**Definition of Closure Property of Real Numbers Addition **

Real numbers are closed with respect to addition and multiplication.

If you add or multiply real numbers the answer is also real.

Let’s go through a couple of examples.

The problem **3 + 6 = 9**** **demonstrates the closure property of real number addition.

Observe that the addends and the sum are real numbers.

The closure property of real number addition states that when we add real numbers to other real numbers the result is also real.

In the example above, 3, 6, and 9 are real numbers.

The problem **5 × 8 = 40**** **demonstrates the closure property of real number multiplication.

Observe that the factors and the product are real numbers.

The closure property of real number multiplication states that when we multiply real numbers with other real numbers the result is also real.

In the example above, 5, 8, and 40 are real numbers.

In general, Closure Property states that:

**When you combine any two elements of the set the result is also in that set.**

Real numbers are closed with respect to addition and multiplication. The examples above illustrate this.

but....what about subtraction and division? Are real numbers closed under subtraction and division too?

Well...subtraction of real numbers is closed but division of real numbers is not closed as we cannot divide by zero.

There are situations when we don’t get a closed system.

For example:

Subtraction of natural numbers is NOT closed.

Consider the natural numbers 7 and 8.

7 – 8 = **– 1**

**Negative 1 is NOT a natural number.**

So, closure property doesn’t work here.

Therefore, the set of natural numbers is not closed under subtraction.

Determine the set that does not satisfy closure property of addition.

A. Real number

B. Irrational numbers

C. Rational numbers

D. Integers

B. Irrational numbers

C. Rational numbers

D. Integers

**Solution:**

**Step 1:** Here, only the set of irrational numbers does not satisfy closure property of addition.

**Step 2:** For example, consider the irrational numbers Ö12 and -Ö12

**Step 3:** Ö12 + (-Ö12) = 0 is a rational number.

**Step 4:** So, the set of irrational numbers does not satisfy the Closure property under addition.

Solved Example on Closure Property of Real Number Multiplication

Determine whether the set {0, 11, - 11} satisfies closure property with respect to multiplication.

**Solution:**

**Step 3: **- 11 × 0 = 0 [0 is an element of the set.]

**Step 4: **- 11 × 11 = - 121, not an element of the given set.

**Step 5:** So, the given set does not satisfy the closure property with respect to multiplication.

**Related Terms for Closure Property of Real Number Addition and Multiplication **

- Real Number
- Sum
- Product