In this post we will try to understand the concept of the modulus and conjugate of the complex number and will also try to solve some example questions for our better understanding. Apart from that we will also look into the important formulas and identities of complex numbers which are important for the point of view of Grade 11
Complex Number Parameters
Modulus of complex number
Suppose you have been provided with the following complex number
z = a + ib
The modulus of the above complex number is then represented as:
\mathbf{\mid z\mid =\sqrt{a^{2} +b^{2}}}
The symbol for modulus of complex number is |z| and the formula is already given above.
Significance of Modulus
Understand that complex number can be represented in two dimension plane with x and y axis as given below. {don’t worry we will study this concept in our next post}
In the above figure, point z is the location of the complex number in two dimensional axis. The Modulus of the complex number |z| is the shortest distance of the number z from origin O.
Since the modulus represents the distance in two dimensional plane, its value can never be negative.
Conjugate of complex number
The conjugate of any complex number is the number with same real part but with opposite sign imaginary part.
For example for complex number z = a+ ib
The conjugate will be \overline{z\ } =a-ib
You can observe that in conjugate the real part (a) is same, but the imaginary part (ib) is of opposite sign.
we can represent the complex number (z) and its conjugate in the two dimensional axis as follows:
You can observe from the above figure that both the complex number z and its conjugate are water image of each other in x-y plane.
Also their modulus |z| (distance from the center) are also equal.
Significance of Conjugate
One important property is that if you multiply complex number with its conjugate you will get a number with no imaginary part.
Suppose we have got complex number z such that
z=a+ib\\ \\
\overline{z\ } =a-ib\\\ \\
z\times \overline{z\ } =( a+ib)( a-ib)\\\ \\
Using\ the\ formula\\ \\
( a+b)( a-b) \ =\ a^{2} -b^{2}\\\ \\
z\times \overline{z\ } =a^{2} -( ib)^{2}\\\ \\
z\times \overline{z\ } =a^{2} +b^{2} \\\ \\
The\ above\ equation\ can\ also\ be\ written\ as\\ \\
z\times \overline{z\ } =\mid z\mid ^{2}
Hence multiplication of complex number with its conjugate results in square of the modulus of the complex number.
Try to remember the above mentioned formula, it will help us solve some questions.
Properties of Modulus of complex number
Below are some of the properties of modulus of complex number. These properties will help you to deepen your understanding about the nature of complex number and will also help you solve exercise questions
1. |z1 z2| = |z1| |z2|
The modulus of multiplication of two complex number is equal to the individual modulus multiplication of the numbers
2. \mid \frac{z1}{z2} \mid =\frac{\mid z1\mid }{\mid z2\mid }
The modulus of division of two complex number is equal to the individual modulus division of the numbers
3. \overline{z1\ z2} =\overline{z1} \ \overline{z2}
4. \overline{z1+z2} =\overline{z1} +\ \overline{z2}
5. \overline{\frac{z1}{z2}} =\frac{\overline{z1}}{\overline{z2}}