Complex numbers

[whitesilk]

To Marblehead, do you think linguistically, algebraically geometrically, spiritually? Causes of this thread are?

 

 

[Marblehead]

z = 2 + 3i

 

 

[wandelaar]

@ Marblehead

 

We have two more functions to explore. Are you ready?

[Marblehead]

1 minute ago, whitesilk said:

To Marblehead, do you think linguistically, algebraically geometrically, spiritually? Causes of this thread are?

 

 

What a set of questions.

 

Well, for sure "bi" is spiritual because we have defined it as being not real.

 

Yes, I suppose we could say that the cause of this thread is spiritual.  I stated an understanding I lacked way back when and you offered to guide me to an understanding.  Of course, I will likely never use this new knowledge I am gaining but I think that just the interaction with words with intent is somewhat spiritual.

 

 

[Marblehead]

Just now, wandelaar said:

@ Marblehead

 

We have two more functions to explore. Are you ready?

I'm ready as long as I don't have to walk too far.

 

Remember, I'm not planning to send a rocket to Pluto so we don't have to get too serious with this.  I suppose that when we reach the point where you say I have a handle on the concept and usage of imaginary numbers it will be enough for me.

 

[Lost in Translation]

I'm still very interested! Imaginary numbers have always been my bete noir so anyone who can help me tame them is welcome.

 

[whitesilk]

For engineers, j is the imaginary unit so a practical application of your formula would be a + bj ?

[wandelaar]

Here is the road map:

 

1. Complex numbers as arrows

2. The real and imaginary part of a complex number

3. The modulus and argument of a complex number

4. The addition of complex numbers

5. The multiplication of complex numbers

6. Examples of complex solutions to quadratic equations

 

Beyond 6. is advanced stuff. We are now on the brink of 3.

[wandelaar]

DEFINITION:  The modulus |z| of a complex number z = a + bi  is the distance between the point (0,0) and the endpoint (a,b) of z.

 

Now I have already posted a lot of pictures of complex numbers as arrows, and I like to see whether some of you can calculate the value of the modulus for some of those complex numbers purely on the basis of the above definition.  

 

 

[Marblehead]

30 minutes ago, wandelaar said:

Here is the road map:

 

1. Complex numbers as arrows

2. The real and imaginary part of a complex number

3. The modulus and argument of a complex number

4. The addition of complex numbers

5. The multiplication of complex numbers

6. Examples of complex solutions to quadratic equations

 

Beyond 6. is advanced stuff. We are now on the brink of 3.

Looks like a great agenda.  If you get me to just understanding #6. I will suggest that you have accomplished the impossible.

 

I hope I don't have to go through all the screw-up again.  I don't have a problem with being wrong but I don't like too much being wrong the same way over and over again.

 

 

 

[Marblehead]

16 minutes ago, wandelaar said:

DEFINITION:  The modulus |z| of a complex number z = a + bi  is the distance between the point (0,0) and the endpoint (a,b) of z.

 

Now I have already posted a lot of pictures of complex numbers as arrows, and I like to see whether some of you can calculate the value of the modulus for some of those complex numbers purely on the basis of the above definition.  

 

 

Bring it on.  No fear here.

 

[wandelaar]

How about |1 + 0i| ?

 

 

[Marblehead]

1 - Real

0 - imaginary

 

 

[wandelaar]

Yes - but that is not the answer to my question. The "|"-signs are not ordinary brackets. According to the definition |1 + 0i| is the modulus of 1 + 0i. And the modulus of 1 + 0i  is the distance between (0,0) and (1,0).

 

 

Edited August 30, 2018 by wandelaar

[Marblehead]

3 minutes ago, wandelaar said:

Yes - but that is not the answer to my question. The "|"-signs are not ordinary brackets. According to the definition |1 + 0i| is the modulus of 1 + 0i. And the modulus of 1 + 0i  is the distance between (0,0) and (1,0).

 

 

Ha.  Nothing new.  I have been subjected to information before of which I had no knowledge.

 

So we are using | as a containment function?

 

Let me make another wild ass guess:

 

|1 + 0i|

 

Let it be known that I have no idea what I am doing at the moment.  But as long as I keep responding and you keep telling me I'm wrong I can keep going until I accidently select a correct answer.

 

[OldDog]

1 hour ago, Lost in Translation said:

Remember basic training?

 

Hold on ... I need to get a rock for my left hand. Of course, it's only an imaginary rock. 

 

 

[wandelaar]

The modulus is a special type of function. You throw a complex number in and you get a non-negative real number out. What you get out depends on what you throw in. If you throw a complex number z in then what you get out in case of the modulus is written as |z| . The modulus is the complex version of the absolute value.

 

Maybe LiT can help some more with explaining the function concept.

Edited August 30, 2018 by wandelaar

[Marblehead]

21 minutes ago, wandelaar said:

The modulus is a special type of function. You throw a complex number in and you get a non-negative real number out. What you get out depends on what you throw in. If you throw a complex number z in then what you get out in case of the modulus is written as |z| . The modulus is the complex version of the absolute value.

 

Maybe LiT can help some more with explaining the function concept.

Well, I apparently I need all the help I can get with this stuff.

 

So we have a function.  1 + 2 is a mathematic function.  Result being three.

 

So in my mind, the function |1 + 0i| results in … what?  1  One what?

 

I think we have more work to do.

 

[wandelaar]

Functions are more abstract objects than numbers or arrows. That's why I didn't want to discuss them. But LiT is right that functions are essential to higher mathematics. However giving a formal definition of functions in terms of set theory is out of the question, that would take a huge topic of its own. We will have to make do with a rough idea of what a function is. The best I can think of now as a concrete example is a calculator that always performs the same operation on whatever is fed into it. For instance it could always add 1 to the number that is fed into it. Or it could always multiply the number that is fed into it by 44. But the operation or sequence of operations could be endlessly more complicated. The only restriction is that the calculator must always perform the same operation or sequence of operations on whatever number is fed into it. Such a calculator would be the physical realisation of a mathematical function. And the mathematical function itself could then be considered as the program that describes what operation or sequence of operations the calculator has to perform. I have no more time today, so see you all again tomorrow.

[OldDog]

8 minutes ago, wandelaar said:

The best I can think of now as a concrete example is a calculator that always performs the same operation on whatever is fed into it.

 

Good analogy.

 

So, we are to think of Im(b) as a function that operates on b and produces an imaginary value. This value can be paired with Re(a) to identify a point on a Cartesian coordinate system, where X axis is real and Y axis is real?

[Marblehead]

Okay.  So the function of |1 + oi| is a set format that is applied to any equation?  This function determines where the arrow ends?

 

X = 1  and Y = 0

 

 

[Lost in Translation]

1 hour ago, OldDog said:

So, we are to think of Im(b) as a function that operates on b and produces an imaginary value. This value can be paired with Re(a) to identify a point on a Cartesian coordinate system, where X axis is real and Y axis is real?

 

Bold added by me.

 

Im(b) does not produce an imaginary number. Rather, it takes a complex number with an imaginary component and returns the imaginary component.

 

 

Here's an analogy. You have a bicycle. The bicycle has many components, e.g. handlebars, frame, pedals. chain, seat, tires, etc. We can store the bicycle as a variable called z.

 

z = bicycle

 

Now imagine we have a function called "frame." The "frame" function takes a whole bicycle and returns the frame component of it.

 

frame(bicycle)

   return information on the frame component of this bicycle

 

If we apply the "frame" function to the bicycle variable we might get something as follows:

 

z = bicycle ("Extra-wide easy-grip handlebars", "Huffy Women's Beach Cruiser, #28, pink", "Ultra EZ pedals", "Tungsten chain", "Faux-Leather Extra-Soft Seat", "Double Knobby sand tires", etc.)

 

frame(z) = "Huffy Women's Beach Cruiser, #28, pink"

 

This function has not produced a bicycle frame. It has merely isolated the frame portion of the bicycle and returned information on it.

 

 

 

So let's take this back to the complex numbers.

 

We have a complex number in the format of "A + Bi", where A equals the X axis on a graph and B equals the Y axis. If we have  a point on the graph (2,3), e.g. X=2, Y=3, then we can express this as 2 + 3i.

 

z = 2 + 3i

 

The function Re(z) isolates and returns the A portion, e.g. Re(z) = 2.

 

The function Im(z) isolates and returns the B portion, e.g. Im(z) = 3.

 

That's all Wandelaar has been saying to this point.

 

 

 

[Marblehead]

Okay.  Function on my friends.

 

Wait a minute.  If we write |A + Bi| we haven't really said anything, have we?   We haven't isolated anything of the original equation?

 

 

 

[Lost in Translation]

4 hours ago, wandelaar said:

DEFINITION:  The modulus |z| of a complex number z = a + bi  is the distance between the point (0,0) and the endpoint (a,b) of z.

 

4 hours ago, wandelaar said:

How about |1 + 0i| ?

 

|1 + 0i] is the distance between (0,0) and (1,0), which is 1.

 

Does this mean that |1 + 1i| is the distance between (0,0) and (1,1), which is the square root of 2?

 

EDIT:

 

Is the modulus |z| another way to say the length of the arrow?

Edited August 31, 2018 by Lost in Translation

[OldDog]

2 hours ago, Lost in Translation said:

If we have  a point on the graph (2,3), e.g. X=2, Y=3, then we can express this as 2 + 3i.

 

Does this mean that any point on an coordinate system where the X and Y axis are real can be expressed with an imaginary (i) component?