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Absolute Value of a Complex Number

The complex numbers are the numbers which is in the form a+ bi. where a and b are real numbers. Here i is the square root of -1. So the complex number has real and imaginary parts. The standard form of a complex number is a+ib where 'a' is the real part and 'b' is the imaginary part. These parts are denoted as real 'z' and Imaginary 'z' respectively.

Two complex numbers are said to be equal if its real part and imaginary parts are equal.

The addition and subtraction of complex numbers are done as same as considering i as a variable.

Absolute value of a complex number: (mod z)

The modulus of a complex number is the distance of it from the origin.

It is denoted by |z| and |z|2 = x2 + y2

∴ |z|= $\sqrt{x^2 +y^2}$

The modulus is also called as the absolute value of a complex number.

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Absolute Value of a Complex Number Properties

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Graphical representation of absolute value :

Absolute Value of a Complex Number

In the graph above the imaginary numbers are marked in the y axis and real numbers are marked in the horizontal axis and a and b are real numbers.The modulus of the number is marked as r, which is the shortest distance of the point (a, b) from the origin.

|z| = $\sqrt{x^2 +y^2}$ (by definition)

|z 1+z2) <= |z1| + |z2| (triangle inequality)

|z 1 z 2| =|z 1| |z2| (which means it is multiplicative)

|z1|z|=|z1| |z2|
|z | = z z Where z is the complex conjugate of the complex number z.

Find the Absolute Value of the Complex Number

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Below are the examples on find the absolute value of the complex number -

Solved Examples

Question 1: Find the absolute value of the complex number 3 + 2i.
Solution:
 
The the absolute value of the complex number 3 + 2i Is |3+2i|

= $\sqrt{3^{2}+ 2^{2}}$

=$\sqrt{ 9+4}$

= $\sqrt{13}$ (Answer)

 

Question 2: Find the absolute value of the complex number 2 + 2i.
Solution:
 
Here the modulus is 22+22

=4+4

=$\sqrt{8}$ (answer)
 

Absolute Value of Complex Number Practice Problems

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Practice Problems

Question 1: Find the absolute value of the complex number -3 + 4i.
Question 2: Find the absolute value of the complex number 3 -i.
Question 3: Find the absolute value of the complex number -3 - 2i.
Question 4: Simplify |3+4i| . |2+3i|
Question 5: Simplify |2+i| / |3+4i|
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