Imaginary Numbers

reply to post by Phage

But what does it have to do with imaginary numbers which do not represent reality in and of themselves?

lol I don't know we've moved so far away from the original line of discussion that I've lost my train of thought. I'm just pointing out that many mathematical concepts have a real world basis and perhaps imaginary numbers might too. I've said before that science is merely the process of finding which equations fit the reality we live in, and if imaginary numbers are so crucial to our physics equations than it should say something to us about the nature of reality.

reply to post by ChaoticOrder

I'm just pointing out that many mathematical concepts have a real world basis and perhaps imaginary numbers might too.

There is no square root of negative one. It's just a trick that happens to make some mathematics work.

Math is not real. Reality is real. While mathematics can provide a representation of some aspects of reality, it is not real.

Mathematics is a creation of the human mind. It is a tool which can be used to help to understand reality by placing it in a context which the human mind can grasp...barely.

Originally posted by ChaoticOrder
reply to post by ImaFungi

 

So couldn't an irrational number represent the distance between us by using some very tiny incremental scale of measurement?

Well the smallest level of measurement you could probably get down to is the planck scale, so you wouldn't need an infinite amount of precision, so the result would never be an irrational number.

But would it be possible to measure that distance exactly ...............now. Or is there an exact finite integer amount of planck lengths between us per smallest unit of planck time?


can the electron be represented in space by an integer (is that why its said to be a point particle, or is that just out of convenience for calculations? Is a wave of energy/wave function represented by integers?

reply to post by Phage

There is no square root of negative one. It's just a trick that happens to make some mathematics work.

Correction: there is no square root of negative one on the real number line. I do agree with what you are saying to a certain degree, and I used to lean much more towards your way of thinking. But over time it has slowly become more and more clear to me that the distinction between reality and numbers isn't really quite as large as you might believe it to be.

reply to post by Bob Sholtz

the biggest problem i have with it is that an infinite quantity cannot exist within time

What if time is infinite? I do find it hard to consider a time when time did not exist. Since time does not have a beginning, therefore time does not have an end.

As for irrational numbers debate, what happens to the imaginary sign as it passes through powers of them and where exactly does the sign move? For example i^0.9 or i^1.1

reply to post by ChaoticOrder

Are you a mathematician?
What has led you to this conclusion?

Originally posted by Phage
reply to post by ChaoticOrder

 

Are you a mathematician?
What has led you to this conclusion?

A mathematician? Hardly... in fact I'm not even really that great at math. I am a programmer actually but I do spend a lot of time learning about theoretical physics just because it fascinates me. The only thing which has led me to my current position on the subject is my personal research, and my reasoning has been provided in this thread.

reply to post by ChaoticOrder

and my reasoning has been provided in this thread.

I see. That's the problem then. I can't follow it to the same conclusion you reach.

Question: How you express i in floating point arithmetic?

reply to post by kwakakev

What if time is infinite? I do find it hard to consider a time when time did not exist. Since time does not have a beginning, therefore time does not have an end. As for irrational numbers debate, what happens to the imaginary sign as it passes through powers of them and where exactly does the sign move? For example i^0.9 or i^1.1

even if time were infinite (i don't believe it is, and i have several good reasons for thinking so), an infinite quantity couldn't exist within time.

you're using a bit of circular reasoning, assuming that time didn't have a beginning, then extrapolating that to it never ending, then using this as evidence of time being infinite. in essence, you assume it is infinite, therefore it is infinite.

as for "i^1.1" etc...it is fully simplified in that state. you could represent it a different way, but it would possess the same value (it is self-defining), and without context it is largely meaningless.

reply to post by Phage

Question: How you express i in floating point arithmetic?

I never claimed to understand the true nature of imaginary numbers or how they can be expressed and operated on. That is the point of this thread, to try and understand the underlying rules of imaginary numbers.

However, as I have already mentioned, I think it's possible that they can be represented like any other number with the only difference being that they don't behave the same way as normal numbers when operated on.

So for example 3.5i can simply be treated as the imaginary version of 3.5. Following the sign shifting rules demonstrated when we take i to the power of n, we can solve the following equations like so:

1.5 x 1.5i = 2.25i
-1.5 x 1.5i = -2.25i
1.5i x 1.5i = -2.25
-1.5i x 1.5i = 2.25

So from those examples it becomes obvious that if you were to ask what is the square root of -9 all you need to do is find the square root of positive 9 and than convert it to the imaginary representation of the same number, which would be 3i, and that is the correct answer to the square root of -9 if you solve it algebraically.

However I haven't properly tested this idea to make sure it works in all cases and I certainly don't have the ability to prove it with some sort of mathematical proof (assuming that hasn't already been done). But if what I just said is a valid way of interpreting and even visualizing imaginary numbers, it certainly makes everything much easier.

EDIT: and of course we could replace the i with a squiggle above the number to indicate it's an imaginary number, but there really is no point because using i next to the number can achieve the same thing and it's already in wide spread use. The great thing is that nothing needs to change in the text books to make this work.

1.5 x 1.5i = 2.25i
-1.5 x 1.5i = -2.25i
1.5i x 1.5i = -2.25
-1.5i x 1.5i = 2.25

You know what's really ironic about this though, assuming it's true. What happens if you ask what is the square root of 1.5i? There is no number on the real or imaginary number line which when multiplied by its self equals an imaginary number. I can see where this is going...

There'll be problems mixing up real & imaginary numbers that way (multiplying values on different axes)

My use of imaginary numbers is simply to define vectors in a 2D space of 4 quadrants, the quadrants being significant in determining the sign of trig functions used to convert the X & Y values to a vector and angle. Simple early school pythagorean trigonometry is all that's required.

Eg a point in that space can be defined as 1.5+j1.5 and the length of that vector is sqrt(1.5^2+1.5^2)=2.12 from the 0,0 origin, the angle being tan^-1(1.5/1.5)=45 degrees.

reply to post by ChaoticOrder

sorry i fail miserably. you would have to designate another symbol to represent a "meta-i"

reply to post by Pilgrum

My use of imaginary numbers is simply to define vectors in a 2D space of 4 quadrants

Yes those vectors represent complex numbers, but again I must stress the difference between imaginary numbers and complex numbers. It's important that you understand the difference between them.

Originally posted by Bob Sholtz
reply to post by ChaoticOrder

 

you would have to designate another symbol to represent a "meta-i"

I wouldn't really agree because i is the same thing as 1i and 1i would be the imaginary version of 1 in this system. Let me show you why this makes sense:

√1 = 1
√-1 = 1i

We must remember that the square root of 1 is still 1. Thus the square root of -1 is imaginary 1 or 1i. The expression 3i is the same as saying 3 imaginary 1's. The only difference with imaginary numbers is that the operators don't apply in the same way, so when you do 1i^2 or 1i^3 you get behavior which is very different to doing 1^2 or 1^3. But note how the result of 1i^n is always 1 in some form, whether negative or positive, real or imaginary.

Originally posted by ChaoticOrder
reply to post by micpsi

 

Imaginary numbers are for children.
There are only four possible division algebras: real numbers(1-dimensional), complex numbers (2-dimensional); quaternions (4-dimensional) and octonions (8-dimensional).

The key point you are overlooking is that imaginary numbers are different from all of those. Tell me, how many dimensions does an imaginary number have?

No, they are not. Imaginary numbers are merely a special case of 2-dimensional complex numbers:
N = X + Yi
where X = 0.
Complex numbers are generalised to 4-dimensional quaternions:
N = a + bi + cj + dk
(i^2 = j^2 = k^2 = -1)
and quaternions are generalised to 8-dimensional octonions:
N = ajej
(e0 = 1, ek^2 = -1 (k=1-7) (letters j =0-7 denote indices).


I am not overlooking ANYTHING. It is you and others here who are overlooking the bigger, mathematical picture.

you could simply have:

1.2247√i

but √i has two values, they both work obviously, but i was thinking that another symbol would be required to denote whether one was referring to the positive or negative value for the sake of graphing

(1/√2)(1+i) and (-1/√2)(1+i)

nevermind. i see my error, it was in neglecting proper distribution.

perhaps i can salvage something of my stupidity. i'm working on something, i'll post it in a bit because i have several things to do.

reply to post by micpsi

I am not overlooking ANYTHING. It is you and others here who are overlooking the bigger, mathematical picture.

What ever bud. You keep on believing what ever it is you want to believe.

1.5i x 1.5i = -2.25

And now let me prove that this statement works algebraically:

1.5i² = 1.5i x 1.5i

We can expand 1.5i to (1.5 x 1i) as we normally would:

1.5i² = (1.5 x 1i) x (1.5 x 1i)

Since we only have multiplication operations this is easy to simplify:

1.5i² = 2.25 x 1i²

And we know that 1i² (or just i²) is equal to -1 so it becomes:

1.5i² = 2.25 x -1

Thus:

1.5i² = -2.25

Applying a radical to both sides we can drop the squared operation on the left side:

1.5i = √-2.25

Proving that the square root of -2.25 is equal to 1.5i.

reply to post by Bob Sholtz


I know the actuality of infinity has not been proven either way, I just provided some of the axioms I use to consider the possibility.

As for i^1.1, are you saying that by being context specific that it could result in a positive, negative or imaginary number? Just depends on what troubles the algorithm is having and what needs tweaking to get consistent and reliable results?