Number Systems

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State whether the following statement is true: There is a number x such that x2 is irrational but x4 is rational. Justify your answer by an example.

: Are there two irrational numbers whose sum and product both are rationals? Justify.

Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.

Every rational number is

(A) a natural number (B) an integer

(C) a real number (D) a whole number

Between two rational numbers

(A) there is no rational number (B) there is exactly one rational number (C) there are infinitely many rational numbers (D) there are only rational numbers and no irrational numbers

Decimal representation of a rational number cannot be

(A) terminating (B) non-terminating (C) non-terminating repeating (D) non-terminating non-repeating

The product of any two irrational numbers is

(A) always an irrational number (B) always a rational number (C) always an integer (D) sometimes rational, sometimes irrational.

Simplify:

(i)  (ii)  (iii) 

(iv) 

Q2. Find:

(i)  (ii)  (iii) 

(iv) 

Find:

(i)  (ii)  (iii) 

Rationalise the denominators of the following:

(i)  (ii) 

(iii)  (iv) 

Represent  on the number line.

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

Simplify each of the following expressions:

(i)  (ii) 

(iii)  (iv) 

1Classify the following numbers as rational or irrational:

(i)  (ii)  (iii) 

(iv)  (v) 2π

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

Visualise 3.765 on the number line using successive magnification.

Classify the following numbers as rational or irrational:

(i)  (ii)  (iii) 0.3796

(iv) 7.478478 (v) 1.101001000100001…

Find three different irrational numbers between the rational numbers and.

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

Look at several examples of rational numbers in the form (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?

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

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

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

(i)  (ii)  (iii) 

You know that. Can you predict what the decimal expansion of  are, without actually doing the long division? If so, how?

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

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

(i)  (ii)  (iii) 

(iv)  (v)  (vi) 

Show howcan be represented on the number line

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, where m is a natural number.

(iii) Every real number is an irrational number.

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.

Find five rational numbers between.

Find six rational numbers between 3 and 4.

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