Closure Property of Real Numbers

Closure Property of Real Numbers

Real numbers are closed with respect to addition and multiplication.

This means:

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

Let’s learn in detail.

Closure Property of Real Number Addition

Take any two real numbers. Add them.  The sum that you get is another real number. This is always true. So we can say that the real numbers are closed under addition.

Let’s look at a couple of examples.

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

The problem 1.5 7.2 = 8.7 demonstrates the closure property of real number addition.

In the examples above, 3, 6, 9, 1.5, 7.2, and 8.7 are real numbers.

Closure Property of Real Number Multiplication

Take any two real numbers. Multiply them.  The product that you get is another real number. This is always true. So we can say that the real numbers are closed under multiplication.

Let’s look at a couple of examples.

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

The problem 3.4 × 5 = 17.0 demonstrates the closure property of real number multiplication.

In the examples above, 5, 8, 40, 3.4, 5, and 17.0 are real numbers.

More about Closure Property

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…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 where 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.

Solved Example on Closure Property of Real Number Addition

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

A. Real number
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 SQRT (12) and –SQRT (12)
Step 3: SQRT (12) [–SQRT (12)] = 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 1: 0, 11, and –11 are the elements of the given set {0, 11, –11}.

Step 2: 0 × 11 = 0          [0 is an element of the set.]

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.