Complex numbers

[wandelaar]

I have cleaned up the Cartesian coordinate system to work with some concrete arrows (= complex numbers). We will use this:

 

[rideforever]

But why are the arrows concrete ?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- only joking 

[wandelaar]

Now what are a and b in the complex number representation a + bi  for the red arrow (= complex number) in the picture below?

 

 

Remember this:

 

Quote

The expression a + bi with a and b real numbers designates the straight arrow starting at the point (0,0) and ending in the point (a,b).

 

There is nothing more to it. Example: the expression 5 + 8i designates the arrow starting at the point (0,0) and ending in the point (5,8).

 

Thus how does it work? First we know that all arrows that are complex numbers start in (0,0), so we only have to know the endpoint of the arrow to be able to draw it. Finding the endpoint goes as follows: given an expression like 5 + 8i than the first real number (5) in the expression gives us the x-coordinate of the endpoint of the arrow, and the second real number (8) in the expression gives us the y-coordinate of the endpoint of the arrow. And thus the endpoint of the arrow is known (in our example it will be (5,8) ). The expression 5 + 8i thus designates the arrow starting at the point (0,0) and ending in the point (5,8).

 

Edited August 28, 2018 by wandelaar

[Marblehead]

4 hours ago, wandelaar said:

Thus these arrows are called complex numbers because - as we will see later - they behave like numbers when we form sums and products of them.

Okay.  In that case, I'm still with you.  I can plot 5,8 on a graph.  I can draw an arrow between 0,0 and 5,8.  I can call the arrow a complex number.

 

I think I'm with you.  We'll se when it's test time.

 

[Marblehead]

4 hours ago, wandelaar said:

Now what are a and b in the complex number representation a + bi  for the red arrow (= complex number) in the picture below?

-2,-1

 

[whitesilk]

5 hours ago, Lost in Translation said:

That's like saying my friend lives 20 miles north and 20 miles east.   Where does he live ?   40 miles away !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

 

 

technically, 20 + 20 == 40, yet the magnitude is square root ( 20 * 20 + 20 * 20) or square root (2 * 20 * 20) == 20 * square root (2)

[wandelaar]

@ Marblehead

 

Yes!!! It's party time. I think you now got it. But first I will post some more tests to see whether you can now handle and recognize all complex numbers as arrows.

[Lost in Translation]

1 minute ago, whitesilk said:

5 hours ago, Lost in Translation said:

That's like saying my friend lives 20 miles north and 20 miles east.   Where does he live ?   40 miles away !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

 

 

technically, 20 + 20 == 40, yet the magnitude is square root ( 20 * 20 + 20 * 20) or square root (2 * 20 * 20) == 20 * square root (2)

 

I never said that. You quoted a quote of Rideforever that I had quoted earlier. That's an issue with the software TDB uses. You're not the first person to do this...  

[Marblehead]

4 minutes ago, wandelaar said:

@ Marblehead

 

Yes!!! It's party time. I think you now got it. But first I will post some more tests to see whether you can now handle and recognize all complex numbers as arrows.

I won't need my compass (protractor), will I?

 

[wandelaar]

2 minutes ago, Marblehead said:

I won't need my compass (protractor), will I?

 

The definitions of the sum and product will involve geometrical constructions with the arrows. But that's no problem for now. First I want to make sure that you now know what a complex number considered as an arrow is and how it can be designated by an expression of the form a + bi. When that is done, then the rest will cause no further problems.

[wandelaar]

 

Can you now represent the red arrow (= complex number) by means of an expression of the form a + bi ?

[Marblehead]

12 minutes ago, wandelaar said:

 

The definitions of the sum and product will involve geometrical constructions with the arrows. But that's no problem for now. First I want to make sure that you now know what a complex number considered as an arrow is and how it can be designated by an expression of the form a + bi. When that is done, then the rest will cause no further problems.

Yes, no problem for now as we aren't talking about it yet.

 

Yes, I think I finally got it.  Complex number is equal to the arrow that begins at 0,0 and ends at a,b (X,Y).  Can't talk about "I" as it is yet undefined.

 

 

[Marblehead]

10 minutes ago, wandelaar said:

 

Can you now represent the red arrow (= complex number) by means of an expression of the form a + bi ?

1,2i  But I will ignore "i" for now so yeah, 1,2(i).

 

That looks strange but I'm sure you will tell me why.

 

 

[wandelaar]

Well - that's almost correct but not quite. The correct answer is: 1 + 2i . Only then is it in the form a + bi. We have to be rigorously correct now, for otherwise the problems will reappear later on. So another test to get it right:

 

 

How do you write the blue arrow in the form a + bi ?

 

 

[Lost in Translation]

13 minutes ago, wandelaar said:

 

 

How do you write the blue arrow in the form a + bi ?

 

2 + (-3i) or do you prefer 2 - 3i?

[wandelaar]

2 minutes ago, Lost in Translation said:

2 + (-3i) or do you prefer 2 - 3i?

 

Formally correct would be 2 + -3i or 2 + (-3)i. But in actual practice one uses 2 - 3i.

 

The last expression can be made mathematically legitimate by considering the as jet undefined expression a - bi as a shorthand for a + (-b)i.

[OldDog]

OK. So we went down this road because imaginary numbers were mentioned in another post. Wandelaar offered to help with understanding imaginary / complex numbers. Many of us were hoping that he would produce an explanation that could be likened to a concept in taoism. So far the explanations are not capturing my/our imagination. 

 

So far, the easiest thing for me to understand  has been that the expression i = square root of (-1) is undefined, hence imaginary. This is easy because we all understand that the square of any number, positive or negative is a positive value; that is there can be no number thst when squared produces a -1.  At least this is true at the level most of us are thinking.

 

Trying to deal with this geometrically, I think, is a bit confusing. Wandelaar has chosen as object of discussion to use the typical Cartesian corrdinate system as an analogy. I say, analogy because for it to actually apply (as in the example a+bi,  the y coordinate must not be real but imaginary. In such a system x coordinate is represented by a and the y coordinate by the imaginary number bi.

 

Where I go of the rails in the geometric example is that the expression a+bi is represented by a straight arrow. Which leads me to the following questions.

 

1. Is the arrow really straight?

 

2. If so, is it because the expression a+bi is like the general straight line expression y=mx +b?

 

3. If a+bi is like y=mx+b ... does that mean i is an imaginary slope?

 

4. What is it that suggests arrowness? The arrow suggests direction ... or was it really meant to represent a line without arrow?

 

5. Can it then be said that any expression (straight or otherwise) which contains an imaginary term be considered  complex?

 

Maybe the answer to these questions will help understand the nature of a+bi. I think they would for me.

 

Oh, and btw ...

 

6. The arrow doesn't always have to emanate from (0,0) ... does it?

[Marblehead]

1 hour ago, wandelaar said:

How do you write the blue arrow in the form a + bi ?

 

 

2 + (-3i)

 

[Marblehead]

16 minutes ago, OldDog said:

OK. So we went down this road because imaginary numbers were mentioned in another post. Wandelaar offered to help with understanding imaginary / complex numbers. Many of us were hoping that he would produce an explanation that could be likened to a concept in taoism. So far the explanations are not capturing my/our imagination. 

Yeah, but we're not done yet.  Give me time, I do know how to go off topic and make a link to something else.

 

[Marblehead]

2 minutes ago, Marblehead said:

2 + (-3i)

 

Isn't this the same as 2 + (-3)I?

 

[wandelaar]

1 hour ago, OldDog said:

1. Is the arrow really straight?

 

Yes. When you define a thing to be such and so, then it is such and so by definition. Otherwise mathematics would become impossible.

 

Quote

2. If so, is it because the expression a+bi is like the general straight line expression y=mx +b?

 

No - the expression a + bi is chosen because the complex numbers will be seen later on (after we have given definitions for the sum and product of two complex numbers) to behave like we would expect for numbers consisting of the sum of a real and an imaginary part. The expression a + bi is suggestive of this.

 

Quote

3. If a+bi is like y=mx+b ... does that mean i is an imaginary slope?

 

No - it isn't.

 

Quote

4. What is it that suggests arrowness? The arrow suggests direction ... or was it really meant to represent a line without arrow?

 

Yes! The angle between the positive x-axis and the arrow will play an important role in the eventual theory of the complex numbers. 

 

Quote

5. Can it then be said that any expression (straight or otherwise) which contains an imaginary term be considered  complex?

 

Yes - the term complex is chosen to refer to the real and imaginary components of a complex number.

 

Quote

6. The arrow doesn't always have to emanate from (0,0) ... does it?

 

Taking the point (0,0) as the starting point of the arrow is by far the easiest thing to do. Why make it more complicated than it already is?

Edited August 28, 2018 by wandelaar

[wandelaar]

@ Marblehead

 

The complex expression a + bi has only two places where numbers may be inserted, and that is on the places of a and b. The brackets in 2 + (-3i) give the false impression that -3i is itself a number. But as yet -3i has no meaning, because only complete expressions of the form a + bi have a meaning as the arrow pointing from (0,0) to (a,b). But we will give a meaning to expressions such as -3i as we go on.

Edited August 28, 2018 by wandelaar

[wandelaar]

As to a possible connection between the complex numbers and Taoism I can mention the application of the complex numbers in the analysis and study of periodic (and other dynamic) phenomena.

Edited August 28, 2018 by wandelaar

[Fa Xin]

Lao Tzus lost book of geometry 

[Marblehead]

12 hours ago, wandelaar said:

@ Marblehead

 

The complex expression a + bi has only two places where numbers may be inserted, and that is on the places of a and b. The brackets in 2 + (-3i) give the false impression that -3i is itself a number. But as yet -3i has no meaning, because only complete expressions of the form a + bi have a meaning as the arrow pointing from (0,0) to (a,b). But we will give a meaning to expressions such as -3i as we go on.

Great.  I needed to clarify that in my mind.  Thanks.