Complex numbers

[OldDog]

I get it ... the definition.

 

So, other than the precise mathematical definition, there is no other significance to actual arg(z) value ... no practical meaning other than within the realm of mathematics.

 

This takes me back to my college days where I just had to accept that learning how to play the game ... apply the rules and manipulate the objects ... and not worry about practical application.

 

At the start of this thread there was a suggestion ... implied perhaps ... that there might be some insight to be gained about Daoism by understanding complex numbers. Do you see a connection? If so, can you elucidate?

 

[wandelaar]

On 27-8-2018 at 5:44 PM, wandelaar said:

The complex numbers are used in quantum mechanics, in electrical engineering, in solving algebraic equations, in Fourier analysis, in Laplace transforms, etc. All of them hugely useful applications.

 

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.

[wandelaar]

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.

 

 

 

 

Edited September 2, 2018 by wandelaar

[wandelaar]

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.) 

 

Edited September 2, 2018 by wandelaar

[Lost in Translation]

4 hours ago, 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 .

 

3 hours ago, wandelaar said:

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:

 

 

And NOW I understand why we have negative angles! (sort of)

 

In my example could I have not said the angle is 288.43 degrees? Would that have not achieved the same result as -71.57?

[wandelaar]

12 minutes ago, Lost in Translation said:

In my example could I have not said the angle is 288.43 degrees? Would that have not achieved the same result as -71.57?

 

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.

Edited September 2, 2018 by wandelaar

[Lost in Translation]

I wrote this:

 

18 minutes ago, Lost in Translation said:

In my example could I have not said the angle is 288.43 degrees? Would that have not achieved the same result as -71.57?

 

before I read this:

 

1 hour ago, wandelaar said:

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.

 

 

 

 

 

I now understand why the angle is negative. If it weren't then w = u + v would not make sense.

 

[Lost in Translation]

Since Old Dog has angle 146.3 and I have angle -71.6, in our instance w = u + v would be w = 146.3 + -71.6, or 74.7. We've now "added" two complex numbers together. Interesting...

 

 

 

 

 

 

 

[wandelaar]

Uh? Could you explain what you are doing?

Edited September 2, 2018 by wandelaar

[wandelaar]

Lets call them u and v. Now what is u + v ?

 

Edited September 2, 2018 by wandelaar

[OldDog]

Except for the pure beauty of mathematics ... I am not really seeing an application.

 

Its been said that mathematics is the language of science. Guess what I was hoping for was a language of philosophy. 

 

I may end up joining Marblehead out on the porch pretty quick.

[Lost in Translation]

18 minutes ago, wandelaar said:

Uh? Could you explain what you are doing?

 

19 minutes ago, Lost in Translation said:

Since Old Dog has angle 146.3 and I have angle -71.6, in our instance w = u + v would be w = 146.3 + -71.6, or 74.7. We've now "added" two complex numbers together. Interesting.

 

Old Dog calculated his arg value as 33.7, which you corrected to be 146.3.

I calculated my arg value as 18.4 which you corrected to be -71.6.

 

2 hours ago, wandelaar said:

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.

 

According to the above definition w = u + v, or in our case w = 146.3 - 71.6, which is 74.7. Am I misunderstanding?

 

[wandelaar]

@ OldDog

 

The only application that is possible with the very elementary theory of complex numbers that we have covered until now is a demonstration that there is a complex number z such that z*z + (1+0i) = 0+0i . Try: z = 0 + 1i .

 

The real number version of this equation would be: z² + 1 = 0, and within the real numbers this equation has no solution. So the complex numbers solve a problem that the real numbers are incapable of. This has many repercussions for the theory of algebraic equations.

 

I cannot speed up this exposition any further because each step has to be fully understood before the next step can be taken. As Euclid said to the ruler Ptolemy I Soter when he asked Euclid if there was a shorter road to learning geometry than through Euclid's Elements:

 

There is no royal road to geometry.

 

Now we don't have to struggle through Euclid's Elements but one cannot reasonably expect to see any applications of the complex numbers before the sum and product of the complex numbers are fully understood. Now I have given the definitions for the sum and product. So if you like go ahead and verify that: (0+1i)*(0+1i) + (1+0i) = 0+0i .

[wandelaar]

48 minutes ago, Lost in Translation said:

Old Dog calculated his arg value as 33.7, which you corrected to be 146.3.

I calculated my arg value as 18.4 which you corrected to be -71.6.

 

According to the above definition w = u + v, or in our case w = 146.3 - 71.6, which is 74.7. Am I misunderstanding?

 

I don't see how that relates to the geometrical construction of u + v following the definition of the sum.

[Lost in Translation]

6 minutes ago, wandelaar said:

I don't see how that relates to the geometrical construction of u + v following the definition of the sum.

 

3 hours ago, wandelaar said:

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.

 

 

 

 

 

I see now. We are moving the starting point of the arrow v from (0,0) to the point specified by the arrow u [ in this case (2,1) ] and redrawing it. I had misunderstood. I thought we were manipulating angles but it seems we're back to working with points.

 

 

Edited September 2, 2018 by Lost in Translation

[wandelaar]

The angles are relevant for the product.

[Lost in Translation]

13 hours ago, wandelaar said:

 

 

4 hours ago, wandelaar said:

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.

 

3 hours ago, wandelaar said:

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.) 

 

 

OK. let's see now.

 

Old Dog:

z = -3 +2i

arg(z) = 146.3 degrees, 2.55 radians

 

LiT:

z = 1 -3i

arg(z) = -71.6 degrees, -1.25 radians

 

 

SUM

The arrow v1 now points to (-2, -1). I can draw a line from (0,0) to the point (-2,-1) and calculate the arg as -153.4 degrees or -2.7 radians.

 

sum = -2.7 radians

 

PRODUCT

arg(w) equals red(2.55 + -1.25) or red(1.3)

 

The following rule seems to apply: 

red(φ) = φ               for  –π <  φ  ≤  π

 

So red(1.3) = 1.3

 

product = 1.3 radians

 

 

 

Edited September 2, 2018 by Lost in Translation

[wandelaar]

The sum and product of two complex numbers are themselves complex numbers (that is: arrows), not angles.

 

Just as the sum and product of two real numbers are always real numbers, we also want that the sum and product of two complex numbers are always themselves complex numbers. That gives us a "closed" number system. Than we can again add or multiply the sum or product of two complex numbers to another complex number, etc. All intermediary results will always be complex numbers, and consequently the (nested) sums and products will always be defined. No special cases need to be considered.

[Lost in Translation]

In that case the sum is -2 -1i.

 

I have no idea how to convert the product into an imaginary number in the format of A + Bi.

[wandelaar]

2 hours ago, Lost in Translation said:

In that case the sum is -2 -1i.

 

Yes.

 

Quote

I have no idea how to convert the product into an imaginary number in the format of A + Bi.

 

Can you first show how you calculate the values of the modulus and argument of the product?

 

As soon as you know the modulus and argument of a complex number z you can use trigonometry to express it in the form z = a + bi .

 

Edited September 2, 2018 by wandelaar

[OldDog]

2 hours ago, wandelaar said:

the sum and product of two complex numbers are always themselves complex numbers.

 

Thought just crossed my mind. Is it not possible, while performing operations such as addition and multiication, on expressions containing complex numbers, to have complex terms cancel out? Thus yielding results without a complex term.

 

 

 

[Lost in Translation]

2 hours ago, wandelaar said:

Can you first show how you calculate the values of the modulus and argument of the product?

 

 

|w| = |u| . |v|

 

|u| = |sq root (2*2+3*3)|

     = |sq root (13)|

     = 3.6

 

|v| = |sq root (1*1 + -3*-3)|

     = |sq root (10)|

     = 3.2

 

|w| = 3.6 * 3.2

|w| = 11.52

 

I don't see where all this is going. It appears we're following a formula with no obvious end in sight. Perhaps you can pull back a bit and explain how all these elements work together?

[wandelaar]

45 minutes ago, OldDog said:

Thought just crossed my mind. Is it not possible, while performing operations such as addition and multiication, on expressions containing complex numbers, to have complex terms cancel out? Thus yielding results without a complex term.

 

You probably mean without an imaginary term? That can be done. But in that case we will write the result in the form a + 0i . And that is still a complex number. Those complex numbers z wherefore  Im(z) = 0 will behave exactly as the corresponding real numbers under addition and multiplication. And that is the reason why the complex number system can be considered as an extension of the real number system.

 

It's essential to the complex numbers that there is a product z*z wherefore Im(z*z) = 0 , because otherwise there would be no solution to the complex equation z*z + (1+0i) = 0 + 0i .

 

The above two points actually form the end point of my explanation. Here we have only reviewed the very basics of the complex numbers. I propose that we go on with this topic till we are able to add and multiply any two complex numbers, and till we understand why the complex numbers can be considered as an extension of the real numbers that gives a complex solution to the equation z*z + (1+0i) = 0 + 0i . After that the theory will become too advanced. But there are great videos on YouTube for further study, and I will be happy to answer any questions you may have as a result of watching those. 

 

(This also hopefully answers the question of LiT about what it all means.)

Edited September 3, 2018 by wandelaar

[OldDog]

2 hours ago, wandelaar said:

in the form a + 0i . And that is still a complex number.

 

Does this mean that i with a coefficient of 0 is a non-zero value thus requiring the term to re retained.

[Lost in Translation]

I "cheated" and looked up some videos on YouTube to help me understand what's going on here.

 

Video showing how to add and subtract complex numbers. I found this video easy to understand.

 

Spoiler

 

 

Here's a video showing the multiplication and division of complex numbers using the FOIL method. I also found this easy to understand.

 

Spoiler

 

 

Lastly here is a video showing how to multiply complex numbers in polar form. This video made no sense to me. The only reason I was able to follow it at all was due to what we had already discussed in this thread.

 

Spoiler