 2 | Page 8

Show how sqrt5 can be represented on the number line.

Q 3 | Page 8

#### Page 14

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

(i) 36/100

(ii) 1/11

(iii) 4 1/8

(iv) 3/13

(v) 2/11

(vi) 329/400

Q 1 | Page 14

You know that 1/7=0.bar142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7  are, Without actually doing the long division? If so, how?

[Hint : Study the remainders while finding the value of 1/7 carefully.]

Q 2 | Page 14

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

(i) 0.bar6

(ii) 0.4bar7

(iii) 0.bar001

Q 3 | Page 14

Express 0.99999 .... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Q 4 | Page 14

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

Q 5 | Page 14

Look at several examples of rational numbers in the form p/q (q≠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?

Q 6 | Page 14

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

Q 7 | Page 14

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

Q 8 | Page 14

Classify the following numbers as rational or irrational :-

(i) sqrt23

(ii) sqrt225

(iii) 0.3796

(iv) 7.478478...

(v) 1.101001000100001...

Q 9 | Page 14

#### Page 18

Visualise 3.765 on the number line, using successive magnification.

Q 1 | Page 18

Visualise 4.bar26  on the number line, up to 4 decimal places.

Q 2 | Page 18

#### Page 24

Classify the following numbers as rational or irrational :-

(i) 2-sqrt5

(ii) (3+sqrt23)-sqrt23

(iii) (2sqrt7)/(7sqrt7)

(iv) 1/sqrt2

(v) 2π

Q 1 | Page 24

Simplify each of the following expressions :-

(i) (3+sqrt3)(2+sqrt2)

(ii) (3+sqrt3)(3-sqrt3)

(iii) (sqrt5+sqrt2)^2

(iv) (sqrt5-sqrt2)(sqrt5+sqrt2)

Q 2 | Page 24

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

Q 3 | Page 24

Represent sqrt9.3  on the number line.

Q 4 | Page 24

Rationalise the denominators of the following :-

(i) 1/sqrt7

(ii) 1/(sqrt7-sqrt6)

(iii) 1/(sqrt5+sqrt2)

(iv) 1/(sqrt7-2)

Q 5 | Page 24

#### Page 26

Find :-

(i) 64^(1/2)

(ii) 32^(1/5)

(iii) 125^(1/3)

Q 1 | Page 26

Find :-

(i) 9^(3/2)

(ii) 32^(2/5)

(iii) 16^(3/4)

(iv) 125^(-1/3)

Q 2 | Page 26

Simplify :-

(i) 2^(2/3).2^(1/5)

(ii) (1/3^3)^7

(iii) 11^(1/2)/11^(1/4)

(iv) 7^(1/2).8^(1/2)

Q 3 | Page 26

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