# What are complex numbers, actually? [duplicate]

What are complex numbers, actually? You can prove $1=-1$ and a complex cosine function can have value greater than $1$ and so on, there are many unexpected results when we use complex numbers. So, what are they actually? Do, they have any physical meaning or are they just a method in mathematics to manipulate numbers?

## marked as duplicate by Dan Rust, Belgi, Anthony Carapetis, Git Gud, mrfSep 22 '13 at 10:12

• How can you prove $1=-1$? – Anthony Carapetis Sep 22 '13 at 9:56
• The point of that question is that it is wrong. Even using complex numbers, there is no way to prove $1=-1$, nor are there other contradictory results. – robjohn Sep 22 '13 at 10:05
A complex number is a number that has both Real and Imaginary Components, It is the the form $$a+bi$$ where $i$ is defined as $$i=\sqrt{-1}$$