Class 8 Maths Rational Numbers Notes

A rational number is any number that can be expressed in the form:$$\frac{p}{q}, \quad p \in \mathbb{Z}, \; q \in \mathbb{Z}, \; q \neq 0$$qp​,p∈Z,q∈Z,q=0

• Here, $p$p = numerator, $q$q = denominator.
• Every integer is a rational number ($n = \frac{n}{1}$n=1n​).
• 0 is a rational number ($0 = \frac{0}{1}$0=10​).
• Not all numbers are rational (e.g., $\sqrt{2}$2​, $\pi$π).

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Standard Form of a Rational Number:

• Denominator is positive.
• Numerator and denominator have no common factor except 1.

Example: $\frac{-4}{6} = \frac{-2}{3}$6−4​=3−2​

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Types of Rational Numbers:

1. Positive Rational Numbers: > 0
2. Negative Rational Numbers: < 0
3. Zero: 0

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Properties of Rational Numbers:

1. Closure:
   • Sum, difference, and product of two rational numbers is a rational number.
   • Quotient of two rational numbers (non-zero divisor) is rational.
2. Additive Inverse:
   • For $\frac{a}{b}$ba​, additive inverse = $-\frac{a}{b}$−ba​
3. Multiplicative Inverse (Reciprocal):
   • For $\frac{a}{b} \neq 0$ba​=0, reciprocal = $\frac{b}{a}$ab​
4. Decimal Representation:
   • Can be terminating (e.g., $\frac{3}{4} = 0.75$43​=0.75)
   • Can be repeating (e.g., $\frac{1}{3} = 0.333…$31​=0.333…)
5. Between Two Numbers:
   • There is always a rational number between any two rational numbers.

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Examples:

• $\frac{2}{3}, -\frac{5}{7}, 0, 4, -3$32​,−75​,0,4,−3 are all rational numbers.
• Non-examples: $\sqrt{2}, \pi$2​,π

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Operations Quick-Formulas:

Operation	Formula	Example
Sum	$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ba​+dc​=bdad+bc​	$\frac{2}{3} + \frac{4}{5} = \frac{22}{15}$32​+54​=1522​
Difference	$\frac{a}{b} – \frac{c}{d} = \frac{ad – bc}{bd}$ba​−dc​=bdad−bc​	$\frac{3}{4} – \frac{1}{2} = \frac{1}{4}$43​−21​=41​
Product	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ba​×dc​=bdac​	$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$32​×43​=21​
Quotient	$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$ba​÷dc​=bcad​	$\frac{5}{6} \div \frac{2}{3} = \frac{5}{4}$65​÷32​=45​

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Tips to Remember:

• All integers = rational numbers.
• Reciprocal of 0 does not exist.
• Negative × Negative = Positive; Negative × Positive = Negative.
• Always simplify fractions to standard form for clarity.

Rational Numbers – MCQ Questions with Answers

Definition & Basic Properties

1. Which of the following is a rational number?
   Answer: Any number that can be expressed as $\frac{p}{q}$qp​, e.g., $\frac{2}{3}$32​.
2. Is 0 a rational number?
   Answer: Yes, $0 = \frac{0}{1}$0=10​.
3. Is every integer a rational number?
   Answer: Yes, $n = \frac{n}{1}$n=1n​.
4. Can every fraction $\frac{p}{q}$qp​ be a rational number?
   Answer: Yes, if $q \neq 0$q=0.
5. Is $\sqrt{2}$2​ a rational number?
   Answer: No.
6. Which fraction is in standard form?
   Answer: $\frac{p}{q}$qp​ with $q > 0$q>0 and gcd(p, q) = 1.
7. What is the additive inverse of $\frac{7}{5}$57​?
   Answer: $-\frac{7}{5}$−57​
8. What is the multiplicative inverse of $\frac{3}{4}$43​?
   Answer: $\frac{4}{3}$34​
9. Can a rational number be negative?
   Answer: Yes.
10. Can a rational number be expressed as a repeating decimal?
    Answer: Yes.
11. Can a rational number be expressed as a terminating decimal?
    Answer: Yes.
12. Is $-\frac{5}{6}$−65​ a rational number?
    Answer: Yes.
13. If $a$a and $b$b are integers, $b \neq 0$b=0, is $\frac{a}{b}$ba​ a rational number?
    Answer: Yes.
14. Is 7 a rational number?
    Answer: Yes, $7 = \frac{7}{1}$7=17​.
15. Which of these is not a rational number: 0.333…, $\sqrt{3}$3​, -2?
    Answer: $\sqrt{3}$3​

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Operations on Rational Numbers

16. The sum of two rational numbers is always:
    Answer: A rational number.
17. The difference of two rational numbers is always:
    Answer: A rational number.
18. The product of two rational numbers is always:
    Answer: A rational number.
19. The quotient of two rational numbers ($b \neq 0$b=0) is always:
    Answer: A rational number.
20. $\frac{2}{3} + \frac{4}{5} = ?$32​+54​=?
    Answer: $\frac{22}{15}$1522​
21. $\frac{3}{7} – \frac{2}{7} = ?$73​−72​=?
    Answer: $\frac{1}{7}$71​
22. $\frac{2}{3} \times \frac{3}{4} = ?$32​×43​=?
    Answer: $\frac{1}{2}$21​
23. $\frac{5}{6} \div \frac{2}{3} = ?$65​÷32​=?
    Answer: $\frac{5}{4}$45​
24. Additive inverse of a rational number $r$r is:
    Answer: $-r$−r
25. Multiplicative inverse of a non-zero rational number $r$r is:
    Answer: $\frac{1}{r}$r1​
26. If $\frac{a}{b}$ba​ is positive, $-\frac{a}{b}$−ba​ is:
    Answer: Negative
27. If a rational number is multiplied by 0, the result is:
    Answer: 0 (rational number)
28. If a rational number is multiplied by 1, the result is:
    Answer: The same rational number
29. If $\frac{p}{q} = \frac{r}{s}$qp​=sr​, then $\frac{p}{q} – \frac{r}{s} = ?$qp​−sr​=?
    Answer: 0
30. Simplify $\frac{-2}{3} + \frac{5}{3} = ?$3−2​+35​=?
    Answer: $\frac{3}{3} = 1$33​=1

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Special Cases & Examples

31. Which rational number lies between 0 and 1?
    Answer: $\frac{1}{2}$21​
32. Which rational number lies between $\frac{1}{2}$21​ and $\frac{3}{4}$43​?
    Answer: $\frac{5}{8}$85​
33. Are repeating decimals always rational?
    Answer: Yes.
34. Can a rational number have a denominator of 1?
    Answer: Yes, that gives an integer.
35. If $\frac{a}{b}$ba​ is negative, what is the sign of $-\frac{a}{b}$−ba​?
    Answer: Positive
36. Can 0 be the denominator of a rational number?
    Answer: No
37. Convert -3 to rational form:
    Answer: $-\frac{3}{1}$−13​
38. Convert 0.75 to a rational number:
    Answer: $\frac{3}{4}$43​
39. Express $-0.6$−0.6 as a rational number:
    Answer: $-\frac{3}{5}$−53​
40. Express 0.333… as a rational number:
    Answer: $\frac{1}{3}$31​
41. Which is larger: $\frac{2}{5}$52​ or $\frac{3}{7}$73​?
    Answer: $\frac{2}{5}$52​
42. Reciprocal of –$\frac{4}{5}$54​ is:
    Answer: –$\frac{5}{4}$45​
43. Product of a rational number and its reciprocal is:
    Answer: 1
44. Additive inverse of 0 is:
    Answer: 0
45. Multiply $\frac{-2}{3} \times \frac{3}{2} = ?$3−2​×23​=?
    Answer: -1
46. Divide $\frac{5}{6} \div \frac{-5}{3} = ?$65​÷3−5​=?
    Answer: –$\frac{1}{2}$21​
47. If numerator = denominator, rational number = ?
    Answer: 1
48. A rational number with numerator 0 = ?
    Answer: 0
49. Are integers positive rational numbers?
    Answer: Only if integer > 0
50. Can a rational number lie between -1 and 0?
    Answer: Yes, e.g., –$\frac{1}{2}$21​
51. Sum of –$\frac{3}{4}$43​ and $\frac{5}{4}$45​ = ?
    Answer: $\frac{2}{4} = \frac{1}{2}$42​=21​
52. Difference between –$\frac{2}{3}$32​ and –$\frac{1}{3}$31​ = ?
    Answer: –$\frac{1}{3}$31​
53. Multiply –$\frac{1}{2}$21​ × –$\frac{4}{3}$34​ = ?
    Answer: $\frac{2}{3}$32​