aim & importance of complex numbers?

Answer 1

Complex numbers become important in differential equations and calculus problems where some or all of the solutions may not lie in the real domain.

It is also highly useful in wave equations and dynamical systems as we can use the Euler substitution e^ix=cos(ix) + i.sin(ix) and map solutions on the complex plane.

The important thing to remember is that it is only the real part of the solutions that prove useful to real problems.

In a way it is like walking from A-B, but the most direct route is impossible because there is a great big canyon in your way. So you have to take a strange route around the canyon to end up where you wanted to be. A bit of a convoluted analogy, I know...

Answer 2

The complex number can into the pictute/

x^2 + 1 =0

this does not have a soplution in real field

This has solution in complex field. You can find more details in source below

Answer 3

The complex numbers are the algebraic closure of the real numbers. One importance of this is that we can always find the roots of any polynomial among the complex numbers.

Answer 4

After the Real Numbers, we find the last set that fills in the set of ALL NUMBERS. In mathematical terms, ALL NUMBERS is termed as COMPLEX NUMBERS, which is the union of the sets of Imaginary Numbers and Real Numbers. With complex numbers, you can write "all" numbers--from the simple integers to the square root of negatives, and somethings like 2^i, (5 + i)^(2 - 3i), etc.

^_^

Answer 5

complex numbers address the need of solving equations involving negative square roots accurately and simplify such equations.

Answer 6

It is another notation for "two-variable calculus". Multivariable calculus is too messy for the sophomore, therefore two-variable calculus is preferred to teach. But calling it complex calculus sounds more intellectual.