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 ?

Question:

Answer:

We often take 22/7 as an approximate value for π. So, π is an irrational number.

∴ π = 22/7 (approx), which is irrational.

c/d = 22/7 (approx), which is irrational.

∵ [π = c/d]

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(All Q's of this Chapter solved)

Simplify : (i) 2^1/3.2 ^1/5

Find : (i) 9^3/2

Find : (i) 64 1/2

Rationalise the denominators each of the following :

Represent √9.3 on the number line.

Simplify each of the following expressions :

Classify the following numbers as rational or irrational: (i) 2 − √5

Classify the following numbers as rational or irrational :

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

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

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).

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.

Express 0.999999.... in the form p/q . Are you surprised by your answer ? With your teacher and classmates discuss why the answer makes sense ?

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

You know that 1/7 = 0.142857 . 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 yes, how ?

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

Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion.

Show how √5 can be represented on the number line.

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.

State whether the following statements are true or false. Justify your answer.

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

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

Find six rational numbers between 3 and 4.

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

Chapter 1

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