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NCERT solutions Mathematics Class 9 chapter 1 Number Systems

Chapters

NCERT Mathematics Class 9

Mathematics Textbook for Class 9

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

NCERT Mathematics Class 9

Mathematics Textbook for Class 9
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