What are some of the practical applications of imaginary numbers?

1. Solving AC circuits that have resistors, inductors, and capacitors in them.


2. Signal processing


3. Solving differential equations.





A smart guy by the name of Euler showed that e^(i*x) = cos(x) + i*sin(x), thus allowing the conversion of a complex number in rectangular form (a + b*i) into polar form (r*e^(i*x)). It also allows us to write sinusoids in terms of complex exponentials, as in:


cos(x) = (1/2)(e^(i*x)-e^(-i*x))


It turns out, perhaps unexpectedly, that complex exponentials are oftentimes easier to deal with then sinusoids. It also turns out, according to a guy by the name of Fourier, that most ';well-behaved'; functions can be written in terms of sinusoids (and thus can be written in terms of complex exponentials as well). This paved the way for a lot of neat applications. Google ';complex exponential'; (or any of the terms or names I dropped here) if you want more details.What are some of the practical applications of imaginary numbers?
For non-engineering application Imaginary numbers are used to prove Central Limit Theorem, which is a cornerstone of Statistics, which is in turn used in social and economic analysis or prediction, stock market trading, testing efficiency of new drugs, etc.What are some of the practical applications of imaginary numbers?
like if you have a baseball team with -16 people and for some reason you want to find out how many teams of the same amount of people you can have as the amount of teams. the answer is 4i
finding the probability in quantum mechanics.


http://hyperphysics.phy-astr.gsu.edu/hba鈥?/a>





electrical circuits and electromagnetic phenomens are much better described by complex numbers


http://www.math.toronto.edu/mathnet/ques鈥?/a>
Engineering. One uses them in doing complex exponentials in modeling.
';Practical'; is often in the eye of the beholder. From the perspective of


algebra, complex numbers give us a complete system for finding the roots of


polynomials. Since numerous applications are based on polynomial models in


theory, complex numbers play a part in all of these. For example, in


electrical engineering we find complex roots in circuit theory where the


polynomial is part of the model equation for simple circuits. Alternating


current relates to complex root cases in the polynomial model. In mechani-


cal engineering, the same type of model relates to vibrations with wavelike


results connected to the complex root cases.