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Complex numbers and phasors
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Represent oscillations as rotating vectors in the complex plane.
e^{iθ} = cosθ + i sinθComplex numbers z = a + ib with i² = −1 can be written as r e^{iθ}. Euler's formula e^{iθ} = cosθ + i sinθ links them to oscillations: a rotating arrow (phasor) projects onto a sinusoid. This simplifies adding oscillations and solving AC circuits.
Adding two AC voltages of the same frequency is just adding their phasors as vectors.
Only the real part (or imaginary part) of the complex expression represents the physical quantity.
In more detail
A complex number has a real part and an imaginary part, and it can be written as a magnitude and a phase. Phasors represent sinusoidal quantities as rotating complex numbers, which simplifies AC circuit analysis.
Real-world uses
- Electrical engineers use phasors to analyse mains power and filter design.
- Quantum mechanics uses complex wavefunctions to describe particles.
- Electrical engineers use phasors to add alternating voltages with different phases, turning a difficult trig problem into vector addition.
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Frequently asked questions
What is the imaginary unit?
i is defined by i squared equals minus one, which lets square roots of negative numbers be written.
Why do physicists use complex numbers if the physical answer is real?
They simplify calculations with sinusoids and waves, and the real part of the final answer is taken at the end.
Why are complex numbers used in circuits if current is real?
They are a convenient way to represent sine waves. You take the real part at the end, so the physical answer is still a real number.
Related topics in Measurement & Math Tools
Gradient, divergence, curl · Differential equations in physics · Taylor series and small angles · Graphs and slopes · Curve fitting and linearisation · Logarithmic scales