Imaginary numbers - represent in the real life

[smslca]

I googled, wikied etc., but I cant understand what it is because, may be i cant understand clearly what they said, or I have these questions in my mind because of little understanding.

What does these complex numbers represent in the real life. Where do they fit in the real 3 spatial (xyz) coordinate system.

Let us consider the real xyz coordinate system. If a person is at the origin. As he moves on x-axis forward(i.e the way he can see) it is +1*x, when he moves back(ie the way he cannot see it is -1*x. Here +1,-1 represents the direction on the line. ""Then what direction does i represent????"".

Even 3/4 (ie fractional) distance exists, and we can approximate the irrational values such as pi,sqr root 2, and show the distance from the origin(ie between 0 and 1, or 0 and -1 etc). But where should we show this 'i' distance from origin.


Answer below if my assumptions are true:
----------------------------------------------------
We know if +1 is forward and +1*-1 represent backward direction, and so on it iterates the direction. By applying similar way i must be a 90 degree direction.
Here i got an another confusion. If the person is moving in perpendicular to his facing side, then what is Y-axis in a 3 dimensional system. If Y-axis is direction of i
why shouldnt we represent every eqn like x+y=0 as x+iy=0.

If above paragraph is true, place a person on y-axis ie iy and x-axis must be real taken from above para. If we move him 90 degrees from y to z-axis then z must be real because i*i=-1 and i*-i=+1. But since we took x as real, if we move him towards z then z must be imaginary. But what is the z- as real axis really n imaginary mean.
-------------------------------------------------------

can any one explain the graph in
https://en.wikipedia.org/wiki/Complex_number

 

[albbg]

Re: Imaginary numbers

Complex numbers have been created to solve equation in which square root of negative numbers are involved. To do this has been set sqrt(-1) = j.

Each complex number is composed by a real part and an imaginary part, f.i.:
n = 1.3 + 2.4j 1.3 is the real part, 2.4 is the imaginary part.

if you write x+jy = 0 this will be the same as: x+jy = 0+j0, then x=0 and y=0
if you write x+y = 0 then x=-y and if x is real also y will be real

Unlike the real numbers you cannot easily sort the complex numbers, so they cannot be represented in a xy coordinate system. You need, instead, a plane in which the abscissa represent the real part and the ordinate the imaginary plane (like the graph at the link you attached)

Representation in tridimensional plane is even more complicate since it requires hypercomplex numbers.

Then complex numbers are a mathematical formalism very useful to solve many engineering (and not only) problems.

 

[calabazas]

Re: Imaginary numbers

What does these complex numbers represent in the real life. Where do they fit in the real 3 spatial (xyz) coordinate system.

Let us consider the real xyz coordinate system. If a person is at the origin. As he moves on x-axis forward(i.e the way he can see) it is +1*x, when he moves back(ie the way he cannot see it is -1*x. Here +1,-1 represents the direction on the line. ""Then what direction does i represent????"".

Even 3/4 (ie fractional) distance exists, and we can approximate the irrational values such as pi,sqr root 2, and show the distance from the origin(ie between 0 and 1, or 0 and -1 etc). But where should we show this 'i' distance from origin.[/url]

Imaginary number/axis is an invention by mathematician. They played with it and found good usage of it, such as for electrical signals. It does not apply well with the "person walk" example you described. But if you swap person with periodic or repetitive electrical signal, that works well.

When a signal is 4+j3 (current or voltage), that means it is with amplitude 5 and angle at 37° with 360° representing a full wave. In this case, imaginary number/axis works well. So, we use simple form of 4+j3 to represent that signal.

Math is tool. If it applies to certain application, then people put it in text book.

 

[Prototyp_V1.0]

Re: Imaginary numbers - explanation needed

Another simolar topic was posted just a few days ago.

**broken link removed**

 

[Element_115]

Re: Imaginary numbers - explanation needed

I had a Prof. say that "j" (√-1) is just a function !....

i.e. to shift to the y-axis.

Of course j is the "Imaginary" number, that does not exist, that is the
square root of -1.

Also here's a little something to confuse the matter just a little bit more !

j^j =0.2078795763507619...

Cheers

 

[Prototyp_V1.0]

Re: Imaginary numbers - explanation needed

I had a Prof. say that "j" (√-1) is just a function !....

The math teacher I had said that the expression j=√-1 is an illegal expression. He stated however that j²=-1 is a legal expression.

 

[albbg]

Re: Imaginary numbers - explanation needed

Yes because, in general, [sqrt(x)]² = ±x
in fact x² = (-x)²

then [sqrt(-1)]² = ±1

but the solution [sqrt(-1)]² = +1 is wrong since j²=-1 by definition.

The use of radicals requires particular care.

Regards

 

[jgruszynski]

Re: Imaginary numbers - explanation needed

The best tutorials I've seen are:

https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

https://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/

 

[Element_115]

Re: Imaginary numbers - explanation needed

I had a Prof. say that "j" (√-1) is just a function !....

The math teacher I had said that the expression j=√-1 is an illegal expression. He stated however that j²=-1 is a legal expression.

I sort of think of it like an atom with 1/2 Spin (As I understand it)

As it spins around once it's "NOT" there, but as it spins a second time it "IS" there.

"j" Does Not Exist, "j^2" Exists....

 

[Prototyp_V1.0]

Re: Imaginary numbers - explanation needed

I made a representation of an imaginary number by Geogebra.

Copy the code below, and paste it in a notepad window. Save it as an html file and run it from your web browser:

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<title>Depict sinus function from an imaginary number - GeoGebra Dynamic worksheet</title>
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<h2>Depict sinus function from an imaginary number</h2>



The blue cross represent ab imaginary number. Drag it around and watch the sine curve. The black sine curve is the sine function of the imaginary number.</p>


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Rotate the cross in a circle around origo. If you make a decent circle, you'll see the sine curve move along the x-axis.</p>


<span style="font-size:small">Prototyp V1.0, Created with [url="https://www.geogebra.org/"]GeoGebra[/url]</span></p></td></tr>
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