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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...
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).
"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.
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.
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.
Are there imaginary numbers other than i? • Physics Forums
Participants examine theoretical constructs, alternative number systems, and the philosophical implications of defining numbers. One participant questions whether there are "imaginary" numbers other than i, suggesting that if utility is the standard for creating numbers, other complex families could exist.
definition - Why can't imaginary numbers be irrational - Mathematics ...
Online Definitions: In mathematics, the irrational numbers are all the real numbers which are not rational numbers In mathematics, a transcendental number is a real or complex number that is not algebraic Is there any specific reason why an imaginary number can't be irrational (other than the definition of irrational).
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?
Is there a third dimension of numbers? - Mathematics Stack Exchange
A good hint that there is no three dimensional numbers is the fact that $ (a_1^3 + b_1^3 + c_1^3) (a_1^3 + b_2^3 + c_2^3)$ doesn't always equal the sum of three cube numbers (the axiom that product of two three cubes is another three cube). The quaternions barely satisfy the condition for four dimensional except for commutativity.
imaginary numbers - how can I understand them - especially as they ...
13 In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I understand that imaginary numbers are "manipulable and usable to do all kinds of things". And thus, "they exist" at least, in the realm of mathematics.
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