Arithmetic Operations on Rational Numbers
In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on rational numbers, say p/q and s/t.
Addition: When we add p/q and s/t, we need to make the denominator the same. Hence, we get (pt+qs)/qt.
Example: 1/2 + 3/4 = (2+3)/4 = 5/4
Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.
Example: 1/2 – 3/4 = (2-3)/4 = -1/4
Adding and Subtracting Rational Numbers
For adding and subtracting rational numbers, we use the same rules of addition and subtraction of integers. Let us understand this with the help of an example.
Example: Solve 1/2 – (-2/3)
Solution: Let us solve this using the following steps:
- Step 1: As we simplify 1/2 – (-2/3), we will follow the rule of addition and subtraction of numbers which says that the subtraction fact can change to an addition fact and the sign of the subtrahend gets reversed. This will make it 1/2 + 2/3
- Step 2: Now, we need to add these fractions 1/2 + 2/3
- Step 3: Using the rules of addition of fractions, we will convert the given fractions to like fractions to get common denominators so that it becomes easier to add them. For this, we need to find the LCM of the denominators 2 and 3 which is 6. Then we will convert the fractions to their respective equivalent fractions which will make them 3/6 + 4/6. This will give the sum as 7/6 which can be written in the form of a mixed fraction 1 1/6.
Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively.
If p/q is multiplied by s/t, then we get (p×s)/(q×t).
Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8
Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs
Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3
Multiplying and Dividing Rational Numbers
The multiplication and division of rational numbers can be done in the same way as fractions. To multiply any two rational numbers, we multiply their numerators and their denominators separately and simplify the resultant fraction. Let us understand this with the help of an example.
Example: Multiply 3/5 × -2/7
Solution: Let us solve this using the following steps:
- Step 1: In order to multiply 3/5 × (-2)/7, we will first multiply the numerators and then multiply the denominators.
- Step 2: In this case, when we multiply the numerators, it will be 3 × (-2) = -6.
- Step 3: When we multiply the denominators, it will be 5 × 7 = 35. Therefore, the product will be -6/35.
When we need to divide any two fractions, we multiply the first fraction (which is the dividend) by the reciprocal of the second fraction (which is the divisor). Let us understand this with the help of an example.
Example: Divide 3/5 ÷ 2/7
Solution: Let us solve this using the following steps:
- Step 1: In order to divide 3/5 ÷ 2/7, we will first write the reciprocal of the second fraction. This will make it 3/5 × 7/2
- Step 2: Now, we will multiply the numerators This will be 3 × 7 = 21.
- Step 3: Then, we will multiply the denominators, it will be 5 × 2 = 10.
Therefore, the product will be 21/10 or
Multiplicative Inverse of Rational Numbers
As the rational number is represented in the form p/q, which is a fraction, then the multiplicative inverse of the rational number is the reciprocal of the given fraction.
For example, 4/7 is a rational number, then the multiplicative inverse of the rational number 4/7 is 7/4, such that (4/7)x(7/4) = 1
Properties of Rational Numbers
The major properties of rational numbers are:
- Commutativity Property
- Associative Property
- Distributive Property
Commutative Property
1. Addition
For any two rational numbers a and b, a + b = b+ a
We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.
2. Subtraction
For any two rational numbers a and b, a – b ≠ b – a. Given are the two rational numbers 53 and 14,
So subtraction is not commutative for rational numbers.
3. Multiplication
For any two rational numbers a and b, a × b = b × a
We see that the two rational numbers can be multiplied in any order. So multiplication is commutative for rational numbers.
4. Division
For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers 53 and 14
We see that the expressions on both the sides are not equal. So division is not commutative for rational numbers.
Associative Property
Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.
In both, groups the sum is the same.
- Addition and multiplication are associative for rational numbers.
- Subtraction and division are not associative for rational numbers.
Distributive Property
Distributive property states that for any three numbers x, y and z we have
x × ( y + z ) = (x × y) +( x × z)
Rational Numbers Properties
Since a rational number is a subset of the real number, the rational number will obey all the properties of the real number system. Some of the important properties of the rational numbers are as follows:
- The results are always a rational number if we multiply, add, or subtract any two rational numbers.
- A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
- If we add zero to a rational number then we will get the same number itself.
- Rational numbers are closed under addition, subtraction, and multiplication.