Properties of Rational Numbers: Complete Guide with Solved Examples

Rational numbers are fundamental elements of mathematics, representing quantities that can be expressed as fractions of two integers. Understanding their properties is crucial for mastering various mathematical concepts and operations. This comprehensive guide delves into the essential properties of rational numbers, providing clear explanations and solved examples to enhance your grasp of the properties of rational numbers. Whether you are a student aiming to solidify your foundation or a math enthusiast seeking deeper insights, this properties of rational numbers guide offers valuable knowledge and practical applications of rational numbers in everyday mathematics.

What are Rational Numbers?

Rational numbers are basically any number that can be expressed as a fraction. More precisely, a rational number is any number that can be written in the form p/q, where:

• p is an integer (whole number, including positive and negative numbers)
• q is an integer, but importantly, q ≠ 0 (because dividing by zero is undefined)

This definition also includes a wide range of numbers, including:

1. Integers: Whole numbers like -3, 0, and 7, which can be expressed as -3/1, 0/1, and 7/1, respectively.
2. Fractions: Proper and improper fractions like ½ and 9/4​.
3. Terminating Decimals: Decimals that end, such as 0.75, which can be written as 3/4.
4. Repeating Decimals: Decimals that repeat a pattern, like 0.333… (which is 1/3​) and 1.666… (which is 5/3).

Rational numbers are part of the larger number system and play an important role in various mathematical operations and real-world applications.

Also Read: Questions of Logical Problems Reasoning

Properties of Rational Numbers

Rational numbers possess several important properties that are fundamental to understanding and working with them in mathematics. Here are the key properties of rational numbers:

1. Closure Property

• Addition: The sum of any two rational numbers is also a rational number. 

• If a/b and c/d are rational numbers, then a/b+c/d is also rational.

• Subtraction: The difference between any two rational numbers is also a rational number. 

• If a/b and c/d are rational numbers, then a/b−c/d is also rational.

• Multiplication: The product of any two rational numbers is also a rational number. 

• If a/b and c/d are rational numbers, then a/b × c/d is also rational.

• Division: The quotient of any two rational numbers (except division by zero) is also a rational number. 

• If a/b and c/d are rational numbers and c/d≠0, then a/b ÷ c/d is also rational.

2. Commutative Property

Addition: The order of addition does not affect the sum.

a/b + c/d = c/d + a/b

Multiplication: The order of multiplication does not affect the product.

a/b x c/d = c/d x a/b

3. Associative Property

Addition: The way in which numbers are grouped in addition does not affect the sum. 

(a/b + c/d) + e/f = a/b + (c/d + e/f)

Multiplication: The way in which numbers are grouped in multiplication does not affect the product.

(a/b x c/d) x e/f = a/b x (c/d x e/f)

4. Distributive Property

Multiplication over Addition: Multiplying a number by a sum is the same as doing each multiplication separately.

a/b + (c/d + e/f) = a/b x c/d x a/b x e/f

5. Identity Property

Additive Identity: The sum of any rational number and zero is the rational number itself.

a/b + 0 = a/b

Multiplicative Identity: The product of any rational number and one is the rational number itself.

a/b x 1 = a/b

6. Inverse Property

Additive Inverse: For every rational number, there exists an additive inverse such that their sum is zero.

a/b + (-a/b) = 0

Multiplicative Inverse: For every non-zero rational number, there exists a multiplicative inverse such that their product is one.

a/b x b/a = 1 (where a ≠ 0 and b ≠ 0)

7. Density Property

Between any two rational numbers, there exists another rational number.

If a/b and c/d are rational numbers and a/b < c/d, then there exists a rational number e/f such that a/b < e/f < c/d.

Properties of Rational Numbers Solved Examples

Here are five solved examples involving rational numbers that demonstrate their properties and operations.

Q1: Add ⅔​ and ⅘​.

Solution: To add two rational numbers, we need a common denominator. The least common denominator (LCD) of 3 and 5 is 15.

⅔ = 2×5/3×5 = 10/15

⅘ = 4×3/5×3 = 12/15 

Now, add the two fractions:

10/15 + 12/15 = 10 + 12/15 = 22/15

so , ⅔ + ⅘ = 22/15

Q2: Subtract 7/10​ from 3/4​.

Solution: Find a common denominator for the fractions. The least common denominator (LCD) of 10 and 4 is 20.

¾ = 3×5/4×5 = 15/20

7/10 = 7×2/10×2 = 14/20

Now, subtract the fractions:

15/20 – 14/20 = 15-14/20 = 1/20

So, ¾ – 7/10 = 1/20.

Q3: Multiply ⅚ and 2/9.

Multiply the numerators and the denominators:

⅚ x 2/9 = 5×2/6×9 = 10/54

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:

10÷2/54÷2 = 5/27

So, ⅚ x 2/9 = 5/27.

Q4: Divide 8/15 by ⅖.

Solution: To divide by a fraction, multiply by its reciprocal.

8/15 ÷ ⅖ = 8/15 x 5/2 = 8×5/15×2 = 40/30.

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 10:

40 ÷ 10/30 ÷ 10 = 4/3

So, 8/15 ÷ ⅖ = 4/3.

Q5: Compare 7/12 and ⅝.

Solution: To compare these fractions, find a common denominator. The least common denominator (LCD) of 12 and 8 is 24.

7/12 = 7×2/12×2 =14/24

⅝= 5×3/8×3 = 15/24

Now compare the numerators:

14 < 15.

So, 7/12 < ⅝.

Also Read: 20+ Questions of Arithmetic Reasoning

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