Oct 6, 2017 · 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 …

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.

May 9, 2014 · 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?

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...

Jul 7, 2023 · 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 …

Jan 2, 2021 · The imaginary unit i i is then just a shorthand for the ordered pair (0, 1) (0, 1). Spivak goes on to say When complex numbers were first introduced, it was understood that real numbers were, in …

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).

Feb 14, 2013 · A complex number z is an order-pair of real numbers (x, y) where we call x the real part of z and y the imaginary part of z. For x = 0 and y = 1, we denote z as i.

Nov 8, 2019 · This is a follow-up to a previous question regarding complex numbers. Many people there compared complex numbers to vectors, and there was disagreement about what the difference was. …

Jan 24, 2013 · 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 …