Complex Numbers: The Phase of a Complex Number

In summary, the phase of a complex number of the form a = C * \exp(i \phi) is not well-defined if C is greater than 0. To get around this, it can be defined as a coset or constrained to a specific interval. If C is less than 0, the sign of C must be absorbed into the phase. If C is equal to 0, the phase is undefined.
  • #1
pamparana
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I just wanted to check something. If I have a complex number of the form

[itex]a = C * \exp(i \phi) [/itex]

where C is some non-complex scalar constant. Then the phase of this complex number is simply [itex]\phi[/itex]. Is that correct?
 
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  • #2
pamparana said:
I just wanted to check something. If I have a complex number of the form

[itex]a = C * \exp(i \phi) [/itex]

where C is some non-complex scalar constant. Then the phase of this complex number is simply [itex]\phi[/itex]. Is that correct?
If ##C > 0## then this is almost correct. However, the phase is not well-defined under this definition, because ##C\exp(i\phi) = C\exp(i(\phi+2\pi n))## for any integer ##n##. You can get around this by defining the phase to be the coset ##\phi + 2\pi \mathbb{Z}## or by constraining it to be in the interval ##[0,2\pi)## or ##[-\pi, \pi)## or some other half-open interval of length ##2\pi##.

If ##C < 0##, then you need to absorb the sign of ##C## into the phase:
$$a = -|C|\exp(i \phi) = |C|\exp(i(\phi + \pi))$$

If ##C = 0## then the phase is undefined.
 
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  • #3
Thank you for this detailed answer!
 

1. What is the phase of a complex number?

The phase of a complex number refers to the angle that the complex number makes with the positive real axis when plotted on a complex plane. It is represented in radians or degrees.

2. How is the phase of a complex number calculated?

The phase of a complex number can be calculated using the arctangent function, where the ratio of the imaginary part to the real part is used as the input. The resulting value is the phase in radians, which can be converted to degrees if desired.

3. What is the range of possible phase values for a complex number?

The range of possible phase values for a complex number is between -π and π radians, or -180 and 180 degrees. This range covers all possible angles on the complex plane.

4. How does the phase of a complex number affect its properties?

The phase of a complex number affects its properties by determining its position on the complex plane and its relationship with other complex numbers. It also plays a role in calculations involving complex numbers, such as multiplication and division.

5. What is the significance of the phase of a complex number in real-world applications?

The phase of a complex number has various applications in fields such as engineering, physics, and signal processing. It is used to represent waves, electrical currents, and other quantities that have both magnitude and direction. The phase also helps in analyzing and manipulating these quantities in practical situations.

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