Q. What are Complex numbers?
Ans.
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is a special number called the imaginary unit, which has the property that i = -1. Notice that i is not a real number.
In the complex number a + bi, the first term a is called the real part, and the second term bi is called the imaginary part.
If a complex number has no imaginary part (or, technically speaking, b = 0), then it's real. Otherwise, it's imaginary. A pure imaginary number is a complex number with no real part.
Notice that every real number can be written in the form of a complex number, since (for example) 10 is equal to 10 + 0i.
The big question students always ask is this: if imaginary numbers are "not real", then why are we bothering to talk about them? The truth is this: at one point in history, negative numbers were not considered to "really exist". At one point in history, nobody would believe that not all numbers were rational--and at one point in history, i was considered to be a figment of a warped imagination, which is why it got the name "imaginary".
But nowadays, mathematicians realize that it's OK to make up new kinds of numbers, as long as their mathematical properties are clearly defined. Imaginary numbers are considered to "exist" just as much as real numbers "exist". And they're even useful, tool