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Do complex numbers really exist? - Mathematics Stack Exchange
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obviou...
Are the reals genuinely a subset of the complex numbers?
With this in mind, I ask the following questions: Can the real numbers be said to be a subset of the complex numbers if complex numbers are defined as ordered pairs? If the complex numbers are constructed in some other way, then is it meaningful to write $\mathbb R \subset \mathbb C$?
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).
What is the dot product of complex vectors?
This complex "dot product" is sometimes called a Hermitian form. This specific separate term serves as a way to make it clear that it might not comply with the usual definition of a dot product, if you don't generalize that definition as shown above.
complex numbers - Why is $ |z|^2 = z z^* $? - Mathematics Stack Exchange
The identity was presented to me baldly along with a sloppy explanation to complex numbers back when I took my intro class. Unfortunately, not all educators are able to present mathematics in an accessible, intuitive way.
"Where" exactly are complex numbers used "in the real world"?
50 Complex numbers are used in electrical engineering all the time, because Fourier transforms are used in understanding oscillations that occur both in alternating current and in signals modulated by electromagnetic waves.
Why do cubic equations always have at least one real root, and why was ...
I don't see why it was necessary to introduce complex numbers due to cubic equations if they were shown to have at least one real solution. Then the real solution could be enough, right?
radicals - How do I get the square root of a complex number ...
The square root is not a well defined function on complex numbers. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number.
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 are There No "Triernions" (3-dimensional analogue of complex ...
Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses...
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