Properties of Rational Numbers

Rational numbers are the building blocks of math that connect whole numbers to integers to real numbers. You encounter them every day in your life, too. Rational numbers are involved in measuring ingredients, tracking money, reading scientific data, and splitting bills. Understanding how rational numbers work and their properties makes diving into advanced concepts easier. Let’s dive in!

1.0What Are Rational Numbers?

In mathematics, rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. The form of a rational number is:

Pq where p,q ∈ Z, q≠0

Here, p is the numerator, and q is the denominator.

Types of rational numbers include:

• Positive and negative fractions (e.g., ¾, −2/5)
• Terminating decimals (e.g., 0.75, -1.2)
• Repeating decimals (e.g., 0.333..., 2.7272...)
• Whole numbers and integers (since 4 = 4/1​)

2.0What Are the Properties of Rational Numbers?

As you now know, a rational number is any number written in the form of p/q, where p and q are both integers and q is not zero.

There are specific rules that all rational numbers follow when performing operations like addition, subtraction, multiplication, and division. By learning these properties, you can recognise patterns, solve problems faster, and tackle tricky questions in algebra and arithmetic. Not only does this help you with your school exams, but it is also crucial for competitive exams like Olympiads and NTSE.

To master rational numbers, there are six key properties of rational numbers that you need to understand:

• Closure property
• Commutative property
• Associative property
• Distributive property
• Identity property
• Inverse property

Once you understand these properties, working with rational numbers in math will feel like second nature to you.

3.0Explaining the Properties (With Examples!)

Closure Property

The closure property says that if you add or multiply any two rational numbers, the result will always be a sensible number.

Addition:

¾ + ¼ = 4/4 = 1 (rational)

Multiplication:

⅔ × ⅗ = 6/15 = ⅖ (rational)

Division isn’t always closed, because:

(5/2) ÷ 0 = undefined

Commutative Property

This property tells us the order doesn’t matter when you add or multiply rational numbers.

Addition:

⅔ + ⅘ = ⅘ + ⅔ 

Multiplication:

⅞ × 3/2 = 3/2 × ⅞ 

Associative Property

It doesn’t matter how you group the numbers; the result will be the same for addition and multiplication.

Addition:

(½ + ⅓) + ⅙ = ½ + (⅓ + ⅙)

Multiplication:

(⅔ × ¾) × ⅘ = ⅔ × (¾ × ⅘)

1. Distributive Property

This property connects multiplication with addition or subtraction.

Formula:

a × (b + c) = a × b + a × c

Example:

½ × (⅔ + ⅙) = ½ × ⅔ + ½ × ⅙ 

LHS: ½ × ⅚ = 5/12 

RHS: ⅓ + 1/12 = 4/12 + 1/12 = 5/12 

2. Identity Property

An identity element doesn’t change the value when added or multiplied.

• Additive Identity = 0
  a + 0 = a
• Multiplicative Identity = 1
  a × 1 = a

Example:

⅚ + 0 = ⅚, ⅚ * 1 = ⅚ 

3. Inverse Property

This property deals with cancelling numbers using their opposite or reciprocal.

• Additive Inverse of a = −a

(3/7) + (-3/7) = 0

• Multiplicative Inverse of a = 1/a (a ≠ 0)

(4/9) * (9/4) = 1

4.0Common Mistakes to Avoid

Even though rational numbers are fairly easy to use, you might trip up if you are not careful. Here are a few common slip-ups students make:

• Thinking division is closed - You cannot divide by zero.
• Mixing up additive and multiplicative inverse - Remember that the additive inverse is the number you add to get zero, and the multiplicative inverse is the number you multiply to get one.
• Confusing associative with commutative - Commutative is all about the order, whereas associative refers to how you group the members.
• Forgetting that subtraction and division aren’t commutative or associative - These two operations don’t have the same rules as multiplication and addition.