Complex numbers

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Abstract: In this article, we attempt to give an intuitive "elementary geometric" idea of the complex numbers. Just as the real numbers can be seen as the points on an axis, the complex numbers can be considered as the points of the two dimensional complex plane. We will attempt to use this representation to illustrate some basic properties of the complex numbers. The treatment is not meant to be mathematically rigorous, though we will strive to provide accurate statements.

Audience: This article is aimed at people with little or no prior knowledge of complex numbers. Prerequisites are only some very basic notions of arithmetics and of elementary geometry, e.g. distances and angles.

Contents 1 The real numbers 2 Beyond the real numbers 3 How to make computations 3.1 Addition 3.2 Multiplication 4 Imaginary numbers

[edit] The real numbers

The real numbers can be seen as the points on an axis, which we will call the real axis. We pick a point on this axis and call it zero or origin. The points left of the origin will be the negative numbers, the points on the right the positive numbers. We also fix a length unit, which will be the distance between two neighbouring integers on the axis. One length unit left of the origin will be -1, and one unit to the right will be +1.

We can characterize each number by two parameters: The modulus, which is its distance from the origin, and the sign, which says if the distance must be taken to the left (negative sign) or to the right (positive sign) of the origin. Thus the number 4 has modulus 4 and positive sign, while the number -4 has also modulus 4 but negative sign.

[edit] Beyond the real numbers

Now we want to extend our number system. However, the axis we have introduced above is already "full", every point of the axis corresponds to a real number. So if we want to introduce further numbers, they cannot be on this axis; they will be "above" or "below" the axis. We extend our number system into a two dimensional plane, which contains the one dimensional real axis as a subset. Real numbers are thus a special case of complex numbers, just like e.g. integers are a special case of real numbers.

Each point of the plane will be a complex number. But how can we make computations with those numbers? First, let's see how we can characterize them. Above we have described the real numbers by their modulus and their sign. The concept of modulus, i.e. the distance from the origin, can be applied to the entire plane; for any point in the plane, just measure its distance from the origin.

The concept of sign however needs to be generalized. "Left" and "right" of the origin are no longer sufficient in a plane, where we can also have "above" and "below" the origin. But this is easy: we simply measure the direction as the angle formed by a given point with the positive real axis. This angle will be called the argument of the complex numbers, and replaces the sign we used for the real numbers.

This may be easier to visualize if we think of a complex number not just as a point, but rather as an arrow extending from the origin to the point. Each number is thus represented as an arrow attached to the origin, and it is uniquely characterized by its length, the modulus, and its direction, the argument. A positive number is thus an arrow pointing to the right, i.e. in the same direction as the positive real axis. Its argument is thus zero. A negative number on the other hand is an arrow pointing in the opposite direction, so its argument will be 180°. But now we can have infinitely many directions, from 0 to 360°.

[edit] How to make computations

[edit] Addition

How can we make computations with our new numbers? Let's first look at the addition. The rule is the following: to add two "arrows", you "shift" the second one (without rotating it!) such as to attach its tail to the tip of the first one. The sum of the two then corresponds to the arrow which extends from the tail of the first arrow (i.e. the origin) to the tip of the "shifted" second arrow. Simply put, two arrows are added by attaching them to each other without rotating them.

Does this match with the result we expect in the case of real numbers? Consider for example

2 + 3

In this case, we attach two arrows pointing in the same direction, one with length 2, the other with length 3. The result is obviously an arrow with lenght 5, still pointing in the positive direction. The result is thus, as expected, +5. In the case

3 - 2 = 3 + (-2)

the first arrow has length 3 and points to the right. To its tip we attach an arrow of length 2 pointing to the left. The arrow from the origin to the tip of the second arrow is thus of length 1 and points to the right. This matches the expected result +1.

[edit] Multiplication

Let's see how we can define the multiplication. Given two numbers, the modulus of the arrow corresponding to the product is obtained by multiplying the moduli (i.e. the lengths) of the two arrows. The argument of the product arrow however is obtained by adding the arguments (i.e. the angles) of the two arrows.

This may sound confusing, but some examples will show that this is quite easy. Let's first consider

3 · 2.

The moduli (lengths) are 3 and 2, respectively, so the modulus of the product will be 3 · 2 = 6. The arguments (angles) are both zero, so the argument of the product will also be zero (0 + 0 = 0). Thus the product is represented by an arrow pointing to the right and with length 6, corresponding to the number +6, as expected. Consider now

3 · (-2).

The moduli (lenghts) are the same as in the previous example, so the product arrow will again have length 6. But this time the second arrow has argument 180°, so the argument of the product will be 0 + 180° = 180°. The product arrow thus corresponds to the number -6, which again matches with the usual product of real numbers.

[edit] Imaginary numbers

So, what has all of this to do with the (in-)famous "square root of negative one"? It turns out that many equations which have no real solutions, do have one or more complex solutions. The classic example is the following:

x² + 1 = 0.

It is known that the square of a real number cannot be negative. Thus, if we add +1 to it, we cannot get zero.

In come the imaginary numbers. Loosely speaking, they correspond to the points of our plane that lie exactly "above" or "below" the origin, i.e. their arguments are 90° or 270°. In particular, the imaginary unit will be one unit length above the origin, i.e. it has modulus 1 and argument 90°. It is frequently denoted by i.

What is the square of the imaginary unit, i.e. its product with itself? Let's recall how to compute it: the moduli must be multiplied; the modulus (length) of the imaginary unit is 1, so we get 1 · 1 = 1. The arguments must be added; the argument of the imaginary unit is 90°, so the argument of the square is 90° + 90° = 180°.

The square of the imaginary unit thus corresponds to an arrow of length 1, and pointing to the left (180°). This arrow corresponds to the real number -1. We thus have

i² = -1,

or equivalently, by adding 1 on both sides of the equation,

i² + 1 = 0.

We have thus found a complex solution to the above equation which had no real solution. By the same reasoning one can check that also -i, the negative of the imaginary unit (modulus 1, argument 270°), is a solution to the same equation. (Take into account that 540° is the same direction as 180°, as the two differ by a full turn of 360°.)

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