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Contents

The Complex Numbers \(\mathbb{C}\)

The real numbers ({{!util.a("r.ci")}}) are more than just a count.

In \(\mathbb{R}\) \(x^2\) only assumes positive values. The equation \(x^2+1=0\) does not have a solution.

So we invent a “number” \(i\) that satisfies \(i^2=-1\).

\(i\) is called imaginary unit and its multiples are called imaginary numbers. \(i\) is like apple or orange. It has nothing to do with the unit \(1\). The imaginary numbers are orthogonal to \(\mathbb{R}\), which means you can choose from these two sets independently. All combinations form a 2-dimensional space, i.e. a plane, the complex plane.

\(z = a + ib \in \mathbb{C}\)

is also a two-dimensional vector: 2 orthogonal directions that can be added independently.

There are two representations

Now consider the following:

Generally: Multiplication with \(i\) produces a rotation by the right angle.

Since two multiplications (\(x^2\)) are supposed to invert (rotate by \(\pi\)) one multiplication should rotate by half of it (\(\pi/2\)).

When multiplying exponentials, the exponent gets added. This gives a hint that there could be a representation that has the angle in the exponent.

In trigonometric addition formulas (e.g. \(\cos(\alpha+\beta)=\cos\alpha\cos\beta-sin\alpha\sin\beta\)), multiplication adds the angles.

Finally developing \(\sin\) and \(\cos\) into a Taylor series and comparing with the \(e^x\) series leads to the Euler Formula:

\(z=re^{i\varphi}\) is a usual way to represent complex numbers.

About \(\sin\) and \(\cos\) we know that the period is \(2\pi\), therefore this is true for \(e^{i\varphi}\). The nth root divides the period up to \(2n\pi\) to below \(2\pi\) and so we have \(n\) different roots.

\[z^{1/n}=r^{1/n}e^{i(\varphi/n+2k\pi/n)}\]

More generally:

In \(\mathbb{C}\) every polynomial of degree n has exactly n roots (fundamental theorem of algebra), if one counts the multiplicity of roots. \(\mathbb{C}\) therefore is called algebraically closed.

This means that not only \(x^2\), but every polynomial maps the whole \(\mathbb{C}\) to the whole of \(\mathbb{C}\).

Note

In function theory one learns that this can be extended to all functions that are infinitely often differentiable (analytic or holomorphic) in all of \(\mathbb{C}\) (entire functions), because they can be developed into a Taylor series.

Further properties:

Applications for \(\mathbb{C}\)

Since \(\mathbb{C}\) is a extension of \(\mathbb{R}\), one can do everything with \(\mathbb{C}\) that you can do with \(\mathbb{R}\). The essentially new is that \(\mathbb{C}\) includes all directions not just \(+\) and \(-\).

What is a direction?

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The complex numbers are used in physics and technology in connection with vibrations and waves and there are many of them:

Basically applications of complex numbers are due to

Many physical systems are described with differential equations. These can be reduced to polynomial and then one gets complex numbers as roots.