What are Rational Numbers? Definition & Examples for Class 8
The concept of Rational Numbers Class 8 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding rational numbers helps Class 8 students easily handle fractions and their operations in higher math and competitive exams.
What Is Rational Numbers Class 8?
A rational number is any number that can be written in the form \( \frac{p}{q} \), where p and q are integers and \( q \neq 0 \). You’ll find this concept applied in areas such as fractions, decimals, and all arithmetic operations in Class 8. Rational numbers include positive numbers, negative numbers, zero, and both terminating and repeating decimals. For example: \( \frac{3}{4} \), -5, 0, 2.25, and \( 0.333... = \frac{1}{3} \) are all rational numbers.
Key Formula for Rational Numbers Class 8
Here’s the standard formula: \( \text{Rational Number} = \frac{p}{q}, \; \text{where} \; p, q \in \mathbb{Z} \; \text{and} \; q \neq 0 \)
Operations on Rational Numbers Class 8
Rational numbers can be added, subtracted, multiplied, or divided very simply using the following rules:
| Operation | Rule | Example |
|---|---|---|
| Addition | \( \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd} \) | \( \frac{1}{2} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4} \) |
| Subtraction | \( \frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd} \) | \( \frac{1}{2} - \frac{3}{4} = \frac{2-3}{4} = \frac{-1}{4} \) |
| Multiplication | \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \) | \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \) |
| Division | \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \) | \( \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{2}{3} \) |
Step-by-Step Illustration
- Add \( \frac{2}{5} \) and \( \frac{3}{10} \ )
Find common denominator (10): \( \frac{4}{10} + \frac{3}{10} \) - Add numerators: \( 4 + 3 = 7 \)
So the answer is \( \frac{7}{10} \)
Properties of Rational Numbers Class 8
| Property | Explanation | Example |
|---|---|---|
| Closure | Addition, subtraction, multiplication of any two rational numbers gives another rational number. | \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \) |
| Commutativity | Order can be changed in addition or multiplication. | \( \frac{2}{3} + \frac{1}{6} = \frac{1}{6} + \frac{2}{3} \) |
| Associativity | Grouping does not change answer for addition/multiplication. | \( (\frac{1}{2} + \frac{1}{3}) + \frac{1}{6} = \frac{1}{2} + (\frac{1}{3} + \frac{1}{6}) \) |
| Identity | 0 is additive, 1 is multiplicative identity. | \( \frac{5}{9} + 0 = \frac{5}{9} \); \( \frac{4}{7} \times 1 = \frac{4}{7} \) |
| Inverse | Additive: \( -x \), Multiplicative: reciprocal | Additive: \( -\frac{2}{5} \); Multiplicative: \( \frac{3}{8} \rightarrow \frac{8}{3} \) |
Relation with Other Number Types
Rational numbers are different from irrational numbers. Irrational numbers cannot be written as \( \frac{p}{q} \) (for example, \( \sqrt{2} \) or \( \pi \)). Every integer and fraction is a rational number, but not every decimal is rational. For a detailed comparison, see the Rational and Irrational Numbers page on Vedantu.
Number Line Representation
Any rational number can be shown on a number line. For example, to represent \( \frac{3}{4} \), divide the space between 0 and 1 into 4 parts. The third part is \( \frac{3}{4} \). Negative rational numbers go to the left of zero. Visualizing numbers on a number line is important for comparison and ordering. Practice this on Vedantu’s Rational Numbers on a Number Line resource.
Speed Trick or Vedic Shortcut
Here’s a handy trick for comparing rational numbers:
- Cross-multiply the two numbers.
- For \( \frac{2}{5} \) and \( \frac{3}{8} \), cross multiply: 2 × 8 = 16; 3 × 5 = 15.
- Since 16 > 15, \( \frac{2}{5} > \frac{3}{8} \).
Tricks like this will save time in exams and are often recommended by Vedantu’s math tutors.
Try These Yourself
- Write five rational numbers between -1 and 1.
- Check if 0 is a rational number.
- Convert 0.7 (repeating) into a rational number in p/q form.
- Is \( \sqrt{7} \) a rational number? Why or why not?
Frequent Errors and Misunderstandings
- Forgetting that the denominator cannot be zero.
- Assuming every decimal is rational (non-terminating, non-repeating decimals are not rational).
- Mixing up positive and negative rational numbers.
- Not simplifying fractions to lowest terms.
Relation to Other Concepts
The topic of Rational Numbers Class 8 is closely connected to integers and fractions. Strengthening this chapter helps students master topics like algebra, proportional reasoning, and number systems in higher classes.
Classroom Tip
A quick way to remember rational numbers: "If you can write the number as a simple fraction, it’s rational." Vedantu’s teachers often use shaded grids and number lines to make this idea clear and easy to visualize for all students.
We explored Rational Numbers Class 8—from the basic definition, operations, properties, and common errors to important tips. Continue practicing with Vedantu’s rational number worksheet and concept resources to become confident in exams and everyday math problems.
Want more practice? Next, review these chapters:
FAQs on Rational Numbers for Class 8 – Concepts, Examples, & Practice
1. What are rational numbers?
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. Examples include ½, -¾, 0, 5, and 2.75 (because it can be written as 11/4).
2. How do you identify a rational number?
A number is rational if it can be written as a fraction of two integers where the denominator is not zero. Alternatively, if its decimal representation either terminates (e.g., 0.75) or repeats (e.g., 0.333…), it’s a rational number.
3. Give five examples of rational numbers.
Here are five examples: ½, -3/4, 0, 5, -2.2 (which is -11/5).
4. Is 0 a rational number?
Yes, 0 is a rational number. It can be expressed as 0/1, 0/2, or any other fraction with 0 as the numerator and a non-zero integer as the denominator.
5. Is √7 a rational number?
No, √7 is not a rational number. It is an irrational number because its decimal representation is non-terminating and non-repeating. It cannot be expressed as a fraction of two integers.
6. How do you add and subtract rational numbers?
To add or subtract rational numbers, find a common denominator. Then, add or subtract the numerators and keep the common denominator. Simplify the result if possible.
7. How do you multiply and divide rational numbers?
To multiply rational numbers, multiply the numerators together and multiply the denominators together. To divide rational numbers, invert the second fraction (find its reciprocal) and then multiply.
8. What is the multiplicative inverse of a rational number?
The multiplicative inverse (or reciprocal) of a rational number a/b is b/a, provided a is not zero. Multiplying a number by its multiplicative inverse always results in 1 (except for 0).
9. What is the difference between rational and irrational numbers?
Rational numbers can be expressed as a fraction of two integers (with a non-zero denominator). Irrational numbers cannot be expressed as such a fraction; their decimal representation is non-terminating and non-repeating.
10. How do you represent rational numbers on a number line?
Divide the number line into equal segments based on the denominator of the fraction. Then, locate the point corresponding to the numerator.
11. What are the properties of rational numbers?
Rational numbers are closed under addition, subtraction, and multiplication. They are also commutative and associative under addition and multiplication. There exist additive and multiplicative identities (0 and 1, respectively), and every non-zero rational number has a multiplicative inverse.
12. How do you convert a repeating decimal to a rational fraction?
Let's say you have a repeating decimal like 0.333... Let x = 0.333... Multiply by 10 (or 100, 1000 depending on repeating digits): 10x = 3.333... Subtract the first equation from the second: 9x = 3. Solving for x gives x = 1/3, which is the rational fraction representation.
