Complex Conjugate
Every complex number has another complex number associated with it, known as the complex conjugate. A complex conjugate of a complex number is another complex number that has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign. The product of a complex number and its complex conjugate is a real number.
A complex conjugate gives the mirror image of the complex number about the horizontal axis (real axis) in the Argand plane. In this article, we will explore the meaning of conjugate of a complex number, its properties, complex root theorem, and some applications of the complex conjugate.
What is a Complex Conjugate?
A complex conjugate of a complex number is another complex number whose real part is the same as the original complex number and the magnitude of the imaginary part is the same with the opposite sign. A complex number is of the form a + ib, where a, b are real numbers, a is called the real part, b is called the imaginary part, and i is an imaginary number equal to the root of negative 1. The complex conjugate of a + ib with real part 'a' and imaginary part 'b' is given by a  ib whose real part is 'a' and imaginary part is 'b'. a  ib is the reflection of a + ib about the real axis (Xaxis) in the argand plane. The complex conjugate of a complex number is used to rationalize the complex number.
Complex Conjugate Definition
The complex conjugate of a complex number, z, is its mirror image with respect to the horizontal axis (or xaxis). The complex conjugate of complex number \(z\) is denoted by \(\bar{z}\). In polar form, the complex conjugate of the complex number re^{ix } is re^{ix}. An easy way to determine the conjugate of a complex number is to replace 'i' with 'i' in the original complex number. The complex conjugate of x + iy is x  iy and the complex conjugate of x  iy is x + iy. As in the image given below, if the complex number z lies in the first quadrant, its image about the horizontal axis, that is, the complex conjugate \(\bar{z}\) lies in the fourth quadrant. Let us consider a few examples: the complex conjugate of 3  i is 3 + i, the complex conjugate of 2 + 3i is 2  3i.
Complex Conjugate of a Matrix
The complex conjugate of a matrix A with complex entries is another matrix whose entries are the complex conjugates of the entries of matrix A. Consider a row matrix A = [1i 4+2i 3+7i], the complex conjugate of matrix A is B = [1+i 42i 37i] where each entry in matrix B is the conjugate of each entry in matrix A. The complex conjugate of matrix A is denoted by \(\bar{A}\). So, B = \(\bar{A}\). Let us consider another example of a matrix with complex entries and determine its complex conjugate.
Multiplication of Complex Conjugate
When a complex number is multiplied by its complex conjugate, the product is a real number whose value is equal to the square of the magnitude of the complex number. To determine the value of the product, we use algebraic identity (x+y)(xy)=x^{2}y^{2} and i^{2} = 1. If the complex number a + ib is multiplied by its complex conjugate a  ib, we have
(a + ib)(a  ib) = a^{2}  (ib)^{2} = a^{2 } i^{2}b^{2} = a^{2} + b^{2}
Let us consider an example and multiply a complex number 3 + i with its conjugate 3  i
(3 + i)(3  i) = 3^{2}  (i)^{2} = 3^{2 } i^{2} = 9 + 1 = 10 = Square of Magnitude of 3 + i
Complex Conjugate Root Theorem
The complex conjugate root theorem states that if f(x) is a polynomial with real coefficients and a + ib is one of its roots, where a and b are real numbers, then the complex conjugate a  ib is also a root of the polynomial f(x).
To understand the theorem better, let us take an example of a polynomial with complex roots. Consider f(x) = x^{3}  7x^{2} + 41x  87. Now, the roots of the polynomial f(x) are 3, 2 + 5i, 2  5i. Here 2 + 5i and 2  5i are the roots of f(x) and conjugates of each other. This implies that nonreal roots, that is, the complex roots of a polynomial come in pairs. Hence, if we know one complex root of a polynomial, then we can say that its complex conjugate is also a root of the polynomial without calculating it.
Properties of Complex Conjugate
Let us now discuss a few properties of complex conjugate which can make our calculations simple and easier. Consider two complex numbers z and w and their complex conjugates \(\bar{z}\) and \(\bar{w}\), respectively.
 The complex conjugate of the product of two complex numbers is equal to the product of the complex conjugates of the two complex numbers, that is, \(\overline{zw} = \bar{z}.\bar{w}\)
 The complex conjugate of the quotient of two complex numbers is equal to the quotient of the complex conjugates of the two complex numbers, that is, \(\overline{(z/w)} = \bar{z} / \bar{w}\)
 The complex conjugate of the sum of two complex numbers is equal to the sum of the complex conjugates of the two complex numbers, that is, \(\overline{z+w} = \bar{z} + \bar{w}\)
 The complex conjugate of the difference of two complex numbers is equal to the difference of the complex conjugates of the two complex numbers, that is, \(\overline{zw} = \bar{z}\bar{w}\)
 The sum of a complex number and its complex conjugate is equal to twice the real part of the complex number, that is, \(z + \bar{z} = 2Re(z)\)
 The difference between a complex number and its complex conjugate is equal to twice the imaginary part of the complex number, that is, \(z  \bar{z} = 2Im(z)\)
 The product of a complex number and its complex conjugate is equal to the square of the magnitude of the complex number, that is, \(z.\bar{z} = z^2\)
 The real part of a complex number is equal to the real part of its complex conjugate and the imaginary part of a complex number is equal to the negative of the imaginary part of its complex conjugate, that is, \(Re(z) = Re(\bar{z})\) and \( Im(z) = Im(\bar{z})\)
Important Notes on Conjugate of a Complex Number
 The complex conjugate of x + iy is x  iy and the complex conjugate of x  iy is x + iy.
 When a complex number is multiplied by its complex conjugate, the product is a real number whose value is equal to the square of the magnitude of the complex number.
 The complex roots of a polynomial come in pairs.
Related Topics on Complex Conjugate
Complex Conjugate Examples

Example 1: Find the complex conjugate of the complex number 4z  i2w, if z = 2  3i and w = 4  7i.
Solution: We will first simplify 4z  i2w = 4(2  3i)  i2(4  7i) = 8  12i + 8i 14 = 6  4i
To determine the complex conjugate of 4z  i2w = 6  4i, we will change the sign of i. Therefore the complex conjugate of 6  4i is 6 + 4i.
Answer: The complex conjugate of 4z  i2w is 6 + 4i.

Example 2: Express the complex number z/w in the form of x + iy if z = 4  5i and w = 2 + 3i, where x and y are real numbers.
Solution: To simplify z/w = (4  5i)/(2 + 3i), we will rationalize the denominator by multiplying z/w by the complex conjugate 2  3i of 2 + 3i.
z/w = ((4  5i)/(2 + 3i)) × ((2  3i )/(2  3i )) = (8  12i + 10i 15)/(2^{2} + 3^{2}) = (23 2i)/13 = (23/13) + (2/13)i
Answer: 4z  i2w = (23/13) + (2/13)i
FAQs on Complex Conjugate
What is a Complex Conjugate in Maths?
A complex conjugate of a complex number is another complex number whose real part is the same as the original complex number and the magnitude of the imaginary part is the same with the opposite sign. The complex conjugate of a + ib with real part 'a' and imaginary part 'b' is given by a  ib whose real part is 'a' and imaginary part is 'b'.
What is the Complex Conjugate of 4  3i?
The complex conjugate of 4  3i is 4 + 3i. To determine the complex conjugate of a complex number a + ib, we change the sign of i, that is, a  ib is the complex conjugate of a + ib.
What is the Complex Conjugate of a Real Number?
A real number has no imaginary part, therefore, the complex conjugate of a real number is the number itself. For example, the complex conjugate of the real number 3 is 3.
How to Find Complex Conjugate of a Fraction?
If z and w are two complex numbers, then the complex conjugate of the fraction z/w is given by: \(\overline{(z/w)} = \bar{z} / \bar{w}\). The complex conjugate of the fraction of two complex numbers is equal to the fraction of the complex conjugates of the two complex numbers.
What Are Complex Conjugate Roots?
The complex conjugate root theorem states that if f(x) is a polynomial with real coefficients and a + ib is one of its roots, where a and b are real numbers, then the complex conjugate a  ib is also a root of the polynomial f(x). This implies that nonreal roots, that is, the complex roots of a polynomial come in pairs.