Magnitude of Complex Number

Introduction to magnitude of complex number:

The complex numbers are formed from the sqrt of (–1) Which is denoted by  i. As only positive numbers have sqrt the sqrt(-1) is an imaginary number.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 of the number ,and and b are real numbers.These parts are denoted as reZ and Im 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 we are considering I as another variable.

Modulus 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|= sqrt(x^2  +y^2)

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

Properties of Magnitude of Complex Number:

|z| = sqrt (x^2+y^2) (by denition)

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

|z1z2| =|z1| |z2| (which means it is multiplicative)

|z1/z2| =|z1| / |z2|

z := z z Where z is the complex conjugate of the complex number z.

Magnitude

Here the magnitude of the complex number is marked as the line from origin.(r)

as magnitude is the shortest distance from the origin to the point,in the complex plane.

Practice Problems on Magnitude of Complex Number:

Problems:

Q 1:  Find the magnitude of complex number 3 + 2i.

Sol: The magnitude of complex number 3 + 2i Is |3+2i|

= sqrt(3^2+ 2^2)

= sqrt(9+4)

= sqrt(13 ) (Answer)

Q 2: Find the magnitude of complex number 2 + 2i.

Here the magnitude of complex number is  sqrt(2^2+2^2 )

sqrt(4+4 )

=sqrt(8) (answer)

Practice problems:

Q 1: Find the magnitude of complex number -3 + 4i.

Q 2: Find the magnitude of complex number 3 -i.

Q 3: Find the magnitude of complex number -3 – 2i.

Q 4: Simplify |3+4i| .|2+3i|

Q 5: Simplify |2+i| / |3+4i|

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