# Number Systems

**Exercise 1**

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

**2.** Find six rational numbers between 3 and 4.

**3.** Find five rational numbers between .

**Exercise 2**

**4.** 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.

**5.** 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.

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

**Exercise 3**

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

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

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

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

**10.** Express 0.99999…. in the form Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmates.

**11.** What can the maximum number of digits be in the recurring block of digits in the decimal expansion of Perform the division to check your answer.

**12.** Look at several examples of rational numbers in the form 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?

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

**14.** Find three different irrational numbers between the rational numbers

**15.** Classify the following numbers as rational or irrational:

(i) 23 (ii) 225 (iii) 0.3796 (iv) 7.478478…

(v) 1.101001000100001…

**Exercise 4**

**16. ** Visualize 3.765 on the number line using successive magnification.

**17.** Visualize on the number line, up to 4 decimal places.

**Exercise 5**

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

**19. **Simplify each of the following expressions:

**20.** 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 g is irrational. How will you resolve this contradiction?

**21.** Represent 9.3 on the number line.

**22. ** Rationalize the denominators of the following:

**Exercise 6**

**23.** Find:

**24.** Find:

**25. ** Simplify: