Modulus and Conjugate of a Complex Number

In the argand plane, the modulus of a complex number is the distance between it and its origin. The non-negative value (x² + y²) is called the modulus of complex number (z = x + iy) if z = x + iy is a complex number with real x and y and I = -1. The absolute value of a complex number is also known as the modulus of the complex number.

In this post, we will learn how to understand the notion of modulus of the complex numbers algebraically and visually, as well as its formula and several solved cases.

Modulus of Complex Number

The square root of the sum of squares of the real and imaginary parts of a complex number is the modulus of the complex number. If z is a complex number,

 its modulus is defined as √[Re(z)]² + [Im(z)]²Â and symbolised by |z|. 

The distance between the origin (0, 0), as well as the point (a, b) in the complex plane, is the modulus of a complex number z = a + ib. Because the distance is the modulus of a complex number, its value is always non-negative.

Formula

The formula |z| = (x² + y²) gives the modulus of a complex number z = x + iy, denoted by |z|, where x is the real component and y is the imaginary part of the complex number z. The conjugate of z can also be used to derive the modulus of a complex number z. We have z. z = (x+iy) (x-iy) = x² + y²Â because z. z = (x + iy) (x – iy) = x² + y².

Graphing the Modulus of a Complex Number

The distance between the complex number’s coordinates and the origin on a complex plane is called the modulus of the complex number when it is shown on a graph. The modulus of a complex number is the distance of a complex number represented as a point on the argand plane (a, b). This distance is defined as r = √ (a² + b²) and is measured from the origin (0, 0) to the point (a, b).

Furthermore, the Pythagoras theorem may be used to calculate the modulus of a complex number, where the hypotenuse is the modulus, the real part is the base, and the imaginary part is the imaginary part. The magnitude (or length) of the vector encoding a + bi is the same as the modulus of a complex number a + bi.

Modulus of Complex Number Properties

Some of the features of a complex number are listed below. When z and w are both complex numbers, we get:

The modulus for complex numbers z & -z is the same, i.e., |z| = |-z|.

|z| = 0 iff z = 0 The modulus of a complex number is 0 if and only if the complex number is zero.

|z.w| = |z|.|w| The modulus of the product of complex numbers is equal to the product of the modulus of complex numbers.

|z/w| = |z|/|w| The modulus of the quotient of 2 complex numbers is equal to the quotient of the modulus of the complex numbers. 

|z|=| z| The modulus of such a complex number is equal to the modulus of the conjugate of the complex number.

The modulus of a complex number’s nth power is equal to the modulus of the complex number’s nth power, so |zⁿ| = |z|ⁿ.

Conjugate of a Complex Number

zÌ… = x – iy represents the conjugate of a complex number z = x + iy. It’s the complex number’s reflection or image on Argand’s plane around the real axis. The conjugate of the complex number is obtained by replacing I with ‘- I.

A complex number z = x + iy has a conjugate of x – iy, which is denoted as z̅.

The conjugate of the complex number z = 3 – 4i, for example, is 3 + 4i.

Take the complex number z = a + ib as an example. The following formulas can be used to do this.

Addition

z + zÌ… = (a +ib) + (a-ib) = 2a

This is a complex number with a zero imaginary portion.

Re (z + zÌ…) = 2a, Im (z+ zÌ…) = 0

Here,

Subtraction

z + zÌ… = (a +ib) – (a-ib) = 2bi

The coordinates of the complex number z = x + iy in the x-axis are the coordinates of zÌ….

The complex conjugate of such a complex number is the number that has the same real part and imaginary portion in magnitude but has the opposite sign. Polynomial roots are discovered using complex conjugates. The complex conjugate root theorem states that if a complex number for one variable with real coefficients is a root of a polynomial, its conjugate is also a root.

Conclusion

A complex number is defined as a number that may be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit in mathematics. The real part of the number is referred to as a, while the imaginary part of the number is referred to as b.

The modulus of a complex number aids in the calculation of the distance of the complex number from the origin in the argand plane, while the conjugate of a complex number aids in the discovery of polynomial roots and the conjugate of a complex number gives the deflection of the real axis in the argand plane.