Angles and Their Measurement

In school, angles are usually measured in degrees (°). But why does a full rotation actually correspond to 360°? Nobody can say for sure anymore. We only know that already the ancient astronomers used it. It may be because 360 ​​is approximately equal to the number of days in a year and has many divisors. In any case, it is a completely arbitrary definition. Equally arbitrary was the definition of 400 gon after the French Revolution.

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Complex Numbers, Part 5—Arithmetic in the Cartesian representation

In part 1 and part 4 we saw different geometric representations of complex numbers and how to “constructively” calculate with them.

In part 3 we discussed different algebraic representations. Now it is time to do arithmetic with complex numbers in the Cartesian representation.

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Complex Numbers, Part 4—an alternative geometric representation

In part 1 we discussed a possible geometrical representation of real numbers as arrows along the real axis. This could be extended to arrows in the plane and thus led to complex numbers.

These arrows fit well with the Cartesian representation from part 3. There we also discussed the polar form, which in a certain sense is “complementary” to the Cartesian one. Here, we will consider the corresponding geometric representation.

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Complex Numbers, Part 3—the confusing variety of algebraic representations

In part 1, we have encountered complex numbers as arrows in the plane. Using geometric constructions, we could add, subtract, multiply, and divide with these arrows. For practical calculations, we now need an algebraic representation with corresponding arithmetical rules. Unfortunately there are several such representations …

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Complex numbers, Part 2—Multiplication, Rotation, and Euler’s formula

Part 1 demonstrated that the multiplication of complex numbers can be seen as a rotation-dilation of the corresponding arrows. As shown in Fig. 1, the rotation-dilation becomes a simple rotation if one of the arrows has length 1.

Fig. 1: The arrow \underline{R}_\alpha with length 1 and angle \alpha to the positive real axis (left). Multiplying the arbitrary arrow \underline{z} by \underline{R}_\alpha results in rotating \underline{z} by the angle \alpha (right).

We have already seen this when we multiplied the arrow \underline{i} by itself to get \underline{i}^2 = -1.

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Complex Numbers, Part 1—A geometric primer


Generally, numbers are very abstract entities. There is, e.g., nothing concrete which you could point to and say that is the number three. There are always either three people, three cows, three cars … Accordingly, there are various concrete representations for the number three, e.g., 3 or III.

Based on our ideas about real numbers, we will first consider one geometric representation of the real numbers and construct the known arithmetic operations geometrically. These representation and constructions we can “easily” expand to represent a new set of numbers—the complex numbers.

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