(Complex) Numbers
This is partly from my time as a teacher, when I made an automatic exercise correcting, course building site called mamchecker based on google appengine.
I go through the different number sets in a not so standard way and conclude with the complex numbers, which have always been fascinating to me, because somehow enigmatic, not obvious, and then because useful and a good example of mathematical thinking.
Natural Numbers (ℕ ) by themselves are a great abstraction. Forget about all qualities but the "how many", the cardinality.
Then, do we add apples or do we take them away? So the sign was invented and the Integers (ℤ ). We write -2, but since this linear direction is a concept by itself, we should write 2( − ) , that is 2 times ( − ) , like we write 2m or 2g . And when we want to say three things (how many, add or remove, what quantity), then we could write 2( − )g . We could also give ( − ) a name like s for subtract (remove) and then we should also have a for add. But to have special signs ( − , + ), not a letters, is a lot better. So these signs make the add and subtract operations part of the natural numbers, making them integers and operations. Instead of 2( − ) we write − 2 , but we should still think of it as two times subtracting. + 3 − 2 is a sequence of such operations, we could emphasize this by writing + 3, − 2 .
The same reasoning we can do with * (multiplying) and its opposite ⁄ (dividing). *2 and ⁄ 2 are elements of the rationals (ℚ ). *3 ⁄ 2*5 is a sequence of such operations (*3, ⁄ 2, *5 ). Most computer programs don't understand *3 ⁄ 2*5 unless the initial * or ⁄ is omitted. For + and − they do understand.
Addition and subtraction are two elements of a set, i.e. of something we can pick one exclusively at a time. I normally call such a thing a variable and the pick a value. This variable is often called after one value: Addition = { + , − } . Same for Multiplication = {*, ⁄ } .
When combining addition and multiplication in a sequence, we have also introduced a convention: multiplication before addition. This operator precedence rule allows us to write 2*3 + 2*4 in one expression instead of a = 2*3, b = 2*4, a + b .
To put multiplication first was a good choice. We have had multiplication above when we wrote 2( − ) to emphasize what − 2 means. There is something very basic about multiplication: Whenever there are two variables, that have nothing in common, that we can pick from independently, that are orthogonal, we simple pick values independently and place them one after the other. When we count the possibilities of combined picks we get |A|*|B| , where || is the cardinality, the information content. This is why we write ab for the area of an a × b rectangle and why physicists combine all kinds of variables via multiplication. These variables are extensive meaning any value of it is a set. E.g. the length of a rectanble side is the set of all its points or of all its unit stretches. I also call such variables quantities here.
Next come the irrational numbers (I ). For example no sequence of multiplication and division with natural numbers will produce the diagonal from the edge of a square (incommensurable, √(2) ), but one can get as close as wanted. This is how they are defined: An irrational is an infinite sequence of rationals. In this sense infinity is an irrational.
Irrationals and rationals make up the real numbers (ℝ ).
Several times numbers were augmented with new information. The same we do with to the real numbers to get to the complex numbers. In a real number we have the number, multiplication operation (* , ⁄ ) and the linear direction ( + , − ). We normally first determine the quantity by comparing to a unit (multiply e.g. meters) and then optionally add it (to 0 on default).
There is another concept, which we can introduce: orthogonality. This "variable" contains 1 (normally omitted) for same direction and i for orthogonal or turned by right angle π ⁄ 2 , counterclockwise by convention. Turning a second time gives i*i = − 1 . This solves x2 + 1 = 0 , starting from which normally i is actually introduced. 2ia is 2a and orthogonal to the direction of a .
i is called imaginary unit and ℝ × {i} are called imaginary numbers. Along this reasoning one should have called them orthogonal numbers. By addition they are independent/orthogonal to the real numbers. Together with the real numbers they make the complex numbers (ℂ ) Orthogonal means that all combinations are possible, which corresponds to a 2-dimensional (2D) plane, the complex plane or Gauss number plane.
z = (a, b) = a + ib ∈ ℂ
- a = Re(z) is the real part
- b = Im(z) is the imaginary part
These numbers are like 2D-vectors: 2 orthogonal directions that can be added independently.
There are three representations
- z = a + ib , i.e. via the components or
- z = r(cosφ + isinφ) via modulus r and argument φ (angle, phase) in radiants.
Note that, by the trigonometric addition formulas, multiplication adds the angles, i.e. multiplication leads to addition. This gives a hint that there could be a representation that has the angle in the exponent. Developing sin and cos into a Taylor series and comparing with the ex series leads to the Euler Formula:
z = reiφ is the third way to represent complex numbers.
- φ = arg(z) is the argument (phase) of z.
- arg(yz) = arg(y) + arg(z)
- arg((y)/(z)) = arg(y) − arg(z)
About sin and cos we know that the period is 2π , therefore this is true for eiφ . The nth root divides the period up to 2nπ to below 2π and so we have n different roots.
More generally:
In ℂ every polynomial of degree n has exactly n roots (fundamental theorem of algebra), if one counts the multiplicity of roots. ℂ therefore is called algebraically closed.
This means that not only x2 , but every polynomial maps the whole ℂ to the whole of ℂ . This closedness is important. All operations are reversible. One can calculate freely.
= re − iφ = a − ib is the complex conjugate of z. = n . We know that there are two orthogonals. We've chosen i to be one of them, so − i is the other.
y combines in itself dot product (Re(y) = |y||z|cosΔφ ) and cross product (Im(y) = |y||z|sinΔφ ). dot and cross are two aspects of orthogonality (=combinability). dot being 0 means there is no dependence, which is saying all combinations are possible, which means the combined variable is of maximum cardinality (area, cross), i.e. a rectangle. The conjugate is important, because it produces the delta angle between y and z . Angle again is another, i.e. third way of expressing dependence. With y dot, cross and angle are beautifully combined.
|z| = √(z) = √(a2 + b2) = r is the absolute value (modulus) of z . Here the imaginary part, the cross has gone, because there is no area in between. And why the √() ? Multiplication brought us to the combined variable, but in this limit case of 0 angle any independent other variable has gone and thus we go back to the the length, the cardinality, the information of z alone.
Why is the pythagorean theorem in there. To understand this we need to understand the fundamentals of the pythagorean theorem. There are two orthogonal variables involved. Let's name their units j and k instead of 1 and i (m and s for length and time,...). z = z1j + z2k is one value of the combined variable. The very fundamental dot of y and z reduces to the dot between j and k via y⋅z = (y1j + y2k)⋅(z1j + z2k) = y1z1j⋅j + y2z2k⋅k + y1z2j⋅k + y2z1k⋅j . Now j and k are orthogonal, so j⋅k = k⋅j = 0 , j and j are the same, thus j⋅j = 1 . Same with k⋅k = 1 . What we get is y⋅z = y1z1 + y2z2 , which is the pythagorean theorem if y = z .
number times unit is a special team: the quantity and the quality. It is also called a vector space. If the number is complex, it is a vector space over complex numbers. A vector can also subsume independent quantities, making it of dimension n > 1. A complex number z already has orthogonality. In a 2D vector space of units j and k it should be ij = k , because there is no other dimension. Actually something like this is done with quaternions. But for a general 2D vector space each component with ℂ becomes 2D itself. So a nD vector space over ℂ becomes kind of 2nD dimensional. What ℂ does is to allow specifying: can be added or cannot be added. This is comparable to a boolean.
To get the orthogonality, the dot, of two elements of such a 2D space over ℂ we do (uj + vk)⋅(yj + zk) = u⋅y + v⋅z = u + v , i.e. we take the conjugates. We need because we know there is orthogonality in a complex number and we've seen above how to deal with it. In general with r, s ∈ ℂn the orthogonality is the real part of r⋅s = ∑ri .
Now, do we need the complex numbers? Yes. The main reason is that orthogonality is a very important concept and so complex numbers find application in many fields. They make notation and calculation so much easier.
One essential point is the comparison of two quantities regarding addability. If they are parallel they are addable, if they are orthogonal, they are not. To this end paramters that have influence on addability, can be mapped to an angle (phase), which with Euler's notation of complex numbers, become a parallel (addable) and a orthogonal (not addable) part.
Examples:
- The time t of a vibration becomes φ = (2π)/(T)t or
- the t , x positions of a wave become φ = (2π)/(λ)x + (2π)/(T)t .
Re(Aeiφ) then represents the addable amplitude.
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