Please help me understand Complex Numbers

  • #1
Hi.
If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1.
1. Is this "another dimension" the third dimension 'Z' that I usually use to describe a three dimensional objects? I ask this question because the above video resource says that the graph shown in the video is in "full two dimensional form". Please seek the video to 1:39 where he says "full two dimensional form".
2. If the parabola drawn is not in three dimensions then what is the new dimension that the person referring to in the video called?
3. If I say that there is a complex number like 3 + 4i then is the real part of this complex number on the x-axis and the imaginary part of the complex number on this new dimension that the person pulls out with his hand from x-axis (at 1:40 seconds into the video)?

My thought:
According to me there are three dimensions: x, y, and the dimension that the person in the video creates with his hand.

4. In the complex number 3 + 4i where does 3 lie and where does 4 lie if I have these three dimensions?
ComplexNumber_zpsg2oc4lrv.png

5. Why does the parabola goes into the new dimension?
6. Why does the parabola looks the way it does in the video when I draw it with the new dimension?

Thanks!
 

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  • #2
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HI,

My problem is that if you don't understand the video, it will be very easy for you to be misguided by anything I write here, so please beware...
and check, ask, check again.

The dimension he pulls out of the paper is not our usual third dimension z. Instead it is a lateral dimension (of ##x##), characterized by this ##i^2=-1##. (whereas the 'normal' ##z^2 \ge 0## ).

The whole pseudo-3D visualization is totally misleading: If you look in the video where ##y=0## you would get the impression that there is a whole parabola of solutions to ##x^2 + 1 =0##, namely where the colored surface crosses the black grid. This is absolutely false (there is this lateral dimension for ##y## as well as for ##x##, only that would make the picture intractable). In the picture there is only one solution visible, namely one unit straight up (the other is straight down ---- under the paper). Thanks to this characteristic ##i^2=-1## : if in the vertical plane (the black grid), the sideways coordinate ##x=0##, and the lateral dimension coordinate of ##x## is 1 (or -1), then and only then you get ##{\bf x}^2 + 1 =0## (where now I use bold face to indicate that ##x## has two dimensions).

i think you are a lot better off trying to read the treatise 'nothing imaginary about complex numbers' by @LCKurtz on his site (even though it is aimed a teachers, it is an excellent introduction to this funny world of not-so-imaginary numbers) where the risk of being misguided by what you see and how to interpret that is much smaller.

But I grant you that the colorful visualization is very nice to look at.
 
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  • #3
Thanks to this characteristic i2 = -1 : if in the vertical plane (the black grid), the sideways coordinate x = 0, and the lateral dimension of x is 1 (or -1), then and only then you get x2 + 1 = 0.
What is lateral dimension?
What is sideways coordinate?

The whole pseudo-3D visualization is totally misleading: If you look in the video where y=0 you would get the impression that there is a whole parabola of solutions to x2 + 1 = 0, namely where the colored surface crosses the black grid. This is absolutely false (there is this lateral dimension for y as well as for x, only that would make the picture intractable).
How do I make the curve x2 + 1 = 0 pass through x-axis?
 
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  • #4
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What is lateral dimension?
What is sideways coordinate?
Lateral is proposed at 2:15 for the coordinate out of the paper. Most of us call it the imaginary part axis for x.
The sideways coordinate is the real part axis of x.
Why is the curve crossing the black grid? It is understandable that to find solutions to a polynomial it should pass through x-axis. So, what should I do to make the equation x2 + 1 = 0 pass through x-axis?
at 1:40 you see the hand pull the parabola upwards, away from the paper. It stays parallel to the paper. At the same time it moves in the minus y direction, the further from the paper, the more down the y-axis it goes.
To make ##x^2 + 1## pass through the x-axis is not the issue.
The issue is to make it pass through ##y=0## on the real y-axis.
May seem the same but is not. It means that if you place another grid vertically out of the paper, this time through the y-axis, the parabola crosses the intersection of the two grids at a point that is at distance 1 from the origin. That point is a solution to ## {\bf x}^2+1= 0 \ \ ## (and so is the mirror point underneath the paper).

A short answer to 'what should I do' is of course: " to make ##x^2 = -1 ## "

If I say that there is a complex number like 3 + 4i then is the real part of this complex number on the x-axis and the imaginary part of the complex number on this new dimension that the person pulls out with his hand from x-axis (at 1:40 seconds into the video)?
Yes!

My grudge remains: much better to read the plane number story in the link.
 
  • #5
.. much better to read the plane number story in the link.
The article emphasizes more on how one could perform 2D arithmetic in a plane which I think is easier to perform if I know what and why the concept of Complex Numbers - in regards to axis and plane, are the way they are.
I have another question: when we speak about Complex Numbers I notice that we don't involve real y-axis when we are trying to get Solutions of a polynomial like x2 + 1 = 0 but instead we replace the real y-axis with imaginary y-axis (the black grid) to plot numbers of form a + ib: 'a' is real x-axis and the black grid is a plane where imaginary y-axis is located with number line in the form: ... -5i, -4i, -3i, -2i, -1i, 0, 1i, 2i, 3i, 4i, 5i ...
So, when somebody draws Complex Number plane, the horizontal number line is real x-axis and vertical number line is the imaginary y-axis (the black grid) and the real y-axis number line disappears? The real y-axis disappears because it's not required to obtain a solution for a graph (question) that has real x-axis and real y-axis?
 
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  • #6
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So, when somebody draws Complex Number plane, the horizontal number line is real x-axis and vertical number line is the imaginary y-axis and the real y-axis number line disappears?
If you will, yes. But more exact would be to say it never existed. The real plane is just a crutch to visualize the complex numbers. It isn't perfect, but better than nothing. It basically means to consider the complex numbers as a two dimensional real (vector) space, which isn't quite the same, as the (field) operations like multiplication and division are lost. To re-install them, the geometric means come into play - again a crutch.
 
  • #7
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So, when somebody draws Complex Number plane, the horizontal number line is real x-axis and vertical number line is the imaginary y-axis (the black grid) and the real y-axis number line disappears?
Yes. We need two axes for representing a single complex number. And, normally, a function of a complex number is also a complex number (that was one of my grudges against the video).
The real y-axis disappears because it's not required
It disappears because we need two of them (one for the real part and one for the imaginary part of ##\bf y##).

So instead we study operations with complex numbers and simple functions of complex numbers by looking at ##\bf x## and ##\bf y## in the same 2D plane. A bit like when you consider the image of real number x under the operation "y = x+2" as a point on the x-axis that is shifted by 2 wrt x.

This is what is treated so nicely in the @LCKurtz link I gave.
 
  • #8
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How do I make the curve x2 + 1 = 0 pass through x-axis?he
The above is not a curve, so it doesn't pass through anything. This is an equation, which either does or does not have solutions, depending on what sort of numbers you're willing to allow.
If x is restricted to the real numbers, then this equation has no solutions. If you square any real number, the result will be at least 0. Adding 1 produces a value that will be at least 1, so there are no real numbers that can be squared, have 1 added to the result, and add up to 0.

If x is allowed to be a complex number, then the equation ##x^2 + 1 = 0## does have solutions; namely ##x = \pm i##, where i is by definition ##\sqrt{-1}##, from which it follows that ##i^2 = -1##.

Your question should have been "How do I make the curve ##y = x^2 + 1## pass through the x-axis?" If you graph this curve on the real plane, its graph is a parabola whose lowest point is at (0, 1), so it does not cross the x-axis at all.
 
  • #9
Mathematicians wanted to plot a Complex Number, so they invented Complex Plane (lateral dimension in this discussion). So, if they have numbers with characteristics that don't match with the characteristics of numbers that are already known then they would invent new planes and dimensions?
 
  • #10
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Mathematicians wanted to plot a Complex Number, so they invented Complex Plane (lateral dimension in this discussion). So, if they have numbers with characteristics that don't match with the characteristics of numbers that are already known then they would invent new planes and dimensions?
Most human beings are attracted to ways of visualizing ideas in pictures. Mathematicians are no exception, so its fair to say that many mathematicians would (and do) attempt to visualize mathematical structures as something that can be plotted in a certain (finite) number of dimensions. However, there is no guarantee that a given mathematical structure can effectively be represented in a finite number of dimensions, even if one is allowed to use a large number of dimensions.

A technical example that specifies limitations on creating "many dimensional numbers" is the Frobenius Theorem:
https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)

If we think of a complicated document like the income tax laws or a complicated computer program, it is unlikely that there is an effective way to represent such structures as simple pictures. Likewise, many mathematical structures cannot be effectively presented and reasoned about using pictures. That's why most mathematical reasoning is conducted in the way one would discuss the income tax laws - namely, it is conducted by using words, not pictures. Of course, mathematical discussions in words are facilitated by using symbols as abbreviations for words. If you count a symbol as a picture then pictures are used in that sense.
 
  • #11
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Mathematicians wanted to plot a Complex Number, so they invented Complex Plane (lateral dimension in this discussion). So, if they have numbers with characteristics that don't match with the characteristics of numbers that are already known then they would invent new planes and dimensions?
'Invented' is probably not the right word. 'Abstracted' ? They simply 'took' a second dimension of something and assigned that to an axis in the plane. The two dimensions of the paper can be used for two conventional space coordinates (##\ ## x and y ##\ ## or ##\ ## x and z ##\ ## or ##\ ## y and z ), but just as well for one conventional and one 'lateral', or for one 'conventional' and one (scalar) function of the conventional -- for example for the speed in the x direction. When you read introductions to special relativity you will encounter graphs with x horizontal and time vertical. The line x = ct is then a representation of the time history of a light pulse at t=0 that expands spherically in space, at the speed of light !

I like the word 'abstraction'... :wink: -- it's what unlimits imagination.

[edit] So 2 dimensions is the maximum of what we can handle on paper (or a screen) in a direct manner. Anything more requires some imagination -- perspective, leaving out, or whatever. I personally like(*) the youtube physics videos by Eugene Khutoryansky very much: he uses time (motion) and the 3d training of our brain to the max to make us 'see' more dimensions. And he does have a 14 minute one on complex numbers (and nicely following the 'plane numbers' path @LCKurtz describes :wink: )

(*) Understatement: I love them. One of the few things that sometimes make me think I was born too early o_O
 
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  • #12
Assume that there is a point P (x,y) (zero dimension). I want to go further and generate a line (one dimension) from this point and then I want to go further more and generate circle (two dimensions) from this line and then I want to go further more and generate three dimensional object from this two dimensional object (circle, triangle, square, rectangle, ...).

I want to learn the words that I encounter during this exercise.
I have two mathematical structures in my mind that I want to create: shapes and waveform/curves - both of them can be described by 'x' and 'y' axes or 'x' and 'y' and 'z' axes.
How do I go about achieving this objective?
 
  • #13
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That's a bit of Monty Python: "and now for something completely different". Perhaps start another thread ? (or watch some of the videos: they actually use the words !)
 
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  • #14
Okay.
What is the system in Math that is equivalent to UML, Flow Chart, or Algorithm that I find in Computer Science?
 
  • #15
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New thread - again !
 
  • #16
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Ok, I recently stumbled across this really cool video explaining imaginary numbers and the complex plane and it's really mind-bending, heck, there're people how even deny them.
Hamilton in his letter of 17 October 1843 to John Graves is very confused about the relationship between i, j, +1, and -1. He asks what are we to do with ij, when i and j are unequal roots of a common square. In fact there is no law of arithmetic which makes ij equal to anything but +1. It is these doubts of Hamilton which are the source of his fallacious theory of the non-commutative properties of the multiplication of imaginary numbers. All multiplication whether of real or imaginary numbers is commutative.

So, my question "Can the Complex Plane Be Expressed as a Z-Axis?" arose when I started thinking about how many zeros (As the zeros of a function) would exist in this new complex "z-axis", and wouldn't there be infinitely many zeros in that axis? and if so, how's that possible? using, for example, the function used in the video
ƒ(x) = x2+1
Wouldn't there be only two? i and -i
 
  • #17
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So in the video they extended the x axis to make a complex number plane. The x values can be complex but the y values are real.

If the y=f(x) allows y to be a complex number then the y axis would have to be extended into a complex number plane meaning you’d need another axis that perpendicular to the y and to the two axes ie x-real and x-imaginary representing the x complex number plane basically you’d need four dimensions not three.

Does that make sense?
 
  • #18
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Assume that there is a point P (x,y) (zero dimension). I want to go further and generate a line (one dimension) from this point and then I want to go further more and generate circle (two dimensions) from this line and then I want to go further more and generate three dimensional object from this two dimensional object (circle, triangle, square, rectangle, ...).

I want to learn the words that I encounter during this exercise.
Okay.
What is the system in Math that is equivalent to UML, Flow Chart, or Algorithm that I find in Computer Science?
Since these questions are unrelated to your initial question in this thread, please start new threads for these questions.
 
  • #19
So in the video they extended the x axis to make a complex number plane. The x values can be complex but the y values are real.
So, how do I write it in Mathematics if
y = f(x) = 'x' values can be complex but the 'y' values are real
y = f(x) = ?
.. some point in three dimensions.
Example?
'Y' in terms of 'x' with three dimensions: (x + i * y) + (0 * x + y) → is this the correct algebraic representation of a point if I have
1. X-Complex Number Plane
2. Y-Real Number Plane

If the y=f(x) allows y to be a complex number then the y axis would have to be extended into a complex number plane meaning you’d need another axis that is perpendicular to the y and to the two axes i.e x-real and x-imaginary representing the x complex number plane basically you’d need four dimensions not three.
So, how do I write it in Mathematics if
... you’d need another axis that perpendicular to the 'y' and to the two axes i.e x-real and x-imaginary representing the x complex number plane basically you’d need four dimensions not three.
y(?) = f(x)(?) = ?
.. some point in four dimensions.
Example?
'Y' in terms of 'x' with four dimensions: (x + i * y) + (i * x + y) → is this the correct algebraic representation of a point if I have
1. X-Complex Number Plane
2. Y-Complex Number Plane

How can a X-Complex Number Plane be both real and imaginary?
How can a Y-Complex Number Plane be both real and imaginary?
(x + i * y) + (i * x + y)?

I think, the word I am looking for is Quadrants?
 
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  • #20
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So, how do I write it in Mathematics if
y = f(x) = 'x' values can be complex but the 'y' values are real
y = f(x) = ?
Example?
You write it ##f\, : \,\mathbb{C} \longrightarrow \mathbb{R}\; , \;f(x)=y## and draw it in a three-dimensional chart
upload_2018-5-15_18-32-38.png

which will give you a somehome mountain like surface.
 

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