# 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 x_{1} 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.

**back to the amount of almond bread**

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