Is 2 + 3i an imaginary number

Complex numbers


The solution to the equation x² = -1 is not a real number because the square of a real number can never be negative. If one wants to find the equation for x1 solve, one extracts the square root of -1. One defines i = square root of -1 as the "Imaginary Unit". So you can get the square root of negative numbers:

A complex number consists of two parts: the real part (Re) and the imaginary part (Im). There are different ways of writing complex numbers. The most common is this form:

z = Re (z) + i * Im (z)

For the above example 3i would be the exact spelling 0 + 3i. Any real number can be written as a complex number by practically omitting the imaginary part: 8 = 8 + 0i.

Another notation for complex numbers is the "pair notation": The real and imaginary parts are simply written as a pair of numbers:

z = (Re, Im)

For z = 2 + 8i would that be z = (2, 8). For the real number 5 would be the pair notation logically z = (5, 0) and for 3i follows z = (0, 3).

Since a complex number consists of a pair of numbers (Re, Im), it cannot be represented on a number line, nor can complex numbers be compared (<,>, =). Due to the pair of numbers, however, complex numbers can be used in a special coordinate system - the " complex plane "- represent. The real part corresponds to the x-coordinate, the imaginary part to the y-coordinate.

 

The magnitude of a complex number

Complex numbers can be represented in a coordinate system. The magnitude of the complex number z represents the distance to the zero point of the coordinate system:

The magnitude of this complex number (z = Re + i * Im) can be calculated using the following formula:

The imaginary unit i falls out of this. So for the example above, the amount would be

 

Multiplication of complex numbers

The two complex numbers z1 = a + i * b and z2 = c + i * d are given and should be multiplied with one another. This happens as usual, taking into account that i² is the negative number -1:

z1 * z2 = (a + i * b) * (c + i * d) = a * c + i * a * d + i * b * c + i² * b * d = a * c - b * d + i * (a * d + b * c)

In general one can say that the product of two complex numbers is again a complex number of the form (a * c - b * d) + i * (a * d + b * c) is.


That's enough for now. In order to be able to represent amounts of Mandelbrot, it is sufficient to calculate the amount of a complex number and to form the product of two complex numbers.

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