wandelaar

In principle you could. One can add or subtract a multiple of 360 degrees (= 2π rad) to an angle without any change in the geometrical situation. That's why we have to specify a range for an angle to isolate only one specific value from all possible values. This is necessary if we want to define a function.

DEFINITION The Product of Two Complex Numbers The product w = u . v of two complex numbers u and v is geometrically defined by the following two properties: 1. |w| = |u| . |v| 2. arg(w) = red( arg(u) + arg(v) ) Where: red(φ) = φ + 2π for φ ≤ -π red(φ) = φ for –π < φ ≤ π red(φ) = φ – 2π for φ > π (The function red( ) forces the angle of the product back into the allowed range (–π, π ] whenever necessary. Adding or subtracting 2π radians changes nothing in the direction of an arrow geometrically speaking.)

DEFINITION The Sum of Two Complex Numbers The sum w = u + v of two complex numbers u and v is geometrically constructed in the following manner: 1. Draw the complex numbers u and v as arrows in the Cartesian coordinate system. 2. Form the arrow v' by parallel transporting the arrow v along the arrow u such that the starting point of this transported arrow falls exactly on the endpoint of arrow u. Note that the arrow v' need not be a complex number itself! This arrow v' is only used for the geometrical construction of the sum u + v of the complex numbers u and v. 3. Draw the arrow from the starting point (0,0) to the endpoint of v'. This arrow, which again is a complex number, is called the sum u + v of the complex numbers u and v.

So as I said before: there are many applications. Why would electrical engineers have bothered to learn complex numbers if they were useless from a practical point of view? Complex numbers hugely simplify calculations on electrical circuits. Complex numbers are also essential to quantum mechanics. Introducing negative numbers leads to a great simplification with many calculations, because when negative numbers are allowed than x - y will always have a value be it positive, zero or negative. Then we no longer need to bother about whether or not there is a non-negative number u such that x - y = u . Something similar applies for the complex numbers. All complex algebraic equations have complex solutions, even: z2 + (1 + 0i) = 0 + 0i (in shorthand: z2 + 1 = 0). Another important result is connected to Euler's formula. Many laws of nature have the form of differential equations. And Euler's formula makes it possible to solve many of those equations with the help of complex numbers. Also interesting from the viewpoint of Taoism is Fourier Analysis, where it is shown how periodic signals can be analysed as the sum of harmonic signals. But let me stop: one can not show how to apply a theory before it is developed.

The definition states what arg( ) means. A definition that isn't applied is mathematically useless. Imagine the following: A person A wants to know arg(z) of your complex number z. Now you happen to know what your complex number z is, and you provide A with the angle you found in the grey triangle. This isn't arg(z) ! But you consider the angle you found as just as significant as its supplement that is equal to arg(z). So you provide A with the angle from the grey triangle nevertheless. Thus A goes home with the wrong value you provided as an answer to his request for arg(z). Now as he gets home he would reason that by definition the argument arg(z) of a complex number z = a + bi is the angle between the positive x-axis and the arrow from (0,0) to (a,b). So using the angle you provided he would wrongly (!) conclude that the complex number z as an arrow must lie on the blue half line in the picture below: (Kept the degrees here for simplicity.) So you see that not following the definitions can lead to big errors in communication.

So the supplementary angle is actually the angle arg(z). The grey triangle was only selected to facilitate application of the sin-1( ) function.

I thought I understood what he meant, until the last sentence!

Your calculator will probably also have a switch that allows you to directly calculate angles in radians.

Degrees are seldom used in higher math. In principle you could indeed use a range of [0,2π ) for the angle arg(z) measured in radians, and in that way we would be able to designate all possible directions. But this would spoil the symmetry of our approach. That's why we have taken (-π , π ] as the range of our angle arg(z). Furthermore, time-dependent harmonic signals S(t) are often written in the form: S(t) = S0 * sin(ωt + φ) . Here ωt + φ is a time-dependent angle that we would like to be meaningful for all values of t. This can only be done when we allow negative angles (in radians). Negative angles are no more weird than negative positions on the x- and y-axis, and they are just as useful.

Yes. You have now calculated the grey angle. But arg(z) is the blue angle:

The angle arg(z) is measured counterclockwise by convention. As in this picture: That means that when the angle goes the other way is has to be given a negative value. It's also more intuitive to have both a and b positive for complex numbers z = a +bi with 0 < arg(z) < π/2 . And our formulae will become more elegant by taking the counterclockwise direction of the angle as positive.

Can you explain what you think is wrong with the cave story?

I think LiT is ready for the next phase. @ OldDog Can you now calculate the modulus and argument for your own complex number?

The correct notation of a complex number is: z = 1 + -3i . Yes. Yes. Yes. Yes. Yes. Looks like you calculated the yellow angle: For arg(z) we then approximately find: arg(z) = -(90 - 18.43) = -71.6 (degrees). And in radians we approximately have: arg(z) = - {π/2 - sin-1(1/(3.16))} = -(1.57 - 0.32) = - 1.25 (rad).

@ OldDog What I am doing here is constructing a model for the complex numbers purely on the basis of elementary and standard math. Simply introducing an undefined "i" for an impossible operation √(−1) isn't legitimate mathematics. That's why I don't want to follow that road. But building up a calculus for complex numbers considered as being arrows is legitimate mathematics, and that's what we are doing here. After the introduction of the sum and product of our complex numbers, it will become clear that our complex numbers as arrow do exactly what we want them to do. The complex numbers will have to include complex numbers that behave exactly as the familiar real numbers and there has to be a complex number z such that z*z = -1 + 0i. And we will see that that's true. Further, because we have only used elementary and standard mathematics in the construction of our complex number system its foundation will be as solid as the elementary and standard math we used to build it. The mystical aura of the complex numbers will be gone. So please use common logic and standard math in this topic.

Yes. Yes. Yes. Yes. And here it goes wrong. This part is still correct: |z| = |a + bi| = |-3 + 2i| . Then by applying the definition of the modulus we get: |z| = "the distance between (0,0) and (-3,2)". This distance can be calculated by means of Pythagoras' Theorem applied to the grey triangle below: We will do that after we have dealt with the modulus.

As properties of a complex number z = a + bi we have discussed: Starting point (0,0) Endpoint (a,b) Real part Re(z) Imaginary part Im(z) Modulus |z| Argument arg(z) See this picture: Now to see whether everything is understood correctly up till now I have prepared two complex numbers: one for OldDog and one for LiT. I like to see whether OldDog and LiT can give me the values of all the above mentioned properties of their very own complex number.

Angles in radians are best visualised in terms of fractions or multiples of π. See the table of LiT.

OldDog & LiT Do both of you know how to measured an angle in radians?

@ OldDog It's just an unhappy coincidence that the word argument is also used for the independent variable of a function, there is nothing more to it then that. But we have to live with this fact because the name "argument" for the angle associated with a complex number (as far as I know) is used everywhere in texts on complex numbers. So indeed the function arg( ) returns the angle of the complex number that is fed in.

@ Marblehead Ah - it's a pity we didn't succeed. But I respect your choice to stop with this topic.

Here is a nice 3D grapher to visualise our functions: https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/

Here's the picture complete with all four functions: Re( ) Im( ) | | arg( )

One more function to go before we will introduce the sum and product for the complex numbers. DEFINITION The argument arg(z) of a complex number z = a + bi is the angle between the positive x-axis and the arrow from (0,0) to (a,b). The argument is usually measured in radians and chosen so that –π < arg(z) ≤ π .

OK Marblehead. Do the others now understand everything in the picture of my previous post?