What are Modulus and Argument?
A complex number can be represented as a point on the complex plane (similar to the Cartesian plane). Besides being a point, we can also view it as a starting from the origin to the point .
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A complex number can be represented as a point on the complex plane (similar to the Cartesian plane). Besides being a point, we can also view it as a starting from the origin to the point .
This vector has a length and a direction. This length and direction are what we call the Modulus and Argument.
The Modulus of a complex number , written as , is the distance from the origin to the point on the complex plane. This is the same as the length of the vector representing .
To calculate the modulus, we can use the Pythagorean Theorem on the right-angled triangle formed by the real part (), the imaginary part (), and the modulus () as the hypotenuse.
Definition of Modulus:
The modulus of the complex number is:
The modulus is always non-negative (never negative) because it represents a distance.
Find the modulus of , with
Find the modulus of , with
Find the modulus of , with
Find the modulus of , with
The Argument of a non-zero complex number , written as or , is the angle formed by the vector with the positive real axis on the complex plane. This angle is usually measured in radians or degrees.
From basic trigonometry on the same right-angled triangle as in the modulus visualization, we know the relationships:
To find , we can use the arctangent function (or ):
Calculators usually give the value in the range or . We need to consider the quadrant where the point lies to determine the correct argument.
Quadrant ():
Quadrant ():
Quadrant ():
Quadrant ():
Often, we are interested in the Principal Argument (written ), which is the argument value in the interval or .
Find the argument of
The point is in Quadrant .
Find the argument of
The point is in Quadrant .
Find the argument of
Find the argument of
The point is in Quadrant .
Find the modulus and argument (in degrees) of the following complex numbers:
For :
(Quadrant ) Modulus:
Argument:
For :
. (Negative real axis) Modulus:
For :
. (Negative imaginary axis) Modulus:
(The modulus of a real number is its absolute value).
or simply
if a negative angle is desired
(Because it's in Quadrant , we use )
The point is in Quadrant .
(Because it's in Quadrant , we use . Principal Argument: or ).
(Because it's in Quadrant , we use . Principal Argument: or ).
Argument: The point is on the negative real axis.
Argument: The point is on the negative imaginary axis.
or (Principal Argument).