Calculate the modulus of complex numbers

Calculate the modulus of a complex number (z = a + bi) is a simple operation but we always tend to forget how to do it, but thanks to our calculator you will no longer have to remember the formula.

You can also use the tool to to extract the absolute value of a complex to check that you have solved the exercises correctly.

How to calculate the modulus of a complex number in Cartesian form

Modulus of a complex number

A complex number in Cartesian or binomial form is of the type z = a + bi and we calculate its modulus by the following formula:

|z| = √(a2 + b2)

Let's see cow to calculate the modulus of a complex number in Cartesian form with a solved exercise in which we get z = 2 + 3i

|z| = √(22 + 32) = √(4 + 9) = √13 = 3,6055

How to calculate the modulus of a complex number in polar form

If we have the complex number in polar form The modulus is already implicit in its expression and we do not have to do any calculation.

A complex number in polar form is of the form z = rαwhere r is the value of the modulus and α the argument.

For example, if we are given the complex number 3120ºthe module is 3.

Calculate the modulus of a complex in Excel

Calculating the modulus of a complex number in Excel

If you have Excel installed on your computer, you can use it to calculate the modulus of any complex number with a simple formula.

If you want to test how it is done, open a new spreadsheet and type the following function in an empty cell:

=IM.ABS("5+3i")

This will return the absolute value or modulus of the complex 5 + 3i.

What you have to do is to modify the formula to get the absolute value of the complex number you want.

Properties of the modulus of a complex number

The following are some of the properties of the modulus of a complex number:

  1. The multiplication of a complex number by its conjugate gives us the modulus of the complex number squared.
  2. The modulus of a complex number is always greater than or equal to zero, therefore, can never be negative. It will be greater than zero if the number is different from zero and will be zero only when the number is zero. This is expressed mathematically as follows:

|z| = 0 if, and only if, z = 0.

  1. The modulus of a complex number is equal to the modulus of the conjugate of the complex number. That is to say:

|z| = |z'|

  1. The modulus of the product of two complex numbers is equal to the product of the moduli of the complex numbers individually, something that is mathematically expressed as follows:

|z1z2| = |z1| |z2|

  1. The above property also applies to division, i.e., the modulus of the quotient of 2 complex numbers is equal to the quotient of the moduli of the complex numbers.

|z1 / z2| = |z1| / |z2|

  1. The real part and the imaginary part of a complex number is less than or equal to the modulus of the complex number.

|Re(z)| ≤ |z|

|Im(z)| ≤ |z|

  1. The triangular inequality tells us that the modulus of the sum of two complex numbers is less than or equal to the sum of the moduli of the complex numbers. This is also true for subtraction.

|z1 + z2| ≤ |z1| - |z2|

|z1 - z2| ≤ |z1| - |z2|

Complex number modulus calculator

Complex numbers module calculator

To make it easier for you to calculation of the modulus of a complex number in binomial form (remember that in polar there is no need to do any operation), we have created a calculator for you to obtain the result automatically.

Its operation is very simple and all you have to do is write the values a and b of the complex number from which you want to obtain its number.

When you have typed it, press the calculate button and you will get the answer. No doubt it will be very useful to save time in operations or to check if the solved exercises you have done are well calculated.

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