Why Are Complex Numbers Needed and What Do They Make Possible Beyond Real Numbers?

i am so much confused with complex exponential.so i need to break that down into basic and rethink about it

1) why do we need complex number why not we stay with real number alone and what are the impossible become after possible after discovery of complex number.Thank you in advance

Hello Bhuvanesh, to answer to that question lets discuss about the "real" world we live in and the mathematics of it.

Lets say if you had 5 apples and you give away 3 to your friend, the equation that will tell you about the remaining apples that you have would be, *a + 3= 5*, where a is the number of apples you have, simple right?

take another example, you order a pizza amongst three of your friends, now if you have to divide the pizza equally among each one of you, you will make an equation of this kind-> *p/3 = I*, which is, an individual gets p/3 slice each, so we have moved from the world of integers to now rational numbers, though we deal with integers in real life.

Complex numbers on the other hand is different, it has helped engineers and scientist to solve equations and realize their meaning, consider an equation z^2 + 1 = 0 which couldn't be solved before, but thanks to Euler, we could extend our imagination to an imaginary axis to solve this kind of equations.

Now being specific, consider an electric circuit which is supplied with a sinusoidal voltage (The one which you get in your wall) V(t), The impedance (not resistance) supplied by the circuit can be written with the help of the equation V(t) = I(t)*Z

Z being the impedance supplied by the circuit, now what is Z??

Z = R + iwL + 1/(iwC)

since we know that impedance provided by the circuit is dependent on the elements inside it with there value as well as the frequency.
To understand the significance of i, what it really means is the "phase".
if you have an inductor in the circuit the current lags the voltage meaning the current is out of phase with the voltage and for a purely capacitive circuit, you have current leading the voltage by some phase.
what is phase in real sense?
Phase is time difference in simple sense, for e.g. if you have voltage leading in a circuit that means current lags behind in "time".

This is how complex numbers has helped scientists and engineers to expand their horizons and give meaning to this kind of equations. complex number is also used widely in quantum mechanics as well.

I hope, I have answered you question.

Cheers!