Operations on Real Numbers

Real Numbers are those numbers that are a combination of rational numbers and irrational numbers in the number system of maths. Real Number Operations include all the arithmetic operations like addition, subtraction, multiplication, etc. that can be performed on these numbers. Besides, imaginary numbers are not real numbers. Imaginary numbers are used for defining complex numbers. To get real numbers, first, we have to understand rational numbers and irrational numbers. Rational numbers are those numbers that can be written as p/q where p is the numerator and q is the denominator and p and q are integers. For example, 5 can be written as 5/1, so it is a rational number and irrational numbers are those numbers that cannot be written in the form of p/q. 

For example, √3 is an irrational number, it can be written as 1.73205081 and continuous to infinity, and it cannot be written in the form of a fraction and is a non-terminating form and non-recurring decimal. And if combine rational numbers and irrational numbers become real numbers.

Example: 12, -8, 5.60, 5/1, π(3.14), etc.

Real numbers can be positive and negative, and it is denoted by R. All the decimals, natural numbers, and fraction come under this category.

Operations on Real Numbers

There are four basic mathematical operations

• Addition
• Subtraction
• Multiplication
• Division

We can easily perform all four mathematical operations with real numbers. Now we will understand these operations on real numbers, i.e. both rational numbers and irrational numbers.

Operation on Two Rational Numbers

When we perform arithmetic operations on two rational numbers like addition, subtraction, division, and multiplication then the result will be rational numbers.

Example

All mathematical operations performed on rational numbers are explained below,

• 0.25 + 0.25 = 0.50 can be written as 50/100 (which is a form of p/q)
• 0.20 – 0.10 = 0.10 can be written as 10/100 (which is a form of p/q)
• 0.4 multiplied by 184 is 73.6 and can be written as 736/10 (which is a form of p/q)
• 0.252 divided by 0.4 is 0.63 and can be written as 63/100 (which is a form of p/q)

Operations on Two Irrational Numbers

When we perform arithmetic operations like addition, subtraction, multiplication, or division on two irrational numbers then the result can be rational numbers or irrational numbers.

Example

All mathematical operations performed on irrational numbers are explained below,

• √2 + √3 = 3.14 can be written as 314/100 which is a rational number.
• √3 – √3 = 0 or 5√4 – 4√3 = 3.07 which can be written as 307/100 which is a rational number.

When √5 is multiplied by √5, we get 5 which is a rational number, or when √3 is multiplied by √5, we get √15 which is an irrational number. When √8 is divided by √8 we get 8 which is a rational number or if √5 is divided by √3 then we get (√5)/(√3) which is an irrational number.

Operation on Rational Numbers and Irrational Numbers

Various operations on a rational number and an irrational number are discussed below:

Addition

When we add an irrational number and a rational number then the result will be an irrational number. When 3 is added to 2√5 then the result will be an irrational number.

Subtraction

When we perform subtraction on an irrational number and a rational number then the result will be an irrational number. When 5√6 is subtracted to 3 then the result will be an irrational number.

Multiplication

When we perform this operation the result can be irrational or rational. When 3 is multiplied by √5 then the result will be 3√5 which is an irrational number and if √12 is multiplied by √3 then the result will be √36 and it can be written as 6 which is a rational number.

Division

If a rational number is divided by an irrational number or vice versa then the result will be always an irrational number. When 4 is divided by √2 then the results will be 4√2 which is an irrational number.

Properties of Real Numbers

We have four properties which are commutative property, associative property, distributive property, and identity property. Consider a, b, and c are three real numbers. Then these properties can be described as

Commutative Property

If a and b are the numbers, then a + b = b + a for addition and a × b = b × a for multiplication.

Addition:

a + b = b + a

For example, 5 + 6 = 6 + 5

Multiplication:

a × b = b × a

For example, 4 × 2 = 2 × 4

Read More about Commutative Property.

Associative Property

If a, b, and c are the real numbers then the form will be

a + (b + c) = (a + b) = c for addition and (a.b)c = a(b.c) for multiplication

Addition: 

a + (b + c) = (a + b) = c

For example, 5 + (3 + 2) = (5 + 3) + 2

Multiplication: 

(a.b)c = a(b.c)

For example, (4×2)×6 = 4×(2×6)

Read More about Associative Property.

Distributive Property

If a, b, and c are the real numbers then the final form will be

a (b + c) = ab + ac

(a + b) c = ac + ab

For example, 5 (2 + 3) = 5×2 + 5×3 the answer will be 25 for both the left and right terms.

Read More about Distributive Property.

Identity Property

There are two identity properties of real numbers, identity property of the addition of a real number states that, on adding a number with zero the result is the number itself, whereas the identity property of multiplication of a real number states that multiplying a number with one gives the numbers itself.

Addition:

a + 0 = 0 (0 is the additive identity)

Multiplication:

a × 1 = 1 × a = 1 (1 is multiplicative identity)

Read More about Identity Property.

Real Numbers

Various numbers are contained inside real numbers and some common numbers that are contained inside real numbers are,

• Rational Numbers: 4/5, 0.82
• Irrational Numbers: √2, π, 0.102012…
• Integers: {… – 3, -2, -1,0,1,2,3…}
• Whole Numbers: {0,1,2,3…}
• Natural Numbers: {1,2,3…}

Operations on Real Numbers Examples

Example 1: Simplify (2√3 + √7) + (3√3 – 4√7)

Solution:

= (2√3 + √7) + (3√3 – 4√7)

= 2√3 + √7 + 3√3 – 4√7

= 2√3 + 3√3 + √7 – 4√7

= 5√3 – 3√7

Example 2: Simplify (-√3) × (- 4√3)

Solution:

= (-√3) × (- 4√3)

= 4(√3)(√3)

= 4 × 3

= 12

Example 3: Simplify (9√5 / 3√5)

Solution:

= (9√5 / 3√5)

= 9√5 / 3√5

= 3

Example 4: Simplify 34(√3) – √3(3 + √3)

Solution:

= 34(√3) – √3(3 + √3)

= 34(√3) – 3√3 – 3

= 31√3 – 3

Sample Problems on Operations on Real Numbers

Problem 1: Show that 7√7 is an irrational number.

Solution:

Let us assume, to the contrary, that 7√7 is rational.

That is, we can find coprime a and b (b ≠ 0) such that 7√7 = ab

Rearranging, we get √7 = ab/7

Since 7, a and b are integers, ab/7 is rational, and so √7 is rational.

But this contradicts the fact that √7 is irrational.

So, we conclude that 7√7 is irrational.

Problem 2: Explain why (17 × 5 × 13 × 3 × 7 + 7 × 13) is a composite number.

Solution:

17 × 5 × 13 × 3 × 7 + 7 × 13 …(i)

= 7 × 13 × (17 × 5 × 3 + 1)

= 7 × 13 × (255 + 1)

= 7 × 13 × 256

Number (i) is divisible by 2, 11 and 256, it has more than 2 prime factors.

Therefore, (17 × 5 × 13 × 3 × 7 + 7 × 13) is a composite number.

Problem 3: Prove that 3 + 2√3 is an irrational number.

Solution:

Let us assume to the contrary, that 3 + 2√3 is rational.

So that we can find integers a and b (b ≠ 0).

Such that 3 + 2√3 = ab, where a and b are coprime.

Rearranging the equations, we get since a and b are integers, we get a²b−32 is rational and so √3 is rational.

But this contradicts the fact that √3 is irrational.

So we conclude that 3 + 2√3 is irrational.

Practice Questions on Operations on Real Numbers

Q1: Add 3√3 and 15√3.

Q2: What should be subtracted from 20√5 to get 7√5

Q3: Divide -13/7 by 26/21.

Q4: Simplify -2/3 ⨯ 15/8 ⨯ -32/105

Q5: Solve 2/5(10/9 + 7/6)