Rational and Irrational Numbers

Q1: Multiple Choice Type

a. Let zero = p/q, where p and q are integers. What additional condition will make 0 = p/q a rational number?

Step 1: Definition of rational number
A number is rational if it can be written as p/q where q ≠ 0
Step 2: Apply condition
Here, 0 = p/q ⇒ valid only when denominator q ≠ 0
Answer: iii. q ≠ 0

b. Every non-terminating decimal number is a:

Step 1: Recall types of decimal numbers
Non-terminating decimals can be either recurring or non-recurring
Step 2: Classification
All such numbers belong to the set of real numbers
Answer: ii. real number

c. 7.478478….. is a:

Step 1: Observe the pattern
7.478478… shows repeating block “478”
Step 2: Identify type
A repeating decimal is called a recurring decimal
Answer: ii. Recurring

d. 71/75 is:

Step 1: Factorize denominator
75 = 3 × 5 × 5
Step 2: Check condition
Denominator has factor other than 2 or 5
Step 3: Conclusion
So decimal expansion is non-terminating
Answer: ii. non-terminating

e. Which of the following is terminating: 13/85, 51/405 and 9/524?

Step 1: Condition for terminating decimal
Denominator should have only prime factors 2 and/or 5
Step 2: Check 13/85
85 = 5 × 17 ⇒ contains 17 → not terminating
Step 3: Check 51/405
405 = 3 × 3 × 3 × 3 × 5 ⇒ contains 3 → not terminating
Step 4: Check 9/524
524 = 2 × 2 × 131 ⇒ contains 131 → not terminating
Answer: iv. None of these

Q2: Are the following statements true or false? Give reasons for your answers.

i. Every whole number is a natural number.

Step 1: Recall definitions
Natural numbers = 1, 2, 3, 4, …
Whole numbers = 0, 1, 2, 3, …
Step 2: Compare sets
Whole numbers include 0 but natural numbers do not include 0
Step 3: Conclusion
Since 0 is a whole number but not a natural number, the statement is false
Answer: False

ii. Every whole number is a rational number.

Step 1: Definition of rational number
A number is rational if it can be written in the form p/q where q ≠ 0
Step 2: Express whole number
Any whole number n can be written as n/1
Step 3: Conclusion
Since n/1 is of the form p/q, every whole number is rational
Answer: True

iii. Every integer is a rational number.

Step 1: Recall integers
Integers include …, -2, -1, 0, 1, 2, …
Step 2: Express integer in rational form
Any integer n can be written as n/1
Step 3: Conclusion
Since n/1 is a rational number, every integer is rational
Answer: True

iv. Every rational number is a whole number.

Step 1: Recall rational numbers
Rational numbers include fractions like 1/2, -3/4, etc.
Step 2: Counter example
1/2 is a rational number but not a whole number
Step 3: Conclusion
Since all rational numbers are not whole numbers, the statement is false
Answer: False

Q3: Arrange \( -\frac{5}{9},\ \frac{7}{12},\ -\frac{2}{3}\) and \( \frac{11}{18} \) in the ascending order of their magnitudes. Also, find the difference between the largest and the smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

Step 1: Convert all fractions into like denominators
LCM of \(9, 12, 3\) and \(18 = 36\) \[ -\frac{5}{9} = \frac{-5 \times 4}{9 \times 4} = \frac{-20}{36} \\ \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36} \\ -\frac{2}{3} = \frac{-2 \times 12}{3 \times 12} = \frac{-24}{36} \\ \frac{11}{18} = \frac{11 \times 2}{18 \times 2} = \frac{22}{36} \]Step 2: Compare numerators \[ -24 < -20 < 21 < 22 \]Step 3: Write ascending order \[ -\frac{2}{3} < -\frac{5}{9} < \frac{7}{12} < \frac{11}{18} \]Step 4: Identify largest and smallest
Largest \(= \frac{11}{18}\)
Smallest \(= -\frac{2}{3}\)
Step 5: Find difference \[ \text{Difference} = \frac{11}{18} – \left(-\frac{2}{3}\right) \\ = \frac{11}{18} + \frac{2}{3} \]Convert: \[ \frac{2}{3} = \frac{12}{18} \\ \text{Difference} = \frac{11}{18} + \frac{12}{18} = \frac{23}{18} \]Step 6: Convert into decimal \[ \frac{23}{18} = 1.277\ldots \]Correct to one decimal place: \[ = 1.3 \]Answer: Ascending order: \( -\frac{2}{3} < -\frac{5}{9} < \frac{7}{12} < \frac{11}{18} \)
Difference \(= 1.3\)

Q4: Arrange \( \frac{5}{8},\ -\frac{3}{16},\ -\frac{1}{4} \) and \( \frac{17}{32} \) in the descending order of their magnitudes. Also, find the sum of the lowest and the largest of these rational numbers. Express the result obtained as a decimal fraction correct to two decimal places.

Step 1: Convert into like denominators
LCM of \(8, 16, 4, 32 = 32\) \[ \frac{5}{8} = \frac{5 \times 4}{8 \times 4} = \frac{20}{32} \\ -\frac{3}{16} = \frac{-3 \times 2}{16 \times 2} = \frac{-6}{32} \\ -\frac{1}{4} = \frac{-1 \times 8}{4 \times 8} = \frac{-8}{32} \\ \frac{17}{32} = \frac{17}{32} \]Step 2: Compare numerators \[ 20 > 17 > -6 > -8 \]Step 3: Write descending order \[ \frac{5}{8} > \frac{17}{32} > -\frac{3}{16} > -\frac{1}{4} \]Step 4: Identify values
Largest \(= \frac{5}{8}\)
Lowest \(= -\frac{1}{4}\)
Step 5: Find sum \[ \text{Sum} = \frac{5}{8} + \left(-\frac{1}{4}\right) \\ = \frac{5}{8} – \frac{1}{4} \]Convert: \[ \frac{1}{4} = \frac{2}{8} \\ \text{Sum} = \frac{5}{8} – \frac{2}{8} = \frac{3}{8} \]Step 6: Convert into decimal \[ \frac{3}{8} = 0.375 \]Correct to two decimal places: \[ = 0.38 \]Answer: Descending order: \( \frac{5}{8} > \frac{17}{32} > -\frac{3}{16} > -\frac{1}{4} \)
Sum \(= 0.38\)

Q5: Without doing any actual division, find which of the following rational numbers have terminating decimal representation:

i. \( \frac{7}{16} \)

Step 1: Condition for terminating decimal
A rational number has a terminating decimal if the denominator has only prime factors \(2\) and/or \(5\)
Step 2: Factorize denominator \[ 16 = 2^4 \]Only factor is \(2\)
Answer: Terminating

ii. \( \frac{23}{125} \)

Step 1: Factorize denominator \[ 125 = 5^3 \]Only factor is \(5\)
Answer: Terminating

iii. \( \frac{9}{14} \)

Step 1: Factorize denominator \[ 14 = 2 \times 7 \]Contains factor \(7\) (not allowed)
Answer: Non-terminating

iv. \( \frac{32}{45} \)

Step 1: Factorize denominator \[ 45 = 3^2 \times 5 \]Contains factor \(3\) (not allowed)
Answer: Non-terminating

v. \( \frac{43}{50} \)

Step 1: Factorize denominator \[ 50 = 2 \times 5^2 \]Only factors are \(2\) and \(5\)
Answer: Terminating