41 questions linked to/from Do complex numbers really exist?
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### difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
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### Why is $i^2$ equal to $-1$? [duplicate]

In this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. Why is this so? What is the intuition behind it?
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### 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, ...
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### Is there a proof for existence of complex and hypercomplex numbers [duplicate]

As mathematics advanced ,mathematicians found out new type of numbers such as complex numbers and hypercomplex numbers . I had been really fascinated by this idea and the uses of these numbers. ...
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### What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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### How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
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### Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
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### How did early mathematicians make it without Set theory?

It is said that Cauchy was a pioneer of rigour in calculus and a founder of complex analysis. Yet if baffles me as set theory was an invention of the 1870s, 20 years after the death of Cauchy. ...
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### Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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### What's the precise meaning of imaginary number?

The same to the title,what's the precise meaning of imaginary number? And on the other hand,how can the imaginary number be reflected in Physics?
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### Refining my knowledge of the imaginary number

So I am about halfway through complex analysis (using Churchill amd Brown's book) right now. I began thinking some more about the nature and behavior of $i$ and ran into some confusion. I have seen ...
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### An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
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### Does a negative number really exist?

Second Update: I see that some answers that reference my image are more closely answering my question. Here is a second image to clarify my point. Take this image representing a checkerboard like ...
I have been in a debate over 9gag with this new comic: "The Origins" And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$". And then, this comment cast a doubt: There is no such ...
### What's bad about calling $i$ “the square root of -1”?
I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a ...