Closure Property(Definition & Examples)

In mathematics, you study different types of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. When you perform arithmetic operations on numbers of a particular category, do you think the result also will be from the same category?

For example, do you get a natural number when you add two natural numbers or the difference is a natural number when you subtract two natural numbers?

The property that answers these questions is called the closure property of numbers.

What is Closure Property?

The closure property means that a set of numbers is closed for some arithmetic operations. To understand the meaning of ‘closed’, consider the following example.

 Let’s consider two natural numbers: $5$ and $9$. Further, let’s add and subtract these two numbers.

 $5 + 9 = 14$,  $9 – 5 = 4$ and $5 – 9 = -4$. 

The results from the above operations are $14$, $4$, and $-4$.

Out of these three results, the first two $14$ and $4$ are natural numbers but the third one, i.e., $-4$ is not a natural number.

Thus, by closed we mean the result of the operation of any two numbers from a set of numbers also belongs to the same set of numbers. In other words, for example, if the two numbers involved in the operation are natural numbers, then the result will also be a natural number or if the two numbers involved in the operation are whole numbers, then the result will also be a whole number.

On the other hand, if the result does not belong to the set from where the numbers involved in the operation are taken, then we say that the operation is not closed.

Note: The same set of numbers may be closed for a particular operation and may not be closed for some other operation.

Now, let’s find out which set of numbers are closed for which operations.

Closure Property of Natural Numbers

Natural numbers are the numbers that start with $1$ and move on to $2$, $3$, $4$, and so on. The set of natural numbers is represented by a set $N = \{ 1, 2, 3, 4, … \}$.

Addition

The set of natural numbers is closed under the operation addition. It means if you add any two natural numbers, you will always get a natural number.

Mathematically, it can be represented as for all $a$, $b \in N$, $a + b \in N$.

For example, $9$ and $4$ are natural numbers and $9 + 4 = 13$ is also a natural number.

Also, $72$ and $48$ are natural numbers and $72 + 48 = 120$ is also a natural number.

Subtraction

The set of natural numbers is not closed under the operation subtraction. It means if you subtract any two natural numbers, you may get a natural number or may not get a natural number.

Mathematically, it can be represented as for all $a$, $b \in N$, $a – b \notin N$.

For example, $9$ and $4$ are natural numbers and $9 – 4 = 5$ is also a natural number, but $4 – 9 = -5$ in not a natural number.

Note: The operation subtraction is not commutative also.

Multiplication

The set of natural numbers is closed under the operation multiplication. It means if you multiply any two natural numbers, you will always get a natural number.

Mathematically, it can be represented as for all $a$, $b \in N$, $a \times b \in N$.

For example, $4$ and $7$ are natural numbers and $4 \times 7 = 28$ is also a natural number.

Also, $19$ and $23$ are natural numbers and $19 \times 23 = 437$ is also a natural number.

Division

The set of natural numbers is not closed under the operation division. It means if you divide any two natural numbers, you may get a natural number or may not get a natural number.

Mathematically, it can be represented as for all $a$, $b \in N$, $a \div b \notin N$.

For example, $12$ and $4$ are natural numbers and $12 \div 4 = 3$ is also a natural number, but $4 \div 12 = 0.333…$ in not a natural number.

Note: 

• The operation subtraction is not commutative also.
• $0.333…$ is a non-terminating but repeating rational number.

Closure Property of Whole Numbers

Whole numbers are the numbers that start with $0$ and move on to $1$, $2$, $3$, and so on. The set of whole numbers is represented by a set $W = \{ 0, 1, 2, 3, …\}$.

Note: The set of whole numbers $W$ has all the numbers that are present in the set of natural numbers $N$ with $1$ extra number $0$.

Addition

The set of whole numbers is closed under the operation addition. It means if you add any two whole numbers, you will always get a whole number.

Mathematically, it can be represented as for all $a$, $b \in N$, $a + b \in N$.

For example, $12$ and $13$ are whole numbers and $12 + 13 = 25$ is also a whole number.

Also, $95$ and $52$ are whole numbers and $95 + 52 = 147$ is also a whole number.

Subtraction

The set of whole numbers is not closed under the operation subtraction. It means if you subtract any two whole numbers, you may get a whole number or may not get a whole number.

Mathematically, it can be represented as for all $a$, $b \in W$, $a – b \notin W$.

For example, $15$ and $3$ are whole numbers and $15 – 3 = 12$ is also a whole number, but $3 – 15 = -12$ in not a whole number.

Note: The operation subtraction is not commutative also.

Multiplication

The set of whole numbers is closed under the operation multiplication. It means if you multiply any two whole numbers, you will always get a whole number.

Mathematically, it can be represented as for all $a$, $b \in W$, $a \times b \in W$.

For example, $8$ and $4$ are whole numbers, and $8 \times 4 = 32$ is also a whole number.

Also, $52$ and $11$ are whole numbers and $52 \times 11 = 572$ is also a whole number.

Division

The set of whole numbers is not closed under the operation division. It means if you divide any two whole numbers, you may get a whole number or may not get a whole number.

Mathematically, it can be represented as for all $a$, $b \in W$, $a \div b \notin W$.

For example, $16$ and $2$ are whole numbers and $16 \div 2 = 8$ is also a whole number, but $2 \div 16 = 0.125$ in not a whole number.

Note: 

• The operation division is not commutative also.
• $0.125$ is a terminating rational number.

Closure Property of Integers (Signed Numbers)

Integers is a set of numbers that consists of positive numbers (i.e., all natural numbers) and negative numbers (i.e., corresponding negative values of all natural numbers) and $0$. The set of integers is represented by a set $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.

Addition

The set of integers is closed under the operation addition. It means if you add any two integers, you will always get an integer.

Mathematically, it can be represented as for all $a$, $b \in Z$, $a + b \in Z$.

For example, $-17$ and $14$ are integers and $-17 + 14 = -3$ is also an integer.

$-32$ and $42$ are integers and $-32 + 42 = -10$ is also an integer.

For addition and subtraction of integers, check here.

Subtraction

The set of integers is closed under the operation subtraction. It means if you subtract any two integers, you always get an integer.

Mathematically, it can be represented as for all $a$, $b \in Z$, $a – b \in Z$.

For example, $15$ and $-3$ are integers and $15 – \left(-3 \right) = 18$ is also an integer.

Similarly, $5$ and $8$ are integers and $5 – 8 = -3$ is also an integer.

Multiplication

The set of integers is closed under the operation multiplication. It means if you multiply any two integers, you will always get an integer.

Mathematically, it can be represented as for all $a$, $b \in Z$, $a \times b \in Z$.

For example, $-8$ and $6$ are integers, and $-8 \times 6 = -48$ is also an integer.

Also, $-20$ and $-15$ are integers and $-20 \times \left(-15 \right) = 300$ is also an integer.

For multiplication and division of integers, check here.

Division

The set of integers is not closed under the operation division. It means if you divide any two integers, you may get an integer or may not get an integer.

Mathematically, it can be represented as for all $a$, $b \in Z$, $a \div b \notin Z$.

For example, $-16$ and $4$ are integers and $-16 \div 4 = -4$ is also an integer, but $4 \div \left(-16 \right) = -0.25$ in not an integer.

Note: 

• The operation division is not commutative also.
• $-0.25$ is a terminating rational number.

Closure Property of Rational Numbers

Rational numbers are the numbers that can be represented in the form $\frac {p}{q}$, where both $p$ and $q$ are integers and $q \neq 0$.

Addition

The set of rational numbers is closed under the operation addition. It means if you add any two rational numbers, you will always get a rational number.

Mathematically, it can be represented as for all $a$, $b \in Q$, $a + b \in Q$.

For example, $2$ and $\frac {1}{2}$ are rational numbers and $2 + \frac {1}{2} = \frac {5}{2}$ is also a rational number.

$-46$ and $58$ are rational numbers and $-46 + 58 = 12$ is also a rational number.

Note: All natural numbers, whole numbers, and integers are rational numbers.

Subtraction

The set of rational numbers is closed under the operation subtraction. It means if you subtract any two rational numbers, you always get a rational number.

Mathematically, it can be represented as for all $a$, $b \in Q$, $a – b \in Q$.

For example, $28$ and $56$ are rational numbers and $28 – 56 = -28$ is also a rational number.

Similarly, $\frac {1}{2}$ and $\frac {3}{2}$ are rational numbers and $\frac {1}{2} – \frac {3}{2} = -1$ is also a rational number.

Multiplication

The set of rational numbers is closed under the operation multiplication. It means if you multiply any two rational numbers, you will always get a rational number.

Mathematically, it can be represented as for all $a$, $b \in Q$, $a \times b \in Q$.

For example, $12$ and $14$ are rational numbers, and $12 \times 14 = 168$ is also a rational number.

Also, $7$ and $\frac {1}{7}$ are rational numbers and $7 \times \frac {1}{7} = 1$ is also a rational number.

Division

The set of rational numbers is closed under the operation division. It means if you divide any two rational numbers, you will always get a rational number.

Mathematically, it can be represented as for all $a$, $b \in Q$, $a \div b \in Q$.

For example, $2$ and $3$ are rational numbers and $2 \div 3 = 0.666…$ is also a rational number.

Note: 

• The operation division is not commutative.
• $0.666…$ is a terminating but recurring rational number.

Closure Property of Irrational Numbers

Irrational numbers are the numbers that cannot be expressed in the form of $\frac {p}{q}$. Also, the numbers that are not rational numbers are called irrational numbers.

Irrational numbers when represented in decimal form give non-terminating and non-recurring numbers.

Generally the numbers with root symbol, such as $\sqrt{2}, \sqrt[3]{5}, \sqrt[4]{3}, \sqrt[5]{9}$ are all irrational numbers.

Addition

The set of irrational numbers is not closed under the operation addition. It means if you add any two irrational numbers, you may get an irrational number or you may not get an irrational number.

Note: If a number is not an irrational number, it is a rational number.

Mathematically, it can be represented as for all $a$, $b \notin \overline{Q}$, $a + b \in \overline{Q}$.

For example, $2 + \sqrt{3}$ and $3 – \sqrt{3}$ are irrational numbers and $\left(2 + \sqrt{3} \right) + \left(3 – \sqrt{3} \right)  = 5$ is not an irrational number.

Also, $7 + \sqrt{5}$ and $6 + \sqrt{5}$ are irrational numbers and $\left(7 + \sqrt{5} \right) + \left(6 + \sqrt{5} \right) = 13 + 2\sqrt{5}$ is an irrational number.

Subtraction

The set of irrational numbers is not closed under the operation subtraction. It means if you subtract any two irrational numbers, you may get an irrational number or you may not get an irrational number.

Mathematically, it can be represented as for all $a$, $b \notin \overline{Q}$, $a – b \in \overline{Q}$.

For example, $5 + \sqrt{2}$ and $7 – \sqrt{2}$ are irrational numbers and $\left(5 + \sqrt{2} \right) – \left(7 – \sqrt{2} \right)  = -2$ is not an irrational number.

Also, $10 + \sqrt{3}$ and $6 – \sqrt{3}$ are irrational numbers and $\left(10 + \sqrt{3} \right) – \left(6 – \sqrt{3} \right) = 4 + 2\sqrt{3}$ is an irrational number.

Multiplication

The set of irrational numbers is not closed under the operation multiplication. It means if you multiply any two irrational numbers, you may get an irrational number or you may not get an irrational number.

Mathematically, it can be represented as for all $a$, $b \notin \overline{Q}$, $a \times b \in \overline{Q}$.

For example, $\sqrt {2}$ and $\sqrt{8}$ are irrational numbers and $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ is not an irrational number.

Note: $\sqrt{16} = 4$ is a rational number.

Also, $\sqrt{3}$ and $\sqrt{5}$ are irrational numbers and $\sqrt{3} \times \sqrt {5} = \sqrt{15}$ is an irrational number.

Division

The set of irrational numbers is not closed under the operation division. It means if you divide any two irrational numbers, you may get an irrational number or you may not get an irrational number.

Mathematically, it can be represented as for all $a$, $b \notin \overline{Q}$, $a \div b \in \overline{Q}$.

For example, $\sqrt {27}$ and $\sqrt{3}$ are irrational numbers and $\sqrt{27} \div \sqrt{3} = \sqrt{9} = 3$ is not an irrational number.

Note: $\sqrt{9} = 3$ is a rational number.

Also, $\sqrt{7}$ and $\sqrt{5}$ are irrational numbers and $\sqrt{7} \div \sqrt {5} = \sqrt{\frac {7}{5}}$ is an irrational number.

Conclusion

The closure property means that a set of numbers is closed for some arithmetic operation. It’s not always that if you pick any two numbers from a set of numbers and you will get a number belonging to the same set of numbers. In all such cases, we say that the set of numbers is not closed for that particular operation.

Practice Problems

1. Which of the following sets are closed under the operation addition?

2. Which of the following sets are closed under the operation subtraction?

3. Which of the following sets are closed under the operation multiplication?

4. Which of the following sets are closed under the operation division?

Recommended Reading

• Commutative Property – Definition & Examples
• Distributive Property – Meaning & Examples
• Associative Property – Meaning & Examples

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