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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?

robjohn
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Rajath Radhakrishnan
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    How can you prove $1=-1$? – Anthony Carapetis Sep 22 '13 at 09:56
  • Here the link-https://brilliant.org/assessment/techniques-trainer/proof-that-1-1-and-1-3/ .Actually the proof has a small mistake(you can easily identify it) but however complex numbers give weird results in some cases. – Rajath Radhakrishnan Sep 22 '13 at 09:59
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    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

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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 $$i=\sqrt{-1}$$

user93089
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  • If you have never heard of "complex numbers", the term "real/imaginary component" is useless. Also, $i:=\sqrt{-1}$ is useless if you have not defined the square root function for negative numbers, except if this is just a notational trick, but this is not what you wrote. – M. Winter Jul 29 '20 at 20:54