Complex numbers

[Lost in Translation]

2 hours ago, wandelaar said:

OldDog & LiT

 

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

 

Radians and Degrees

Let us see why 1 Radian is equal to 57.2958... degrees:

In a half circle there are π radians, which is also 180°

π radians =180°

So 1 radian =180°/π

=57.2958...°

 (approximately)

To go from radians to degrees: multiply by 180, divide by π

To go from degrees to radians: multiply by π, divide by 180

Here is a table of equivalent values:

Degrees	Radians (exact)	Radians (approx)
30°	π/6	0.524
45°	π/4	0.785
60°	π/3	1.047
90°	π/2	1.571
180°	π	3.142
270°	3π/2	4.712
360°	2π	6.283

 

https://www.mathsisfun.com/geometry/radians.html

 

I haven't used radians in decades but the concept seems simply enough.

 

 

 

 

 

[OldDog]

Yeah, I am not conversant in radians but I looked it up and understsnd the concept. I can visualize degrees but not radians very well except  6 radians is nust short of a circle.

[wandelaar]

4 hours ago, OldDog said:

I can visualize degrees but not radians very well except  6 radians is nust short of a circle.

 

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

[wandelaar]

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. 

[Marblehead]

I'm glad I didn't get an arrow.

 

[OldDog]

OK, I'll give it a shot ...

 

Starting point (0,0)  

Endpoint (-3,2)

Real part Re(z) = -3

Imaginary part Im(z) = 2

Modulus |z| = |a + bi| = |-3 + 2i| = sqrt of (-3 + 2) = sqrt of (-1)  ... and we all know what that means. Did you do this on purpose or did I take a wrong turn?

Argument arg(z) ... alas I am at a loss here. Don't know how to calc an actual value in radians.

[wandelaar]

10 minutes ago, OldDog said:

Starting point (0,0)  

 

Yes.

 

10 minutes ago, OldDog said:

Endpoint (-3,2)

 

Yes.

 

10 minutes ago, OldDog said:

Real part Re(z) = -3

 

Yes.

 

10 minutes ago, OldDog said:

Imaginary part Im(z) = 2

 

Yes.

 

10 minutes ago, OldDog said:

Modulus |z| = |a + bi| = |-3 + 2i| = sqrt of (-3 + 2) = sqrt of (-1)  ... and we all know what that means. Did you do this on purpose or did I take a wrong turn?

 

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:

 

 

 

10 minutes ago, OldDog said:

Argument arg(z) ... alas I am at a loss here. Don't know how to calc an actual value in radians.

 

We will do that after we have dealt with the modulus.

[OldDog]

Well, I thought about Pythagoras but for some reason felt like standard math, geometry, etc was not legal in this discussion.

 

Similarly could resort to trig functions to get degrees of angle and then resort to some sort of conversion formula to convert to radians. But again, I am not sure what's allowable and what is not. Have we firmly established that standard math is applicable to non-real numbers?

[wandelaar]

@ 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.

Edited September 1, 2018 by wandelaar

[Lost in Translation]

11 hours ago, wandelaar said:

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. 

 

I feel like I'm back in high school... OK, here goes.

 

z = (1,-3)

 

Starting point: (0,0)

Ending point: (1,-3)

Re(z): 1

Im(z): -3

Modulus(z)  = square root of 1*1 + -3*-3, which is the square root of 10, or approximately 3.16

Arg(z) = sin-1(1/3.16) or approximately 18.43 degrees.

 

 

 

 

[wandelaar]

7 minutes ago, Lost in Translation said:

z = (1,-3)

 

The correct notation of a complex number is:  z = 1 + -3i .

 

7 minutes ago, Lost in Translation said:

Starting point: (0,0)

 

Yes.

 

7 minutes ago, Lost in Translation said:

Ending point: (1,-3)

 

Yes.

 

7 minutes ago, Lost in Translation said:

Re(z): 1

 

Yes.

 

7 minutes ago, Lost in Translation said:

Im(z): -3

 

Yes.

 

7 minutes ago, Lost in Translation said:

Modulus(z)  = square root of 1*1 + -3*-3, which is the square root of 10, or approximately 3.16

 

Yes.

 

7 minutes ago, Lost in Translation said:

Arg(z) = sin-1(1/3.16) or approximately 18.43 degrees.

 

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).

[wandelaar]

I think LiT is ready for the next phase.

 

@ OldDog

 

Can you now calculate the modulus and argument for your own complex number?

[Lost in Translation]

6 hours ago, wandelaar said:

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).

 

The arg(z) number is negative because the Im(z) is negative?

[OldDog]

Sorry, back. Got busy today.

 

OK, since standard math is allowed

 

|z|  = sqrt (a*a + b*b)  = sqrt (13) = 3.606

 

and

 

Arg(z) = sin-1(2/3.606) = sin-1(0.555) = 33.7 deg 

 

More or less. Guess I thought we could only use methods disclosed in discussion.

[OldDog]

Oops, forgot radians

 

33.7 deg = 0.559 rads ... where 1 deg = 0.0175 radians

 

[Lost in Translation]

At least I get to use the "shift" keys on my calculator. That's not something I do every day. 

[wandelaar]

2 hours ago, Lost in Translation said:

The arg(z) number is negative because the Im(z) is negative?

 

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.

[Lost in Translation]

15 minutes ago, wandelaar said:

 

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.

 

I don't understand. How can an angle have direction? In your diagram above you draw arg(z) as a curved line, which is not a measure of degrees but rather a portion of the circumference of a circle. Is this why you brought radians into the picture? Are we describing a measure of degrees or are we describing an arc upon the circle's circumference?

 

[wandelaar]

3 hours ago, OldDog said:

OK, since standard math is allowed

 

|z|  = sqrt (a*a + b*b)  = sqrt (13) = 3.606

 

Yes.

 

3 hours ago, OldDog said:

Arg(z) = sin-1(2/3.606) = sin-1(0.555) = 33.7 deg 

 

More or less. Guess I thought we could only use methods disclosed in discussion.

 

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

 

 

[wandelaar]

50 minutes ago, Lost in Translation said:

I don't understand. How can an angle have direction? In your diagram above you draw arg(z) as a curved line, which is not a measure of degrees but rather a portion of the circumference of a circle. Is this why you brought radians into the picture? Are we describing a measure of degrees or are we describing an arc upon the circle's circumference?

 

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. 

Edited September 2, 2018 by wandelaar

[wandelaar]

3 hours ago, Lost in Translation said:

At least I get to use the "shift" keys on my calculator. That's not something I do every day. 

 

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

[OldDog]

22 hours ago, wandelaar said:

triangle below:

 

 

 

I assumed we were talking about the triangle you described above. The angle of interest was the angle formed by the two adjacent sides with the common point (0,0).

 

The other angle is the supplement of the angle I calculated. So ...

 

180 - 33.7 = 146.3 deg

and

146.3 * 0.0175 = 2.56 rad

 

But, now that you mention it, please explain the significance of the supplementary angle to this discussion.

[wandelaar]

On 31-8-2018 at 5:52 PM, wandelaar said:

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 .

 

 

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

Edited September 2, 2018 by wandelaar

[OldDog]

OK, I see the definition. But why is that angle any more significant than its supplement? 

[wandelaar]

59 minutes ago, OldDog said:

OK, I see the definition. But why is that angle any more significant than its supplement? 

 

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.

 

Edited September 2, 2018 by wandelaar