R s aggarwal solutions mathematics class 9 Solutions for Chapter 1 number systems

Is zero a rational number? Justify.

Represent each of the following rational numbers on the number line:

(i)?/?

(ii)?/?

(iii) -??/?

(iv) 1.3

(v) -2.4

Find a rational number between

(i)?/? and ?/?

(ii) ?. ? ??? ?. ?

(iii) -1 and ?/?

(iv) -3/? and – ?/?

(v) ?/? and ?/?

Find three rational number lying between ?/? and ?/? .

How many rational numbers can be determined between these two numbers?

Find four rational numbers between ?/? and ?/?.

Find the six rational numbers between 2 and 3.

Find five rational numbers between ?/? and ?/?.

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

(i) Every natural number is a whole number.

(ii) Every whole number is a natural number.

(iii) Every integer is a whole number.

(iv) Every integer is a rational number.

(v) Every rational number is an integer.

(vi) Every rational number is a whole number.

Express 2.̅?̅̅?̅ + 0.̅?̅̅?̅ as a fraction in simplest form.

Is the product of two irrationals always irrational? Justify your answer.

Examine whether the following numbers are rational or irrational.

(i) 3 + √?

(ii) √? – 2

(iii) √? ? × √?? ?

(iv) √? × √???

(v) √ ??/???

(vi) √? × √?

Insert a rational and an irrational number between 2 and 2.5.

How many irrational numbers lie between √? and√?? Find any three irrational numbers lying between √? and√?.

Find two rational and two irrational numbers between 0.5 and 0.55.

Find the three different irrational numbers between the rational numbers ?/? and ?/??.

Find two rational numbers of the form ?/? between the numbers 0.2121121112 …. And 0.2020020002…. .

Find two irrational numbers between 0.16 and 0.17.

State, in each case, whether the given statement is true or false.

(i) The sum of two rational numbers is rational.

(ii) The sum of two irrational numbers is irrational.

(iii) The product of two rational numbers is rational.

(iv) The product of two irrational numbers is irrational.

(v) The sum of a rational number and an irrational number is irrational.

(vi) The product of a non-zero rational number and an irrational number is a rational number.

(vii) Every real number is rational.

(viii) Every real number is either rational or irrational.

(ix) ? is irrational and ??/? is rational.

Multiply

(i) 3√? by 2√?

(ii) 6√?? by 4√?

(iii) 2√? by 3√?

(iv) 3√? by 3√?

(v) √?? by √??

(vi) 3√?? by 2√?

Divide

(i) 16 √? by 4 √?

(ii) 12 √?? by 4 √?

(iii) 18 √?? by 6 √?

Examine whether the following numbers are rational or irrational:

(i) (5 – √?) (5 +√?)

(ii) (√? + ?) ?

(iii) ?√??/(?√??−?√???)

(iv) √? + ?√?? − ?√?

Represent √? on the number line.

Represent √?. ? geometrically on the number line.

Represent √?. ?? geometrically on the number line.

Represent (1 +√?. ?) on the number line.

Visualize the representation of 3.765 on the number line using successive magnification.

It being given that √? = 1.414, √? = 1.732, √? = 2.236 and √?? = 3.162, find the value to three

places of decimals, of each of the following.

(i)?/√?

(ii)(?− √?)/√?

(iii)(√??− √?)/√?

Find the values of a and b if

(?+?√?)/(?+√?)–(?−?√?)/(?−√?)= a + b√?

If x =(?− √??)/?, find the value of x +?/?.