Comparision of Rational Numbers

Comparison of Rational Numbers: Rational numbers are crucial for problem-solving across various disciplines. They help us solve equations, analyze data, make predictions, and model real-world scenarios in science, engineering, economics, and finance. Comparing rational numbers is similar to comparing fractions and integers.

These numbers are compared using the numerators and denominators. It is compared using either the decimal or the same denominator method.

The article will detail the rational numbers and the methods to compare them with examples.

What are Rational Numbers?

Rational numbers can be written as a fraction a/b, where 'a' and 'b' are integers, and 'b' is not zero.

In simple terms, a rational number is any number that can be shown as one whole number divided by another whole number, as long as you are not dividing by zero.

For example, 4/8, -2/14, -11/-5, 7/-9, 7/-15, and 6/-11 are rational numbers.

In a fraction a/b, 'a' is the numerator, and 'b' is the denominator.

Properties of Rational Numbers

The properties of rational numbers are simplified into simple points:

• Multiplying, adding, or subtracting any two rational numbers always gives a rational number.
• A rational number remains unchanged if both its numerator and denominator are multiplied or divided by the same factor.
• Adding zero to a rational number leaves the number unchanged.
• Rational numbers maintain their "rationalness" when subjected to addition, subtraction, and multiplication.

Rules For Comparing Rational Numbers

There are some rules to compare rational numbers; the simple rules are listed below:

• Positive rational numbers are always bigger than 0.
• Negative rational numbers are always smaller than 0.
• Positive rational numbers are bigger than negative rational numbers.
• On a number line, a rational number is bigger than any number to its left and smaller than any number to its right.

Step-by-Step Method to Compare Rational Numbers

To compare two rational numbers, follow these steps:

Step 1: Start with the two rational numbers you want to compare.

Step 2: Make sure both rational numbers have positive denominators.

Step 3: Find the Least Common Multiple (LCM) of the positive denominators.

Step 4: Rewrite each rational number with the LCM as the new denominator.

Step 5: Compare the numerators of these new fractions. The one with the larger numerator is the larger rational number.

Comparing Rational Numbers with Decimal Method

Decimal method is one of the easiest ways to compare rational numbers. The steps for comparing rational numbers using the decimal method:

Step 1: Take a rational number and divide the numerator by the denominator to obtain its decimal representation.

Step 2: Once both rational numbers are in decimal form, visually compare the decimal values to determine which one is greater.

Rational number corresponding to the larger decimal value is considered greater, while the one with the smaller decimal value is lesser.

This method simplifies the comparison process by transforming rational numbers into decimals, making it easier to visually compare and determine their relative magnitudes.

Related Articles
Natural Numbers	Whole Numbers
Prime Numbers	Imaginary Numbers

Examples of Comparing Rational Numbers

Example 1. Which is greater, -1/2 or – 1/5 of the two rational numbers?

Solution:

Given rational numbers are -1/2 and – 1/5

Taking the LCM of denominators, which are 2 and 5

LCM comes out to be 10

Expressing the rational numbers with the same denominator using the LCM obtained.

-1/2 = (-1×5)/(2×5)= -5/10

-1/5 = (-1×2)/(5×2) = -2/10

-2 > -5

Therefore, – 1/5 is greater than -1/2.

Example 2. Of the two rational numbers, which is greater, 2/5, or -3/4 is greater?

Solution:

To compare ⅖ and -¾

We know that between a positive rational number and a negative rational number, a positive rational number is always greater.

Therefore, 2/5 is greater than -3/4.

Example 3. Of the two rational numbers, which is greater, 2/3 or 5/7?

Solution:

To Compare the Rational Numbers ⅔ and 5/7

Taking the LCM of the denominators 3 and 7, which comes out to be 21

Rational numbers with the same denominator using the LCM obtained, we get

2/3 = (2×7)/(3×7) = 14/21

5/7 = (5×3)/(7×3) = 15/21

See the numerators of both the rational numbers obtained, i.e., 14/21, 15/21

Since 15 is greater, the rational number 5/7 is greater.

Conclusion

Rational numbers, written in the ratio of two integers, are easy to compare, like fractions using numerators and denominators. Decimal method is followed only when it is easy to convert the fraction into decimals,with the help of either method we can compare the rational numbers.

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