# Number System

## Exercise 1.3 Part 3

Question 5: 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.

**Answer:**

= 0.0588235294117647

Thus, maximum number of digits in the repeating block is 17.

Question 6: 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.

**Answer:** For having terminating decimal expansions, the denominator should have either 2 or 5 or both as factor. So, q must have either 2 or 5 or both.

Examples:

`1/2=0.5` terminating

`1/3` = 0.3 non-terminating repeating

`1/4=0.25` terminating

`1/5=0.2` terminating

`1/6` = 0.16 non-terminating repeating

`1/7` = 0.142857 non-terminating repeating

`1/8=0.125` terminating

`1/9` = 0.1 non-terminating repeating

Question 7: Write three numbers whose decimal expansions are non-terminating non-recurring.

**Answer:** Non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q≠0.

Following are the possible numbers:

0.72012001200012000001………

0.73013001300013000001…………

0.7501500150001500001………..

Question 8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.

**Answer:** 5/7 = 0.714285714285…….. and 9/11 = 0.8181818………

Possible irrational numbers between them can be as follows:

0.72012001200012000001………

0.73013001300013000001…………

0.7501500150001500001………..

Note: Non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q≠0. Numbers given above cannot be expressed in the form of p/q and hence are irrational.