Rationalization of Complex Numbers
Last Updated :
10 Jan, 2024
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.
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
Similar Reads
Comparision of Rational Numbers
Comparison of Rational Numbers: Rational numbers are crucial for problem-solving across various disciplines. They help us solve equations, analyze data, make predictions, and model real-world scenarios in science, engineering, economics, and finance. Comparing rational numbers is similar to comparin
5 min read
Rationalization of Denominators
Rationalization of Denomintors is a method where we change the fraction with an irrational denominator into a fraction with a rational denominator. If there is an irrational or radical in the denominator the definition of rational number ceases to exist as we can't divide anything into irrational pa
8 min read
Polar Representation of Complex Numbers
Complex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum
9 min read
Square Root of Complex Numbers
A complex number is a number that has two components: or equally divided into a real part and an imaginary part. The general form of a complex number is:z = a + biwhere:a is the real part.b is the imaginary part, and i is the imaginary unit, where i 2 = -1. Square Root of Complex NumbersSquare Root
8 min read
Decimal Expansions of Rational Numbers
Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. So in this article let's discuss some rational and irrational numbers an
6 min read
Complex Numbers Questions with Solutions
Complex numbers are fundamental mathematical concepts with wide-ranging applications in science and engineering. Understanding them is essential for solving advanced problems in fields like physics, electrical engineering, and signal processing.This article presents a variety of important complex nu
7 min read
Multiplication of Rational Numbers
For the multiplication of rational numbers, we take the numerators multiplication and the denominators multiplication and divide the numerator multiplication by denominator multiplication. Simplify the obtained result to get the multiplication of rational numbers. In this article we will cover the m
5 min read
Division of Rational Numbers
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Some examples of rational numbers are 1/2, -3/2, 5, etc. 5 is whole number which can be written as 5/1 in the form of a fraction. Hence, we can say that a whole number is also a ratio
6 min read
Trigonometric Form of a Complex Number
In mathematics, particularly in complex number theory, the trigonometric form of the complex number plays a crucial role. This representation is not only elegant but also simplifies many operations involving complex numbers such as multiplication and division. The trigonometric form is an essential
5 min read
Algebraic Operations on Complex Numbers
A complex number is a number that includes both a real and an imaginary part. It is written in the form:z = a + biWhere:a is the real part,b is the imaginary part,i is the imaginary unit, satisfying i2 = â1.Algebraic operations on complex numbers follow specific rules based on their real and imagina
7 min read