Complex numbers: how complex are they?

The "history" of the integer numbers is a simple one. From the natural numbers 1, 2, 3, ... we move to the positive integers 0, 1, 2, 3, ... then we add the negative integers ... -3, -2, -1, 0, 1, 2, 3, ....

Then we escalate to fractions and decimals and non-terminating decimals (although historically was not in this order) . The ladder continues to the irrational numbers and to the algebraic and transcendental numbers. This is the "world" or universe of the real numbers.

But mathematics is a product of our minds, so this "universe" or field can be further expanded to suit our needs.

The next heaven after the real numbers field are the imaginary numbers; numbers that in combination with the reals make the complex numbers field.

But how complex are the complex numbers? Curiously, they are simple as the "preceding" ones.

The negative numbers haunted the mathematicians and philosophers for many centuries; no wonder the misnomer "negative". Even the number zero took a long time before it was accepted in the kingdom of the mathematics (in Europe, where it was lately accepted.) It was unacceptable to count "backwards".

The imaginary numbers suffered the same fate: no wonder the epithet of "imaginary". The square root of minus 1 was impossible to compute because no number times itself is equal to minus 1.

Take a read at this article: The imaginary numbers are not so imaginary and the complex numbers are not so complex and you will see how easily and beautifully the complex numbers emerge out of the real numbers.