Closure Property

The closure property is one of those concepts of math that sound simple but are incredibly powerful. At its core, it helps us figure out whether a set of numbers we perform an operation on has the result in the same set or not. If the result is within the set, it is closed under that operation. This property helps you with algebra, number systems, and even advanced math. Every time you crunch numbers, this property ensures predictability and consistency. Let’s learn more about it.

1.0Closure Property in Math

The closure property in math tells us whether or not a set stays complete when we do operations like addition, subtraction, multiplication, and division. What you need to do is take two elements from a set, apply an operation, and if the result is from the same set, the set will be closed under that operation.

What this tells us is whether a set will behave consistently with that particular operation. It becomes especially useful when working with real numbers, rational numbers, integers, and other number systems.

Closure Property of Addition

Addition is perhaps the most commonly used operation in mathematics. And it is closed in many number sets.

Sets where addition is closed:

• Integers
• Rational Numbers
• Real Numbers
• Whole Numbers

Let’s break it down with some closure property of addition examples:

• 6 + 13 = 19 (Real number + Real number = Real number)
• (1/3) + (5/2) = 17/6 (Rational + Rational = Rational)
• 5√5 – 2√5 = 3√5 (Real + Real = Real)

Exception: Natural numbers are not closed under subtraction. For example, 3 − 5 = –2, which is not a natural number.

Closure Property of Multiplication

Just like addition, multiplication often keeps numbers within the same set. The closure property of multiplication tells us that multiplying any two numbers from a set should also give a number in that set. 

Sets closed under multiplication: 

• Whole Numbers
• Integers
• Rational Numbers
• Real Numbers

Closure property of multiplication examples:

• 3 × (–4) = –12 (Integer × Integer = Integer)
• (1/2) × (3/5) = 3/10 (Rational × Rational = Rational)
• √3 × √5 = √15 (Real × Real = Real)

Even zero plays nice here: 8 × 0 = 0

Closure Property of Integers

Integers, positive, negative, and zero, follow the closure property for addition, subtraction, and multiplication.

Addition:

• (–8) + 6 = –2
• 11 + 9 = 20

Subtraction:

• 19 – 6 = 13
• (–6) – (–3) = –3

Multiplication:

• 3 × (–9) = –27
• (–7) × (–9) = 63

Division:

The closure property does not hold for division.

• (–16) ÷ 4 = –4 
• (–4) ÷ (–16) = 1/4 (Not an integer)

Closure Property of Rational Numbers

Rational numbers (fractions of integers) are incredibly consistent under addition, subtraction, and multiplication.

Addition:

• (5/6) + (2/3) = 3/2
• –(1/2) + (1/4) = –1/4

Subtraction:

• (7/8) – (3/8) = 1/2
• (6/7) – (–3/7) = 9/7

Multiplication:

• (3/2) × (2/9) = 1/3
• (–7/4) × (5/2) = –35/8

Division: Division by 0 is where closure breaks.

• (1/2) ÷ (0) = (Undefined)

Closure Property of Whole Numbers

Whole numbers, 0, 1, 2, 3, and so on, are closed under addition and multiplication, but not under subtraction or division.

Addition:

• 12 + 0 = 12
• 9 + 7 = 16

Subtraction:

• 13 – 14 = –1 (Not a whole number)
• 4 – 0 = 4 

Multiplication:

• 4 × 6 = 24
• 0 × 7 = 0

Division:

• 18 ÷ 4 = 4.5 (Not a whole number)
• 22 ÷ 2 = 11 

Closure Property Under Division

While we often assume division "just works," it doesn’t always follow the closure rule.

• The division is not closed for natural numbers, integers, or whole numbers.
• The real numbers are almost closed under division, except when the divisor is 0.

Examples:

• 9 ÷ 3 = 3
• (3/4) ÷ (1/2) = 3/2
• (–8) ÷ 0 = (Undefined)