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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?

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

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    $\begingroup$ How can you prove $1=-1$? $\endgroup$ – Anthony Carapetis Sep 22 '13 at 9:56
  • $\begingroup$ Here the link-brilliant.org/assessment/techniques-trainer/… .Actually the proof has a small mistake(you can easily identify it) but however complex numbers give weird results in some cases. $\endgroup$ – Rajath Krishna R Sep 22 '13 at 9:59
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    $\begingroup$ 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. $\endgroup$ – robjohn Sep 22 '13 at 10:05
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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 $$i=\sqrt{-1}$$

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