Students can go through AP 8th Class Maths Notes Chapter 1 Rational Numbers to understand and remember the concepts easily.
Class 8 Maths Chapter 1 Notes Rational Numbers
→ Rational numbers are closed under the operations of addition, subtraction and multiplication.
→ The operations addition and multiplication are
- commutative for rational numbers.
- associative for rational numbers.
→ The rational number 0 is the additive identity for rational numbers.
→ The rational number 1 is the multiplicative identity for rational numbers.
→ The additive inverse of the rational number \(\frac{\mathrm{a}}{\mathrm{~b}}\) is \(\frac{\mathrm{a}}{\mathrm{~b}}\) and vice-versa.
→ The reciprocal or multiplicative inverse of the rational number \(\frac{\mathrm{a}}{\mathrm{~b}}\) is \(\frac{\mathrm{c}}{\mathrm{~d}}\) if \(\frac{\mathrm{a}}{\mathrm{~b}}\) × \(\frac{\mathrm{c}}{\mathrm{~d}}\) = 1.
→ Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac
→ Rational numbers can be represented on a number line.
→ Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.
→ We are familiar with numbers that are used to count something’s like coins, days, pages and stars etc.
These count starts from 1 and goes up 1, 2, 3 These numbers are called Natural Numbers and is denoted by (N).
→ These Natural Number set is enough to solve some linear equations like
x + 7 = 10, 2x + 5 = 11, 4x – 5 = 35 etc.,
whose solutions are 3,3, 10 respectively.
But to solve some equations like 2x + 10 = 10, 4x + 15 = 15, we need to add zero (0) also to above Natural number system (N).
→ After adding this Zero to (N) the new Number System can be written as : 0,1, 2, 3, ………
→ This number system is called Whole Numbers and is denoted by “W”.
Now, we are able to solve above linear equations.
2x + 10 = 10 (Solution is zero)
4x + 15 = 15 (Solution = 0)
→ But in the daily life, we are to answer some questions like
- If the cost of 2 pizzas is 95 Rs/-, then what is cost of 1 pizza ?
- If a bowler gives 31 runs in four overs, then what will be the run rate ?
→ To solve above questions, we need to introduce rational numbers.
Before that, let us have a look about the numbers to count the vehicles on right side as well as left side of signal post.
→ So to indicate the negative direction we add the symbol (-).
→ Now, this entire nurpber system can be expressed as
Negative Numbers + Whole Numbers
Or
Negative Numbers + Zero + Natural Numbers
→ These are called “Integers” and is denoted by “Z”.
→ Integers are helpful to solve the follow¬ing linear equations.
x + 7 = 3 (Solution = – 4),
x + 5 = 0 (Solution = – 5),
x – 4 = 3 (Solution = 7)
→ But to solve linear equations like 2x – 3 – 6, we won’t have solution in Integers. For this Rational Numbers are introduced. These are denoted by “Q”.
Definition :
A number which can be written in the form of \(\frac{\mathrm{p}}{\mathrm{q}}\) where ‘p’ is any integer and ‘q’ is also any integer but not equal to zero (q ≠ 0) is called a rational number.
Example: \(\frac{-2}{3}\) (\(\frac{\mathrm{p}}{\mathrm{q}}\) form p = – 2, q = 3 are integers q ≠ 0 so \(\frac{-2}{3}\) is rational number)
→ Is ‘6’ a rational number ?
Solution:
Yes. ‘6’ can be written as 6 = \(\frac{120}{20}\), \(\frac{30}{5}\), \(\frac{-12}{-2}\), \(\frac{6}{1}\) which are all in the form of \(\frac{\mathrm{p}}{\mathrm{q}}\) satisfying p, q are integers and q ≠ 0, hence’6’is rational number.
→ Is ‘0’ a rational number ?
Solution:
Yes. Zero is a rational number, because
0 = \(\frac{0}{5}\), \(\frac{0}{10}\), \(\frac{0}{6}\) (Satisfies the definition of rational number)
Thus we can say all integers are rational numbers.
More over they can expressed in decimal form also.
For example \(\frac{3}{2}\) = 1.5 is also a rational number.
Properties : Closure property in Whole numbers, Integers, Rational numbers.
i) In Whole numbers:
| Operation | Numbers a, b | Result number a Θ b | Remarks | Conclusion |
| Addition (+) | 2, 0 | 2 + 0 = 2 | Whole number | Closed under addition |
| Subtraction (-) | 5, 9 | 5 – 9 = -4 | Not a whole number | Not closed under subtraction |
| Multiplication (×) | 5, 10 | 5 × 10 = 50 | Whole number | Closed under multiplication |
| Division (÷) | 3, 4 | \(\frac{1}{2}\) = 0.75 | Not a whole number | Not closed under division |
ii) In Integers :
| Operation | Numbers a, b | Result number a Θ b | Remarks | Conclusion |
| Addition (+) | -2, -3 | – 2 – 3 = -5 | Integer | Integers are closed under addition |
| Subtraction (-) | -7, 5 | – 7 – 5 = – 12 | Integer | Integers are closed under subtraction |
| Multiplication (×) | 3, 0 | 3 × 0 = 0 | Integer | Integers are closed under multiplication |
| Division (÷) | 4, 10 | 4 ÷ 10 = 0.4 | Not an Integer | Integers are not closed under division |
iii) In Rational numbers :
