what are the applications of complex numbers?

Question

I'm sorry to ask a question like this. But i'm really wondering what are the uses of complex numbers. I've figured the use of negative numbers - they are used in modern accountancy. I've even figured the use of logarithm - it's used in graphs so that an exponential increase can be plotted neatly. But i just dont understand the uses of complex numbers. I mean, why would anyone wana care about the square root of -1? Tell me if u're with me. Or else tell me how complex numbers are used for.

Answer 1

Complex numbers are used to make things simpler which may otherwise become very very complex without complex numbers.

applications of complex numbers:

Control Theory: In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The systems poles and zeros are then analysed in the complex plane.

Signal analysis: Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value i.e mod z is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.

Improper integrals: In applied fields, complex analysis is often used to compute certain real-valued improper integrals by means of complex valued functions.

Quantum mechanics: The complex number field is also of utmost importance in quantum mechanics since the underlying theory is based on (infinite dimensional) Hilbert spaces.

Relativity: In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.

Applied mathematics: In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system.

Fluid dynamics: In fluid dynamics complex functions are used to describe potential flow in 2d.

Fractals: Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

Answer 2

I use them all the time in structural engineering. How a building shakes during an earthquake is related to complex numbers. It is a bit difficult to explain, but here is an easy way see the application. Complex numbers and be transformed into a sin and cos function. Buildings sway back and forth similar to a sin or cos function.

Answer 3

Complex numbers have broad applications in physics and higher math. Complex numbers are used widely for analysis of electrical systems and particle physics ( quantum physics) uses it very heavily. There are just some systems that are nicely represented and analyzed using complex number systems.

Answer 4

I'm sure there are lots of uses, but I used them all the time in circuit analysis in electrical engineering. It turns out that you can solve for voltage and current in AC circuits containing resistors, capacitors, and inductors by assuming that capacitors have a negative complex impedance and that inductors have a positive complex impedance.

There are also many applications in signal processing.

Answer 5

A lot of physical problems are described by complex functions, e.g. the schroedinger equation, which describes quantum mechanics. So you need complex numbers to evaluate these functions.

Answer 6

Electronics is one. in case you concentrate on capacitive reactance as –i ohms, and inductive as +i ohms, the position i = ?(–a million) as an get mutually, a 100 ohms resistor in sequence with a 2 hundred ohms equivalent inductor and 50 ohms equivalent capacitor, then the impedance is two hundred + i200 – i50 = 2 hundred + i150