NCERT solutions for Mathematics [English] Class 9 chapter 1 - Number Systems [Latest edition]

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

Find six rational numbers between 3 and 4.

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

State whether the following statement is true or false. Give reasons for your answer.

Every natural number is a whole number.

State whether the following statement is true or false. Give reasons for your answer.

Every integer is a whole number.

State whether the following statement is true or false. Give reasons for your answer.

Every rational number is a whole number.

State whether the following statement is true or false. Justify your answer.

Every irrational number is a real number.

State whether the following statement is true or false. Justify your answer.

Every point on the number line is of the form `sqrt m`, where m is a natural number.

State whether the following statement is true or false. Justify your answer.

Every real number is an irrational number.

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.

Show how `sqrt5` can be represented on the number line.

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 OP₁ of unit length. Draw a line segment P₁ P₂ perpendicular to OP₁ of unit length. Now draw a line segment P₂ P₃ perpendicular to OP₂. Then draw a line segment P₃ P₄ perpendicular to OP₃. Continuing in this manner, you can get the line segment P_n–1P_n by drawing a line segment of unit length perpendicular to OP_n–1. In this manner, you will have created the points P₂, P₃,...., P_n,.... ., and joined them to create a beautiful spiral depicting `sqrt2, sqrt3, sqrt4,` ...

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

`36/100`

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

`1/11`

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

`4 1/8`

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

`3/13`

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

`2/11`

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

`329/400`

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.]

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

`0.bar6`

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

`0.4bar7`

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

`0.bar001`

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.

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.

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?

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

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

Classify the following number as rational or irrational:

`sqrt23`

Classify the following number as rational or irrational:

`sqrt225`

Classify the following number as rational or irrational:

0.3796

Classify the following number as rational or irrational:

7.478478...

Classify the following number as rational or irrational:

1.101001000100001...

Classify the following numbers as rational or irrational:

`2-sqrt5`

Classify the following number as rational or irrational:

`(3+sqrt23)-sqrt23`

Classify the following number as rational or irrational:

`(2sqrt7)/(7sqrt7)`

Classify the following number as rational or irrational:

`1/sqrt2`

Classify the following number as rational or irrational:

2π

Simplify the following expression:

`(3+sqrt3)(2+sqrt2)`

Simplify the following expression:

`(3+sqrt3)(3-sqrt3)`

Simplify the following expression:

`(sqrt5+sqrt2)^2`

Simplify the following expression:

`(sqrt5-sqrt2)(sqrt5+sqrt2)`

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 π is irrational. How will you resolve this contradiction?

Represent `sqrt9.3`  on the number line.

Rationalise the denominator of the following:

`1/sqrt7`

Rationalise the denominator of the following:

`1/(sqrt7-sqrt6)`

Rationalise the denominator of the following:

`1/(sqrt5+sqrt2)`

Rationalise the denominator of the following:

`1/(sqrt7-2)`

Find:-

`64^(1/2)`

Find:-

`32^(1/5)`

Find:-

`125^(1/3)`

Find:-

`9^(3/2)`

Find:-

`32^(2/5)`

Find:-

`16^(3/4)`

Find:-

`125^((-1)/3)`

Simplify:-

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

Simplify:-

`(1/3^3)^7`

Simplify:

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

Simplify:

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