Phase of a Complex Number in MATLAB
In mathematics, the phase of a complex number is the angle between the positive real axis and the vector representing the complex number in the complex plane. In MATLAB, the phase of a complex number can be calculated using the angle
function. This function returns the phase in radians.
For example, consider the complex number z = 3 + 4i
. The phase of this number can be calculated as follows:
>> z = 3 + 4i;
>> phi = angle(z);
>> phi = atan(4/3)
phi = 0.9330
In this example, the phase of the complex number z
is approximately 0.9330
radians.
It is important to note that the angle
function returns the phase in the interval (π, π]
, meaning that the phase may be negative. To ensure that the phase is always positive, we can add 2π
to the result if it is negative.
>> z = 3 + 4i;
>> phi = angle(z);
>> if phi < 0
phi = phi + 2*pi;
end
>> phi = atan(4/3) + 2*pi
phi = 4.7124
In this example, the phase of the complex number z
is approximately 4.7124
radians.
Another way to calculate the phase of a complex number is to use the cart2pol
function. This function converts a complex number from Cartesian coordinates to polar coordinates. The second output argument of this function is the phase.
For example, consider the complex number z = 3 + 4i
. The phase of this number can be calculated as follows:
>> z = 3 + 4i;
>> [r, phi] = cart2pol(real(z), imag(z));
>> phi = atan(4/3)
phi = 0.9330
In this example, the first output argument of the cart2pol
function is the magnitude of the complex number z
, and the second output argument is the phase. The phase of the complex number z
is approximately 0.9330
radians.
It is also possible to calculate the phase of a complex number in degrees instead of radians. To do this, we can simply multiply the result by 180/π
.
>> z = 3 + 4i;
>> phi = angle(z);
>> if phi < 0
phi = phi + 2*pi;
end
>> phi = (atan(4/3) + 2*pi) * 180/pi
phi = 270.0000
In this example, the phase of the complex number z
is approximately 270.0000
degrees.
In conclusion, the phase of a complex number in MATLAB can be calculated using the angle
or cart2pol
function. The phase is the angle between the positive real axis and the vector representing the complex number in the complex plane. The result is given in radians by default, but it can be converted to degrees if desired.
Polar Representation of Complex Numbers
In addition to the Cartesian representation, complex numbers can also be represented in polar coordinates. The polar representation of a complex number is given by its magnitude and phase. The magnitude is the distance from the origin to the complex number in the complex plane, and the phase is the angle between the positive real axis and the vector representing the complex number in the complex plane.
In MATLAB, the polar representation of a complex number can be obtained using the cart2pol
function. This function converts a complex number from Cartesian coordinates to polar coordinates. The first output argument of this function is the magnitude of the complex number, and the second output argument is the phase.
For example, consider the complex number z = 3 + 4i
. The polar representation of this number can be calculated as follows:
>> z = 3 + 4i;
>> [r, phi] = cart2pol(real(z), imag(z));
>> r = 5
r = 5
>> phi = atan(4/3)
phi = 0.9330
In this example, the first output argument r
is the magnitude of the complex number z
, and the second output argument phi
is the phase. The magnitude of the complex number z
is approximately 5
, and the phase is approximately 0.9330
radians.
Cartesian Representation from Polar Representation
In addition to converting from Cartesian to polar coordinates, it is also possible to convert from polar to Cartesian coordinates. This is useful when we have the polar representation of a complex number and need to obtain its Cartesian representation.
In MATLAB, the Cartesian representation of a complex number can be obtained from its polar representation using the pol2cart
function. This function converts a complex number from polar coordinates to Cartesian coordinates. The first output argument of this function is the real part of the complex number, and the second output argument is the imaginary part.
For example, consider the polar representation of a complex number z = 5 * exp(0.9330i)
. The Cartesian representation of this number can be calculated as follows:
>> r = 5;
>> phi = 0.9330;
>> [x, y] = pol2cart(phi, r);
>> x = 3
x = 3
>> y = 4
y = 4
In this example, the first output argument x
is the real part of the complex number z
, and the second output argument y
is the imaginary part. The real part of the complex number z
is approximately 3
, and the imaginary part is approximately 4
.
Complex Conjugates
In mathematics, the complex conjugate of a complex number is obtained by changing the sign of the imaginary part. The complex conjugate of a complex number z = x + yi
is given by z* = x  yi
.
In MATLAB, the complex conjugate of a complex number can be obtained using the conj
function. This function returns the complex conjugate of a complex number.
For example, consider the complex number z = 3 + 4i
. The complex conjugate of this number can be calculated as follows:
>> z = 3 + 4i;
>> z_conj = conj(z);
>> z_conj = 3  4i
z_conj = 3  4i
Popular questions

What is the phase of a complex number in MATLAB?
The phase of a complex number in MATLAB is the angle between the positive real axis and the vector representing the complex number in the complex plane. It can be obtained using theangle
function or theatan2
function in MATLAB. 
How do you convert a complex number from Cartesian to polar coordinates in MATLAB?
To convert a complex number from Cartesian to polar coordinates in MATLAB, use thecart2pol
function. This function takes in the real and imaginary parts of the complex number and returns the magnitude and phase in polar coordinates. 
How do you convert a complex number from polar to Cartesian coordinates in MATLAB?
To convert a complex number from polar to Cartesian coordinates in MATLAB, use thepol2cart
function. This function takes in the magnitude and phase of the complex number in polar coordinates and returns the real and imaginary parts in Cartesian coordinates. 
How do you obtain the complex conjugate of a complex number in MATLAB?
To obtain the complex conjugate of a complex number in MATLAB, use theconj
function. This function takes in a complex number and returns its complex conjugate by changing the sign of the imaginary part. 
What is the difference between the
angle
function and theatan2
function in MATLAB?
Theangle
function in MATLAB returns the phase of a complex number in radians. Theatan2
function in MATLAB also returns the phase of a complex number in radians, but it takes into account the signs of the real and imaginary parts of the complex number to determine the quadrant in which the phase lies. This makes theatan2
function more robust for use in certain applications.
Tag
Complexity