7. Complex Numbers

In the previous paragraph we saw that we can solve sums like x2 = -4, if we allow that numbers do not have to lie exclusively on the number line, but that they can also lie in a number plane. In this paragraph we will look at this in more detail. First a few conventions: Numbers in this plane are called complex numbers Since we are now dealing with a plane, it is useful to introduce two "coordinate" axes. The number line we have always worked with up until now is called the real axis and the axis perpendicular to it through the number 0 is called the imaginary axis. The unit along the imaginary axis is called i The advantage of these axes is that we can then record complex numbers in two ways. By the length of the arrow and the angle that the arrow makes with the positive real axis. By "decomposing" the arrow and giving the length of the real part and the imaginary part. We write the real part of c as Re(c) and the imaginary part as Im(c). Suppose we find for a complex number Re(c) = 3 and Im(c) = 2. The number is then often written as follows: 3 + 2i The solutions to the above sum with this notation are: ±2i , because these numbers were on the imaginary axis and therefore have no real part. Another example: x2 - 6x + 13 = 0 Applying the abc formula yields, after simplification, that the solutions to this quadratic equation given by 3 ± √-4, so by 3 ± 2i Finally, a remark: Be careful not to confuse the real and imaginary axes with the x-axis and the y-axis in the graphs of functions. The input of one specific x-value yields one specific function value and that is plotted along the y-axis. In the next paragraph, we will study our familiar parabola again, but now with starting values that are complex numbers. The input then consists of two numbers and the function value will in general also be a complex number. Drawing a graph of a function is then no longer possible. High time to go back to the fractals. We are almost there now! Numbers/font> ← Table of Contents → Julia Fractals