This GSEB Class 8 Maths Notes Chapter 1 Rational Numbers covers all the important topics and concepts as mentioned in the chapter.
Rational Numbers Class 8 GSEB Notes
Natural numbers:
- The counting numbers are known as natural numbers. 1, 2, 3, 4, … are natural numbers. The collection of all-natural numbers is denoted by ‘N’
N = {1, 2, 3, 4,…} - The smallest natural number is 1.
- There are infinite natural numbers.
Whole numbers:
- If ‘0’ is included in the collection of natural numbers, then the collection of numbers 0, 1, 2, 3, … is known as whole numbers. It is denoted by ‘W’
W = {0, 1, 2, 3, 4,…} - The smallest whole number is 0.
- There is no greatest whole number.
Rational numbers:
- A number of the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0 is called a rational number.
\(\frac{0}{5}, \frac{3}{7}, \frac{(-5)}{9}, 19, \frac{7}{(-13)}\), etc. are all rational numbers. - \(\frac{0}{5}\) is a rational number but \(\frac{5}{0}\) is not a rational number. (∵ q = 0)
- 0 is a whole number but not a natural number. Every natural number is a whole \number but every whole number is not a natural number.
- Zero is a rational number because we can divide zero by a non-zero number.
- There are infinite rational numbers between two rational numbers.
Basic operation :
- Addition
- Subtraction
- Multiplication
- Division
Properties of addition :
- Closure property
- Commutative property
- Associative property
- Property of zero OR additive identity
- Property of additive inverse
1. Closure property: The sum of two rational numbers is a rational number. If x and y are two rational numbers, then (x + y) is also a rational number.
2. Commutative property: Addition of rational numbers is commutative. If x and y are two rational numbers, then x + y = y + x.
3. Associative property: The addition of any three rational numbers is associative. If x, y and z are any three rational numbers, then (x + y) + z = x + (y + z).
4. Additive identity: The sum of a rational number and zero (0) is the same rational number. If x is a rational number, then x + 0 = 0 + x = x.
Property of additive inverse:
- If we add two same rational numbers having opposite signs, the sum is zero.
- If x is rational number, then
x + (- x) = (- x) + x = 0 - The negative of x is denoted by (- x) and vice versa.
Properties of subtraction:
- Closure property: If x and y are two rational numbers then x – y is a rational number.
- Commutative property: Commutative property does not hold for subtraction of rational numbers,
x – y ≠ y – x - Associative property: The subtraction of rational numbers is not associative. If x. y and z are any three rational numbers, then (x- y) – z ≠ x- (y – z).
Properties of multiplication:
- Closure property
- Commutative property
- Associative property
- Multiplicative identity
- Distributive property of multiplication (over addition)
1. Closure property: If x and y are two rational numbers, then x × y is also a rational number.
2. Commutative property: For any two rational numbers x and y,
x × y = y × x
3. Associative property: If x, y and z are any three rational numbers, then
(x × y) × z = x × (y × z)
4. Multiplicative identity: If x is any rational number, then
x × 1 = 1 × x = x
∴ 1 is called multiplicative identity.
5. Distributive property of multiplication (over addition) : If x, y and z are any three rational numbers, then
x × (y + z) = x × y + x × z
Properties of division:
If x and y are any two rational numbers and y ≠ 0, then x ÷ y is always a rational number.
For any rational number x
x ÷ 1 = x and x ÷ (- 1) = (-x)
For every non-zero rational number
x ÷ x = 1
x ÷ (-x) = (- 1)
(-x) – x = (- 1)
Property of multiplicative inverse:
If \(\frac{p}{q}\) is a rational number (p, q ≠ 0), then \(\frac{q}{p}\) is the multiplicative inverse (reciprocal) of \(\frac{p}{q}\).
Their product is always 1.
\(\frac{p}{q} \times \frac{q}{p}\) = 1
Note:
- We can also represent a rational number on a number line. Let’s say the rational number Is If you want to plot It accurately on the number line, divide the number line between two whole numbers between which \(\frac{x}{y}\) lies Into y equal parts and plot It on the xth part between those two numbers.
e.g.. To represent on a number line, make five equal parts between 0 and 1. - Between two rational numbers x and y, there Is a rational number \(\frac{x+y}{2}\)
- We can find as many rational numbers between x and y as we want.