Comparing rational numbers involves determining which number is greater or smaller. In Class 8, you typically learn several methods for doing this. Here's a breakdown:
1. Understanding Rational Numbers:
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. They can be positive, negative, or zero.
2. Basic Comparison Rules:
- Positive vs. Negative: Any positive rational number is always greater than any negative rational number.
- Rational Number vs. Zero: Any positive rational number is greater than zero, and any negative rational number is less than zero.
3. Comparing Rational Numbers with the Same Denominator:
If two rational numbers have the same denominator, the number with the larger numerator is greater. For example, 5/7 > 3/7 because 5 > 3.
4. Comparing Rational Numbers with Different Denominators:
This is usually done by finding a common denominator:
- Find the Least Common Multiple (LCM) of the Denominators: This will be your common denominator.
- Convert each rational number: Multiply the numerator and denominator of each rational number by a suitable factor to obtain the common denominator.
- Compare the Numerators: Once the denominators are the same, compare the numerators. The rational number with the larger numerator is the greater rational number.
Example: Compare 3/5 and 2/3.
- Find the LCM: The LCM of 5 and 3 is 15.
- Convert the rational numbers:
- 3/5 = (3 3) / (5 3) = 9/15
- 2/3 = (2 5) / (3 5) = 10/15
- Compare the numerators: 10 > 9, so 10/15 > 9/15, which means 2/3 > 3/5.
5. Using a Number Line:
Rational numbers can be plotted on a number line. Numbers to the right are always greater than numbers to the left. This provides a visual way to compare them.
Example:
Imagine -1/2 and 1/4 on a number line. 1/4 is to the right of -1/2, so 1/4 > -1/2.
In summary, you compare rational numbers by first ensuring they have a common denominator and then comparing their numerators. Remember that positive numbers are always greater than negative numbers, and the number line provides a visual aid.
