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What are imaginary numbers? - Mathematics Stack Exchange
At school, I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$. When I...
"Where" exactly are complex numbers used "in the real world"?
There are plenty of applications of complex numbers, but from what I have seen they are typically used to simplify solving a math equation, and the end result is still a real number. Or in some cases (lile quantum) a 2d vectors are represented with complex numbers, but could be represented with 2d vectors.
Can a complex number ever be considered 'bigger' or 'smaller' than a ...
It would seem that the 'sizes' of numbers of any type (real, rational, integer, natural, irrational) can be compared, but once imaginary and complex numbers come into the picture, it becomes a bit counter-intuitive for me. So, does it ever make sense to talk about a real number being 'more than' or 'less than' a complex/imaginary one?
Are there imaginary numbers other than i? • Physics Forums
The discussion revolves around the existence of "imaginary" numbers beyond the standard unit i in complex analysis. Participants explore the concept of extending the number system, mentioning quaternions and octonions as higher-dimensional analogs that introduce additional square roots of -1, though they lose properties like commutativity and associativity. The hyperreals and dual numbers are ...
notation - What is the symbol for imaginary numbers? - Mathematics ...
$\\mathbb Q$ is used to represent rational numbers. $\\mathbb R$ is used to represent real numbers. Is there an accepted symbol for imaginary numbers?
How to find complex eigenvectors from complex eigenvalues?
The same way you usually do. Complex numbers aren't that different from real numbers, after all.
complex numbers - What is $\sqrt {i}$? - Mathematics Stack Exchange
The square root of i is (1 + i)/sqrt (2). [Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers.
Why do complex numbers lend themselves to rotation?
First of all, complex numbers are two-dimensional, having independent x (real) and y (imaginary) components. This makes it possible to define a “rotation”, which you can't really do with one-dimensional real numbers (unless you count flipping the sign).
Understanding imaginary exponents - Mathematics Stack Exchange
Imaginary exponents are just the same. i, 2i, 3i are just like 1, 2 ,3: identity, square, and cube. They just need to run into another imaginary exponent to manifest their value, or you are carrying a lot of extra stuff you don't see. The reason we like e is because it's derivative is the same as it's function's output.
Quaternions: why does ijk = -1 and ij=k and -ji=k
I think the geometric algebra interpretation of complex numbers and quaternions is the best, since it reveals more directly the fact that the "imaginary numbers" can be seen as encodings of rotations/reflections. Here is a pretty straightforward explanation.
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