Properties of Real Numbers

The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

A Number system or numeral system is defined as elementary system to express numbers and figures. It is the unique way of representation of numbers in arithmetic and algebraic structure.

What are Properties of Real Numbers?

The Properties of Real Numbers are as follows:

• Closure Property 
• Commutative Property 
• Associative Property 
• Distributive Property
• Identity Element Property 
• Inverse Element Property

Let’s understand these properties in detail.

Closure Property

The closure Property of Real Numbers states that the addition and multiplication of any real number results in a real number. Let’s consider the following examples for addition as well as multiplication.

Closure Property for Addition 

2 + 5 = 7 and √2 + 5√2 = 6√2

Where, 2 and 5 (both real numbers) added to get 7(real number) and √2 and 5√2(both real numbers) to get 6√2 (real number).

Closure Property for Multiplication

6× 5 = 30 and √3 ×2√5 = 2√(15)

Where, 6 and 5 (both real numbers) multiply together to get 30(real number) and √3 & 2√5 (both real numbers) are multiplied together to get 2√(15)(real number).

Commutative Property

This property states that the order of operation for addition and multiplication doesn’t matter as it will give us the same result in any order we take the two values for which the operation is defined. In other words, if we have two real numbers a and b, then either we add them a + b or b + a, the result remains the same. Similarly for multiplication, either ab or ba the result remains the same.

This property is only valid for addition and multiplication not for subtraction and division.

For addition x + y = y + x

Example: If we add 6 in 2 or add 2 in 6 results will be the same i.e.,

6 + 2 = 8 = 2 + 6

For multiplication x×y = y×x

Example: If we multiply both the real number (6 and 5) the results will be same i.e.,

6 × 5 = 30 = 5 × 6 

Where, x and y are any two Real Numbers.

Associative Property

This property states that when three or more numbers are added or multiplied then the result is the same regardless of the grouping of the components.

The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property.

That is, rearranging the numbers in such a manner that will not change their value.

For addition (x + y) + z = x + (y + z)

Example: If we add 6 in 2 or add 2 in 6 results will be the same i.e.,

(4 + 5) + 6 = 15 = 4 + (5 + 6)

For multiplication (x×y)×z = x×(y×z)

Example: If we multiply both the real numbers (6 and 5) the results will be the same i.e.,

(2 × 5) × 6 = 60 = 2 × (5 × 6)

Where, x, y, and z are any three Real Numbers.

Distributive Property

This property helps us to simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.

• x × (y + z) = x × y + x × z
• x × (y – z) = x × y – x × z  

Example: Simplify 4 × (5 + 6)

Solution:

4 × (5 + 6) = 4 × 5 + 4 × 6

⇒ 4 × (5 + 6) = 20 + 24

⇒ 4 × (5 + 6) = 44

Identity Element Property

This is an element that leaves other elements unchanged when combined with them. The identity element for the addition operation is 0 and for multiplication is 1.

For addition, 0 is the identity element for the Real Numbers i.e.,

a + 0 = a = 0 + a

For multiplication, 1 is the identity element for the Real Numbers i.e.,

a×1 = a = 1×a

Note: 0 is the additive identity and 1 is the multiplicative identity.

Inverse Element Property

The inverse of an element is the number in the given set which when operated on the element gives the identity. In other words, in addition for any real number a, the inverse of a is such a number when added with results in identity (0) or for multiplication for any real number b, the inverse is such a real number when multiplied with b results in identity (1).

For addition: a + (-a) = 0, i.e., -a and a are inverses of each other under Addition.

Example: Find the additive inverse of 1/5.

Solution:

Let the inverse of 1/5 be x, then using the property of inverse 

1/5 + x =0

x = -1/5

Similarly, 7 & -7 are the additive inverse of each other.

For Multiplication: a×1/a = 1 i.e., a and 1/a are inverse of each other under multiplication.

Example: Find the multiplicative inverse of √2.

Solution:

Let the inverse of √2 be x, then using the property of inverse 

√2 × x = 1

x = 1/√2

 Similarly, 7 & 1/7 are the multiplicative inverse of each other.

Read More

• Complex Numbers
• Irrational Numbers
• Operations on Real Numbers

Sample Problems on Properties of Real Numbers

Problem 1: Give some examples of Commutative properties.

Solution:

For addition

• 8 + 3 = 3 + 8 = 11
• 26 + 11 = 11 + 26 = 37

 For multiplication

• 12 × 5 = 5 × 12 = 60
• 2 × 5 = 5 × 2 = 10

Problem 2: Simplify 70 × (20 + 9) by distributive property.

Solution:

As per the distributive property 

⇒ 70 × (20 + 9) = 70 × 20 + 70 × 9

⇒ 70 × (20 + 9) = 1400 + 630

⇒ 70 × (20 + 9) = 2030

Problem 3: Verify the associative property for the following:

(30 + 60) + 7 = 30 + (60 + 7)

Solution:

To verify the given expression, calculate the LHS and RHS separately, 

LHS = (30 + 60) + 7 

⇒ LHS = 90 + 7 

⇒ LHS = 97 

RHS = 30 + (60 + 7)

⇒ RHS= 30 + 67

⇒ RHS= 97

⇒ LHS = RHS [Hence, verified.]

FAQs on Properties of Real Numbers

Q1: Define Real Number.

Answer:

A real number is a number that can be expressed as a finite or infinite decimal representation. Examples of real numbers include 2, -3, 0.5, √2, and π.

Q2: What are the Properties of Real Numbers?

Answer:

Some properties of real numbers are as follows:

• Commutativity Property
• Associativity Property
• Distributivity Property
• Identity Property
• Inverse Property

Q3: What is the Additive Inverse of a Real Number?

Answer:

The additive inverse of a real number x is the number -x, such that x + (-x) = 0.

Q4: What is the Multiplicative Inverse of a Real Number?

Answer:

The multiplicative inverse of a nonzero real number x is the number 1/x, such that x×(1/x) = 1.

Q5: What is the Absolute Value of a Real Number?

Answer:

The absolute value of a real number x is the distance between x and 0 on the number line. It is denoted by |x| and is always nonnegative. 

[Tex]\bold{|x| =\begin{cases} x,& x\geq0 \\ -x & x<0\end{cases}} [/Tex]

Q6: What is the Order Property of Real Numbers?

Answer:

The order property of real numbers states that any two real numbers can be compared using the inequality symbols < (less than), > (greater than), or = (equal to). In other words, real numbers can be arranged in a linear order.