|
complex numbers - Parametrizing shapes, curves, lines in $\mathbb {C ...
I've been struggling with parametrizing things in the complex plane. For example, the circle |z − 1| = 1 | z 1 | = 1 can be parametrized as z = 1 +eiθ z = 1 + e i θ.
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.
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...
"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.
radicals - How do I get the square root of a complex number ...
To find a square root of a given complex number z z, you first want to find a complex number w w which has half the argument of z z (since squaring doubles the argument).
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).
complex numbers - Why is $ |z|^2 = z z^* $? - Mathematics Stack Exchange
I've been working with this identity but I never gave it much thought. Why is $ |z|^2 = z z^* $ ? Is this a definition or is there a formal proof?
Why is the complex number - Mathematics Stack Exchange
That is, the collection of complex numbers is a two-dimensional real vector space, and multiplication by a + bi a + b i is a real-linear map of C C to itself, so, with respect to any R R -basis of C C, there'll be a corresponding matrix.
Is the existence of quaternions as an extension of complex numbers an ...
Then 1 + i 1 + i is both an equation that shows how to "get" from real and imaginary numbers to complex numbers, and a representation of the final complex number itself.
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 $\mathbb R \subset \mathbb C$?
|
