Division of Rational Numbers – Conditions

To ensure that the division of rational numbers is valid and accurate, the following conditions must be satisfied:

(1) Non-zero Divisor

Condition: The divisor must not be zero.

Reason: Division by zero is undefined in mathematics.

For two rational numbers, a/b and c/d c≠0 and d≠0.

The divisor c/d cannot be zero because it would involve dividing by zero, which is not allowed.

(2) Rational Form

Condition: Both the dividend and the divisor must be in rational form, meaning they can be expressed as fractions where the numerators and denominators are integers, and the denominators are non-zero.

Reason: Rational numbers are defined as fractions of the form a/b. Where a and b are integers. This ensures that the numbers are well-defined and can be handled mathematically in a consistent manner.

Detailed Conditions

To divide rational numbers:

Ensure both numbers are in rational form with non-zero denominators.
Verify that the divisor is non-zero.
Multiply the dividend by the reciprocal of the divisor.
Simplify the resulting fraction if possible.
By following these steps and conditions, we can correctly and safely divide rational numbers.

Check if Divisor is Zero:

Ensure that the rational number we are dividing by (c/d) is not zero. This implies both c and d must not be zero simultaneously, though typically, we are only concerned with c being non-zero since d is already assumed to be non-zero as part of being a rational number.

Reciprocal of the Divisor:

To divide c/d, we actually multiply by its reciprocal, which is d/c. For this step to be valid, c ≠ 0.

Example: Dividing Rational Numbers with Conditions
Let’s go through another example with a focus on the conditions.

Find the Reciprocal of the Divisor:

a/b ÷ c/d

The reciprocal of c/d is d/c

This step is valid if c≠0

Multiply the Dividend by the Reciprocal of the Divisor:

= ad/bc

Simplify the Resulting Fraction:

Simplify the fraction ​by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this GCD.

Example1: Valid Division

Divide 6/7 by 2/3

Identify the Rational Numbers:

6/7 and 2/3

​Ensure Non-zero Divisor:

2/3≠0

Find the Reciprocal of the Divisor:

3/2

​Multiply the Dividend by the Reciprocal of the Divisor:

6/7÷2/3

= 6/7×3/2 =(6×3)/(7×2)=18/14

Simplify the Fraction:

The GCD of 18 and 14 is 2, so

18/14 = (18÷2)/(14÷2)=9/7

​Therefore, 6/7÷2/3=9/7

Example2: Division by Zero (Invalid Division)
Attempt to divide

5/4 by 0/3

Identify the Rational Numbers:

5/4 and 0/3

​Check the Divisor:

0/3 =0, which means the divisor is zero.

Non-zero Divisor Condition: The divisor is zero, so the division 5/4÷0/3 is not valid.

Example3: Valid Division

Divide 5/8 by 3/4:

Identify the rational numbers: 5/8 and 3/4.

Ensure the divisor is not zero: 3/4 ≠ 0

Find the reciprocal of the divisor: 4/3.

Multiply the dividend by the reciprocal of the divisor:

5/8 ÷ 3/4 = 5/8 × 4/3 = (5×4)/(8×3)=20/24

Simplify the fraction:

The GCD of 20 and 24 is 4. So,

(20÷4) ÷ (24÷4) = 5/6

Therefore, 5/8÷3/4 =5/6.

Example4: Division by Zero (Invalid Division)

Divide 7/5 by 0/2

Identify the rational numbers:

7/5 and 0/2

Check the divisor: 0/2=0

which means the division is undefined.

In this case, since the divisor is zero, the operation
7/5 ÷ 0/0 is not valid.

Summary

Ensure both numbers are in rational form with non-zero denominators.
Verify that the divisor is non-zero.
Multiply the dividend by the reciprocal of the divisor.
Simplify the resulting fraction if possible.
By following these steps and conditions, we can correctly and safely divide rational numbers.