Modulus of a Complex Number – Many maths students have come across the terminology, modulus. Basically, it is denoted by two vertical lines that shows the distance of the number from the origin to argand plane. Those who don’t know anything about the Argand plane, they must read some articles about it and then read this one. Also, many of you have heard about the complex number. It is a mixture of real and imaginary parts. For instance, if z = x + iy complex number is there where x and y are real but i= √-1 the modulus of a complex number will be √(x2 + y2).
To put simply, modulus of a complex number is also considered as complex numbers’ absolute value. In this article, we will look at some concepts related to the modulus of a complex number. Follow this guide to get a better understanding and after reading this post start solving some questions related to triangles as well as this modulus of complex numbers too.
Table of Contents
Know all about the Modulus of Complex Number
In simpler words the modulus of a complex number will be the real part and imaginary part sum’s square root. Did you understand? No worries, for instance, if z is a complex number( the number which we have mentioned above) , the modulus of that number will be √{[Re(z)]2 + [Im(z)]2}. And we also talked about two vertical lines, the symbol will be |z|for denoting the modulus.
In this example, the complex number is z = a + ib. It shows the distance between origin points (0,0) and the coordinate points (a,b) in the complex plane. And always remember that the modulus cannot be negative because it is portraying the distance. And we all know that the distance cannot be negative. Thus, remember this fundamental when you are solving modulus related questions.
Formula of the Modulus of Complex Numbers
As you have understood about that the complex number is denoted by |z|. For that the z is a complex number which is z = x + iy. In this x is the real part and y is the imaginary part. The absolute modulus formula will be |z| = √(x2 + y2). If you are looking for some alternatives, understand that the modulus can also be calculated using the z conjugate.
Properties of Modulus of Complex Number
Below we have discussed some of the essential modulus of a complex number’s properties. Suppose, z and w are two complex numbers, then properties will be
- Modulus of the complex number z and –z will be equal. It means |z| = |-z|
- Modulus of the complex number can be zero only if the complex number value is zero. It means if z= 0 the modulus |z| will also be 0.
- If you want to derive the product of a complex number, The product will be equal to the complex numbers’ product. That is |z.w| = |z|.|w|
- As far as division of two modulus of complex numbers are concerned the final quotient will be z/w| = |z|/|w|. It is somewhat similar to the multiplication property also.
- The modulus and the conjugate of the complex number will be equal. It means |z|= |z|
- The modulus of the nth power of the complex number will be equal to the nth power of the modulus of a complex number. This is what we are trying to say |zn| = |z|n
Final Words
The modulus of a complex number is very much important to understand. There are several crucial questions based on it. However, calculating the square root of the sum of the squares of the real part and the imaginary part requires proper attention.