Closure Property

Closure property states that when a set of numbers is closed under any arithmetic operation such as addition, subtraction, multiplication, and division and is performed on any two numbers of the set with the answer being another number from the set itself. This property is applicable for real numbers, whole numbers, integers, and rational numbers. Let us learn more about the closure property regarding each form of numbers and solve a few examples.

Closure property can be defined as the closure of a set of numbers by using arithmetic operations and is completed by these operations. In other words, when the operation is performed on any two numbers from the set and the result obtained will be the number from the set itself is considered to be closed. A set either has or lacks closure depending on the given operation. The closure property is applicable for addition and multiplication for most of the forms of numbers. However, for a few subtraction and division as the result might not be the same form of the number. For example, 4 + 5 = 9, here all the numbers are natural numbers.

Let us see the closure property of each arithmetic operation in brief.

Closure property of addition states that when any two real numbers are added, the result will be a real number also. For example, 2 + 5 = 7, where all the three numbers are real numbers. The formula is a + b = R, where a, b and R are real numbers. Let us see this property for all the real numbers:

• Natural Numbers: a + b = N (a, b and N = natural numbers)
• Whole Numbers: a + b = W (a, b, and W = whole numbers)
• Integers: a + b = C (a, b, and C = integers)
• Rational Numbers: a + b = Q (a, b, and Q = rational numbers)

Closure property of multiplication states that if any two real numbers a and b are multiplied, the product will be a real number as well. For example, 5 × 2 = 10. This property is applicable for natural numbers, whole numbers, integers, and rational numbers. Let us see this property for all the real numbers:

• Natural Numbers: a × b = N (a, b and N = natural numbers)
• Whole Numbers: a × b = W (a, b, and W = whole numbers)
• Integers: a × b = C (a, b, and C = integers)
• Rational Numbers: a × b = Q (a, b, and Q = rational numbers)

Closure property of subtraction states that if any two real numbers a and b are subtracted from each other, the difference or result will be a real number as well. For example, 9 - 4 = 5. This property is applicable only for integers and rational numbers.

• Integers: a - b = C (a, b, and C = integers)
• Rational Numbers: a - b = Q (a, b, and Q = rational numbers)

Closure property formulas use all four operations where each of them results in their respective numbers. If two real numbers a and b are given, then the closure property formula of numbers is given as,

Note: Closure property for subtraction is applicable only for integers and rational numbers.

☛Related Topics

Listed below are a few topics that are related to closure property.

• Properties of Whole Numbers
• Properties of Natural Numbers
• Properties of Integers
• Properties of Rational Numbers

What is Closure Property with Examples?

Closure property states that when any two real numbers are solved using any arithmetic operations, the result will also be a real number. This property is applicable under addition and multiplication for natural numbers, whole numbers, integers, and rational numbers. The closure property of subtraction is only applicable for integers and rational numbers whereas division is not applicable. For example, 12 + 10 = 22, here all the three numbers are real numbers.

What is Closure Property Under Addition and Subtraction?

Closure property under addition and subtraction states that if two real numbers a and b are added and subtracted the result will also be a real number. a + b = c and a × b = c. For example, 4 and 6 are real numbers, 4 + 6 = 10 and 4 × 6 = 24. Here, both 10 and 24 are real numbers.

Is Closure Property True for Subtraction?

Closure property for integers is true for addition, multiplication, and subtraction. Whereas for natural numbers and whole numbers, subtraction is not applicable.

What is the Closure Property of Whole Numbers?

The closure property of the whole number states that addition and multiplication of two whole numbers is always a whole number. For example, consider whole numbers 7 and 8, 7 + 8 = 15 and 7 × 8 = 56. Here 15 and 56 are whole numbers as well. This property is not applicable on subtraction and division.

What is the Closure Property of Integers?

The closure property of integers states that the addition, subtraction, and multiplication of two integers always results in an integer. However, this property does not hold true for the division as the division of two integers may not always result in an integer.

Are Natural Numbers Closed Under Subtraction?

Closure property of natural numbers is applicable only for addition and multiplication. The subtraction of two natural numbers does not necessarily create another natural number, for example, 5 - 11 = -6.

What is the Closure Property of Rational Numbers with Examples?

The closure property of rational numbers states that when any two rational numbers are added, subtracted, or multiplied the result of all three cases will also be a rational number. For example, consider two rational numbers 1/3 and 1/4, 1/3 + 1/4 = (4 + 3)/12 = 7/12, 7/12 is a rational number.

What is the Formula for Closure Property?

Closure property is not applicable under subtraction and division for many cases. Addition and subtraction result in real numbers. Consider two real numbers a and b, then the closure property formula of numbers is given as,

• a + b = c
• a × b = c
• a - b ≠ c
• a ÷ b ≠ c