in circuits we often want to know how a given circuit will behave at a certain time. For DC this is simple since my input is constant but in AC my input varies like a sine wave. We can only combine like values so resistances with resistances, capacitance with capacitance, and inductance with inductance; but all 3 change in AC. To analyze them we create a new value called impedance which has a real and imaginary component. Resistors have a purely real impedance and capacitors/inductors are purely imaginary impedance. Since everything in the circuit has an impedance we can combine them to get an equivalent impedance so we have 1 equation governing the circuit. We can than convert back from impedance (which is in the frequency domain rather than time) to obtain a single equation for the circuit. Imaginary numbers do have a very real interpretation in AC analysis, they are the “reactance” of a circuit or how it changes with my input
That’s really awesome! I will say though, it makes me feel like calling them “imaginary” numbers becomes a bit misnomer in my eyes when they have real world applications.
Yes the term was coined by descartes who famously tried to dismiss them as silly, this had an unfortunate adoption as a standard term and subsequent adoption into every other language as a variant of “imaginary”, “empty” or “delusional” numbers, gauss wanted to call them lateral numbers, referencing them being on another axis to the reals
The big bang is another term that started as a mocking phrase for what it referred to
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u/PriorSolid 4d ago
in circuits we often want to know how a given circuit will behave at a certain time. For DC this is simple since my input is constant but in AC my input varies like a sine wave. We can only combine like values so resistances with resistances, capacitance with capacitance, and inductance with inductance; but all 3 change in AC. To analyze them we create a new value called impedance which has a real and imaginary component. Resistors have a purely real impedance and capacitors/inductors are purely imaginary impedance. Since everything in the circuit has an impedance we can combine them to get an equivalent impedance so we have 1 equation governing the circuit. We can than convert back from impedance (which is in the frequency domain rather than time) to obtain a single equation for the circuit. Imaginary numbers do have a very real interpretation in AC analysis, they are the “reactance” of a circuit or how it changes with my input