SameDistance

Asked Perplexity AI how to find the distance between two points on the complex plane:

The distance between two complex numbers z1 = a + bi and z2 = c + di on the complex plane is given by the formula:

d = √[(c - a)^2 + (d - b)^2]

Where:

- z1 = a + bi is the first complex number 

- z2 = c + di is the second complex number

- a and c are the real parts

- b and d are the imaginary parts

To find the distance d between the two points z1 and z2, you take the difference of the real parts (c - a) and square it, add to the square of the difference of the imaginary parts (d - b), and then take the square root of the sum.[1][2][3]

For example, to find the distance between 3 + 2i and -1 + 4i:

a = 3, b = 2, c = -1, d = 4

d = √[(-1 - 3)^2 + (4 - 2)^2]

d = √[(-4)^2 + 2^2]

d = √(16 + 4)

d = √20

d = 2√5[2]

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No, the formula for calculating the distance between two points on the complex plane is the same as the formula for calculating the distance between two points on the standard Cartesian plane.

The distance formula on the complex plane is:

d = √[(c - a)^2 + (d - b)^2]

Where:

- z1 = a + bi is the first complex number

- z2 = c + di is the second complex number 

This is essentially the same as the distance formula on the Cartesian plane:

d = √[(x2 - x1)^2 + (y2 - y1)^2]

The only difference is that on the complex plane, the real parts (a and c) correspond to the x-coordinates, while the imaginary parts (b and d) correspond to the y-coordinates.[1][3][4]

So the process of finding the distance between two points is identical, regardless of whether you are working on the complex plane or the standard Cartesian plane. You simply take the differences of the real and imaginary (or x and y) parts, square them, add them together, and take the square root.[1][3]

For example, to find the distance between 3+2i and -1+4i on the complex plane, the calculation is:

d = √[(-1 - 3)^2 + (4 - 2)^2] = √(16 + 4) = 2√5

Which is the same result you would get using the Cartesian distance formula with the points (3,2) and (-1,4).

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From Perplexity:

https://www.youtube.com/watch?v=spUNpyF58BY&t=2s why does the center mass not always coincide with the origin when we map the cycles on the complex plane

YouTube

According to the video, the center of mass does not always coincide with the origin when we map the cycles on the complex plane because the center of mass is influenced by the frequency of the signal.

The video explains how complex numbers are used to represent the center of mass of a graph that has been wrapped around a circle. The winding frequency determines how the graph is wrapped around the circle, and this in turn affects the location of the center of mass. When the winding frequency matches the frequency of the signal, the center of mass will be located far from the origin. This is because all of the high points of the graph will be on one side of the circle, and all of the low points will be on the other side. On the other hand, when the winding frequency is very different from the frequency of the signal, the center of mass will be located closer to the origin. This is because the high points and low points of the graph will be more evenly distributed around the circle.

The video introduces the Fourier transform as a mathematical tool that can be used to decompose a signal into its constituent frequencies. The Fourier transform takes a signal as an input and outputs a function that describes the strength of each frequency in the signal. The center of mass calculation described in the video is an important step in understanding how the Fourier transform works.

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Fourier T in simple terms:

https://www.jezzamon.com/fourier/