Class 8 — Mathematics (NCERT)
Rational Numbers
Class 8
- ✓By the end of this lesson students will be able to define rational numbers and identify them.
- ✓By the end of this lesson students will be able to understand and apply the properties of rational numbers (closure, commutativity, associativity, distributivity).
- ✓By the end of this lesson students will be able to identify additive and multiplicative identities and inverses for rational numbers.
- ✓By the end of this lesson students will be able to represent rational numbers accurately on a number line.
- ✓By the end of this lesson students will be able to find rational numbers between any two given rational numbers.
Key concepts
A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero (q ≠ 0). The set of rational numbers includes natural numbers, whole numbers, and integers. For example, 1/2, -3/4, 5 (which can be written as 5/1), and 0 (which can be written as 0/1) are all rational numbers.
Rational numbers exhibit several important properties under different operations:
A set of numbers is closed under an operation if performing that operation on any two numbers in the set always results in a number that is also in the set.\n\n1. **Addition**: Rational numbers are closed under addition. If a and b are rational numbers, then a + b is also a rational number.\n2. **Subtraction**: Rational numbers are closed under subtraction. If a and b are rational numbers, then a - b is also a rational number.\n3. **Multiplication**: Rational numbers are closed under multiplication. If a and b are rational numbers, then a × b is also a rational number.\n4. **Division**: Rational numbers are closed under division, provided the divisor is not zero. If a and b are rational numbers and b ≠ 0, then a ÷ b is also a rational number.
An operation is commutative if changing the order of the operands does not change the result.\n\n1. **Addition**: Addition is commutative for rational numbers. For any two rational numbers a and b, a + b = b + a.\n2. **Subtraction**: Subtraction is NOT commutative for rational numbers. For example, 1/2 - 1/3 ≠ 1/3 - 1/2.\n3. **Multiplication**: Multiplication is commutative for rational numbers. For any two rational numbers a and b, a × b = b × a.\n4. **Division**: Division is NOT commutative for rational numbers. For example, 1/2 ÷ 1/3 ≠ 1/3 ÷ 1/2.
An operation is associative if the grouping of the operands does not change the result.\n\n1. **Addition**: Addition is associative for rational numbers. For any three rational numbers a, b, and c, (a + b) + c = a + (b + c).\n2. **Subtraction**: Subtraction is NOT associative for rational numbers.\n3. **Multiplication**: Multiplication is associative for rational numbers. For any three rational numbers a, b, and c, (a × b) × c = a × (b × c).\n4. **Division**: Division is NOT associative for rational numbers.
Zero (0) is the additive identity for rational numbers. When 0 is added to any rational number, the sum is the rational number itself. For any rational number a, a + 0 = 0 + a = a.
One (1) is the multiplicative identity for rational numbers. When any rational number is multiplied by 1, the product is the rational number itself. For any rational number a, a × 1 = 1 × a = a.
For every rational number a/b, there exists another rational number -a/b such that their sum is 0. -a/b is called the additive inverse or negative of a/b. For example, the additive inverse of 2/3 is -2/3, because 2/3 + (-2/3) = 0.
For every non-zero rational number a/b, there exists a rational number b/a such that their product is 1. b/a is called the reciprocal or multiplicative inverse of a/b. For example, the reciprocal of 2/3 is 3/2, because (2/3) × (3/2) = 1. Note that 0 has no reciprocal.
For any three rational numbers a, b, and c:\n\n1. **Over Addition**: a × (b + c) = (a × b) + (a × c)\n2. **Over Subtraction**: a × (b - c) = (a × b) - (a × c)
To represent a rational number p/q on a number line:\n\n1. Draw a number line and mark integers (0, 1, -1, etc.).\n2. **For positive rational numbers (p/q, where p, q > 0)**: Divide the unit length between 0 and 1 (or 1 and 2, etc., depending on the value) into 'q' equal parts. Then, count 'p' parts from 0 to the right.\n * Example: To represent 3/4, divide the segment between 0 and 1 into 4 equal parts. The third mark from 0 represents 3/4.\n3. **For negative rational numbers (-p/q, where p, q > 0)**: Divide the unit length between 0 and -1 (or -1 and -2, etc.) into 'q' equal parts. Then, count 'p' parts from 0 to the left.\n * Example: To represent -2/5, divide the segment between 0 and -1 into 5 equal parts. The second mark from 0 to the left represents -2/5.
There are infinitely many rational numbers between any two given rational numbers. To find rational numbers between two given rational numbers, we can use the following methods:\n\n1. **Common Denominator Method**: Convert the given rational numbers to equivalent fractions with a common denominator. If there aren't enough integers between the numerators, multiply both the numerator and denominator by a suitable number (e.g., 10, 100) to create more 'space' between them.\n2. **Mean Method**: The mean (average) of two rational numbers a and b, which is (a+b)/2, is always a rational number lying between a and b. This method can be repeated to find more rational numbers.
Key facts to remember
- 1A rational number can be written as p/q, where p and q are integers and q ≠ 0.
- 2Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero).
- 3Addition and multiplication are commutative and associative for rational numbers.
- 4Zero (0) is the additive identity for rational numbers.
- 5One (1) is the multiplicative identity for rational numbers.
- 6Every rational number a/b has an additive inverse -a/b such that a/b + (-a/b) = 0.
- 7Every non-zero rational number a/b has a multiplicative inverse (reciprocal) b/a such that (a/b) × (b/a) = 1.
- 8Multiplication is distributive over addition and subtraction for rational numbers: a(b+c) = ab + ac and a(b-c) = ab - ac.
Worked examples
Example 1
Using suitable properties, evaluate: (2/5) × (-3/7) - (1/14) - (3/7) × (3/5)
Answer
-1/2
Always look for common factors or terms that can be grouped to simplify calculations using properties like distributivity.
Example 2
Represent -7/4 and 5/3 on the number line.
Answer
A number line with -7/4 marked between -1 and -2 (at the third division from -1 towards -2) and 5/3 marked between 1 and 2 (at the second division from 1 towards 2).
It is helpful to convert improper fractions to mixed fractions first to identify the integers between which the rational number lies.
Example 3
Find five rational numbers between 1/4 and 1/2.
Answer
Five rational numbers between 1/4 and 1/2 are 11/40, 12/40, 13/40, 14/40, 15/40 (or any other five valid rational numbers).
There are infinitely many rational numbers between any two given rational numbers. The choice of the multiplier (e.g., 10 in this case) depends on how many numbers you need to find.
Common mistakes
- ✗Forgetting that the denominator 'q' in p/q cannot be zero when defining rational numbers.
- ✗Confusing additive inverse with multiplicative inverse (reciprocal).
- ✗Incorrectly representing negative rational numbers on the number line (e.g., placing -3/4 between 0 and 1 instead of 0 and -1).
- ✗Errors in arithmetic operations with rational numbers, especially with signs or finding common denominators.
- ✗Assuming that division by zero is possible or results in a defined number.
Exam tips
- ★Memorise all the properties of rational numbers (closure, commutativity, associativity, distributivity, identities, inverses) as they are frequently tested.
- ★Practice representing various types of rational numbers (positive, negative, proper, improper) on the number line until you can do it accurately and quickly.
- ★When simplifying expressions, always look for opportunities to apply properties like distributivity to make calculations easier.
- ★Show all steps clearly in your solutions, especially when applying properties or performing multi-step calculations, to avoid losing marks for working.
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