Rationalization of Complex Numbers

A rational number is of the form p/q where p and q are integers and q is not equal to zero. It is the ratio of two integers. The basic difference between a fraction and a rational number is that rational numbers can be positive or negative whereas fractions are always positive. A fraction becomes a rational number when the numerator and denominator are integers. All fractions are rational numbers. The rational number can be expressed in a simplified form. The decimal of a rational number terminates after a finite number of decimal places and can be recurring. The set of rational numbers includes integers, whole numbers, and natural numbers. The symbol ‘Q’ is used to define the set of rational numbers. There are different types of Rational Numbers. Some of them are:

• Integers like 1, and -4 are rational numbers since they are expressed in the form 1/1 and -4/1.
• Fractions like 2/3, 8/9.
• Decimals like -0.9, 0.847474, etc.
• 0 is a rational number.
• 1.313131…. is a rational number.

Irrational numbers are those numbers that cannot be expressed in the form of p/q. Their decimal expansion is non-recurring and non-terminating. They can only be expressed in terms of roots. Although they are real numbers they are not expressed in a ratio. There are different types of Irrational Numbers. Let ‘R’ be the set of real numbers and ‘Q’ is the set of rational numbers. Then ‘R’ – ‘Q’ is the set of irrational numbers often denoted by ‘P’. Some of them are:

• π is an irrational number
• 3.42325390538929213465768… is an irrational number
• √2 is an irrational number.

Properties of Rational and Irrational numbers

Let’s take a look at some properties of rational and irrational numbers,

• The sum or difference of two rational numbers is rational. For example: 2 + (-3) = -1 is rational.
• The product of two rational numbers is always rational. For example: 0.2 × 0.06 = 0.012
• The sum or difference or division of two irrational numbers can be rational or irrational.
• The sum or difference of rational and irrational numbers is always irrational. For example: 2 + √2 is irrational.
• The product of a rational and irrational number is always irrational. For example: 2 × √2 = 2√2 is irrational

Complex Numbers

Complex Numbers are of the form x + iy where x and y are real numbers and i is the iota which is used to represent the imaginary number. It is the combination of real and imaginary numbers. For example: Let 2 + 5i be a complex number. The real part of the complex number is 2 and the imaginary part is 5i. The ‘i’ have also known as iota is the square root of -1. As we all know, the square roots of negative numbers cannot be represented on the number line so they are represented by ‘i’. The value of i is given by √-1.

• i² = -1
• i³ = -i
• i⁴ = 1

Operations of Complex numbers

• Addition of Complex Numbers: The addition of two complex numbers is the addition of real parts and imaginary parts separately. Let a+ib and x+iy be two complex numbers. The result is (a + x) + i( b + y).
• Difference between Complex Numbers: The difference between two complex numbers is a difference between the real part and imaginary parts separately. Let a+ib and x+iy be two complex numbers. The result is (a – x) – i(b – y).
• Multiplication of complex Numbers: Let a + ib and x + iy be two complex numbers. The multiplication of two complex numbers is (a + ib) × (x + iy) = ax + iay + ibx – by
• Conjugate of Complex Number: The conjugation of complex numbers is nothing but changing the sign of the operator. Let a + ib be a complex number. The conjugate is a – ib
• Division of complex numbers: In division we rationalise the denominator by multiplying it with its conjugate.  Let a+ib and x + iy be two complex numbers. Therefore (a + ib)/c + id = [(a + ib) × (c – id)]/(c² + d²)

Rationalization of complex numbers

To make the denominator free from radicals we multiply the numerator and the denominator with an irrational number. The irrational number that we multiply is the radical that is present in the denominator. Rationalization is used to simplify the denominator so that a denominator is a whole number. It is done to simplify the fraction. Let us illustrate with the help of an example:

Let x = 1/√2, we multiply √2 in the numerator and the denominator. Therefore the result becomes √2/2. The above method is applicable when there is only one term in the denominator. But when the denominator is in the form of expression, then we multiply with the conjugate of the denominator. 

For example: Let us rationalize, 1/(1 – √2). Since the sign is negative we will multiply with (1 + √2) in the numerator and the denominator.

1/(1 – √2) × (1 + √2)/(1 + √2) 

= -(1 + √2)

Similar Problems

Question 1: Find the value of (2 + √5i)(2 – √5i).

Solution:

This is of the form (a – b)(a + b) = a² – b²

Here, a = 2 b = √5

Value of i² = -1

The value is 2² – (√5i)²

= 4 +  5 = 9

Question 2: Simplify 

•  1/5√5i 
• (9 + 2√5)/√5i

Solution: 

• We multiply 5√5i with the numerator and denominator

1/5√5i × (5√5i/5√5i) 

= -5√5i/125 

= -√5i/25

• We multiply √5 with the numerator and denominator

(9 + 2√5)/√5i × (√5i/√5i) 

= (9√5i – 10)/5

Question 3: Simplify (1 + √5)/(2 + √5)

Solution: 

Multiplying with the conjugate of the denominator we get,

(1 + √5)/(2 + √5i) × (2 – √5i)/(2 – √5i) 

= (1(2 – √5i) + √5(2 – √5i))/(9) 

= (2 – √5i + 2√5  -5i)/9

Question 4: Multiply (√2+√5)/(√7 – √3i) with its conjugate.

Solution: 

Multiplying with the conjugate of the denominator we get,

(√2 + √5) / (√7 – √3i) × (√7 + √3i)/(√7 + √3i) 

= (√2(√7 + √3i) + √5(√7 + √3i) )/10 

Question 5: Find a simplified version of 1/√2345.

Solution:

Multiplying the denominator with √2345 we get,

1/√2345i × √2345i/√2345i 

= -√2345/2345

Question 6: Find a simplified value of √8/(√6 + √2i).

Solution:

√6 = √2  ×  √3

Taking √2 from numerator and denominator we get  2/{√3 + i}

Rationalizing the denominator we get,

2/(√3 + i) × (√3 – i)/(√3 – i) 

= (√3 – i)/2

Question 7: Rationalise 1/(√3 + 4i).

Solution:

Rationalizing the denominator by multiplying √3 – 4i in numerator and denominator we get,

1/(√3 + 4i) × (√3 – 4i)/(√3 – 4i) 

= (√3 – 4i)/((√3² – 4i²) 

= (4 – √3)/25