What are the different properties of Complex Numbers. ?
I have doubt on real life use of complex numbers. Where and in what conditions do we use complex numbers in our day to day life.
My main focus is to know apart from Electrical ENgineering where it is used. Daily Life use. which can be understood by layman
Why are real numbers useful? Most people can think of many reasons they are useful, they allow people to encode information into symbols that most anyone can understand. Its the same case with complex numbers. Most examples give highly specific and niche uses for complex numbers, but in reality, they could be used anywhere. The simplest way to understand complex numbers is to realize that $i \cdot i=-1$, $-1 \cdot i=-i$, and $-i \cdot i=1$. You'll notice that multiplying something by $i$ repeatedly results in eventually getting back the number you started with. In addition, note that complex numbers are made from both real and imaginary components. Replace real with x and imaginary with y, and it becomes apparent that complex numbers can be plotted on x-y graphs. This also means that repeatedly multiplying by $i$ corresponds to rotation. So complex numbers allow us to encode more "complicated" information. I'll leave you with a question. A question for you, what are some uses of x-y graphs and rotation?
Compilation Of all -
Complex numbers are mainly used in electrical engineering techniques all the time, because Fourier transforms are used in understanding oscillations and wave behaviour that occur both in AC Current and in modulated signals.
Two-dimensional problems involving Laplace's equation (e.g. heat flow, fluid flow, electrostatics) are often solved using complex analysis, in particular conformal mapping
Complex numbers are used in real world literally EVERYWHERE.
Complex analysis (transformation or mapping) is also used when we launch a satellite and here on earth we have z-plane but in space we have w-plane as well. So to study various factors we use transformation.
