This is a model that explain the interactions of the universe without gravity. This is not a decay model, It is an interactive information model. Inorder to realize a need for a new model requires disproving certain understandings that permiate cosmology. A field (E) cannot equal a scalar (mass) * a vector (C) ^2. It also describes baryonic expression of information. Comments are welcome, Anonymous comments with non-relevant links will not be posted.
So, the video makes a very basic mistake. Mathematicians do not define multiplication of complex numbers in the same way as multiplication of real numbers.
First, some definitions. Let me define ** to mean multiplication of complex numbers, in the same way that * is multiplication of "normal" real numbers, and let a complex number be defined as a + bi, where a is the real part, and i is the imaginary part.
With complex numbers, we define ** to mean (a + bi)**(c + di) -> (a*c - b*d) + (b*c + a*d)i. That shows how to convert a ** operation into a set of simpler * operations.
So complex numbers are more than just numbers, part of their definition includes how the basic arithmatical operations interact with them, and ** is the defined multiplication operation.
So back to the video, we can more fully describe i by the complex number 0 + 1i, since the real part is zero. Then we have:
Note that when we are using the complex ** operation, and not the normal * operation. Why? Because that is the only multiplication operation defined between complex numbers. i*i is not valid. And so we apply the above rule about how to evaluate **, and proceed to get -1, which is consistent.
The mistake made in the video is clear: they tried to multiply two complex numbers using the wrong rules of multiplication. Instead of using the multiplication operation defined for complex numbers (**), the used the operation defined for real numbers (*). Mathematicians would be happy to agree that that produces the wrong result.
So why is ** defined the way it is, and why is that "correct"?
Because we say so :)
That's actually true, in a way. We created them, and we're free to define how they work. We could have defined complex numbers to work an any number of ways, but the way described above is the one way that actually conforms to the algebraic laws that real numbers also conform to, so it's the one that is actually useful. The other possible definitions are certainly possible, but why would anyone use them if they don't have useful properties from which one could derive useful results?
And in fact, complex numbers are a little less arbitrarily defined than I make them out to be. It turns out you can derive the "right" implementation by starting with the algebraic laws that apply to real numbers, and use them to determine how complex numbers should behave in order to conform to those rules. A consistent definition would follow naturally from that process, and that's how we ended up with the particular definition of complex numbers we use today, and is how the ** operation was derived.
So, the video makes a very basic mistake. Mathematicians do not define multiplication of complex numbers in the same way as multiplication of real numbers.
First, some definitions. Let me define ** to mean multiplication of complex numbers, in the same way that * is multiplication of "normal" real numbers, and let a complex number be defined as a + bi, where a is the real part, and i is the imaginary part.
With complex numbers, we define ** to mean (a + bi)**(c + di) -> (a*c - b*d) + (b*c + a*d)i. That shows how to convert a ** operation into a set of simpler * operations.
So complex numbers are more than just numbers, part of their definition includes how the basic arithmatical operations interact with them, and ** is the defined multiplication operation.
So back to the video, we can more fully describe i by the complex number 0 + 1i, since the real part is zero. Then we have:
Note that when we are using the complex ** operation, and not the normal * operation. Why? Because that is the only multiplication operation defined between complex numbers. i*i is not valid. And so we apply the above rule about how to evaluate **, and proceed to get -1, which is consistent.
The mistake made in the video is clear: they tried to multiply two complex numbers using the wrong rules of multiplication. Instead of using the multiplication operation defined for complex numbers (**), the used the operation defined for real numbers (*). Mathematicians would be happy to agree that that produces the wrong result.
So why is ** defined the way it is, and why is that "correct"?
Because we say so :)
That's actually true, in a way. We created them, and we're free to define how they work. We could have defined complex numbers to work an any number of ways, but the way described above is the one way that actually conforms to the algebraic laws that real numbers also conform to, so it's the one that is actually useful. The other possible definitions are certainly possible, but why would anyone use them if they don't have useful properties from which one could derive useful results?
And in fact, complex numbers are a little less arbitrarily defined than I make them out to be. It turns out you can derive the "right" implementation by starting with the algebraic laws that apply to real numbers, and use them to determine how complex numbers should behave in order to conform to those rules. A consistent definition would follow naturally from that process, and that's how we ended up with the particular definition of complex numbers we use today, and is how the ** operation was derived.
I thought I would do a little complex number theory while I recover from a reboot.
I now understand how the ** is derived.
This problem stems from the real problem, x^2 + 1 = 0 x^2 = -1 x = sqrt(-1)
Using complex multiplication, I get a scalar + a vector or a+bi. Now I have a 2-dimensional problem. In some disciplines in particular, electrical engineering, where i is a symbol for current[1].
The reason I am playing with this is because I see this problem occurring in only one place in 1 dimensional Real number line. It is an asymptote. It produces a different result than the divide by zero asymptote. The result of this operation produces another number line with its own functions and sets.
It doesn't occur in just one place. x^2 + 2 = 0 is also imaginary, where x = sqrt(-2).
First, a + bi is just a notation. It's not a mixed-dimensional formula... it's just how a single complex number is written by convention. The "a" part is a single-dimensional quantity, and the "bi" part is a single-dimensional quantity.
And as to x^2 + 1 = 0, we're not mixing 2-dimensional complex numbers and 1-dimensional real numbers in some invalid way. Instead, the reals are to be considered complex numbers themselves, so x^2 + 1 = 0 involves only complex numbers. x = 0 + 1i, 2 = 2 + 0i, 1 = 1 + 0i, and 0 = 0 + 0i.
One can then think of all of the traditional equations involving only real numbers as mere simplifications of the same equations using complex numbers.
The key then is that you don't "mix" complex numbers and real numbers. You instead think of real numbers AS complex numbers. It just so happens that ** reduces to * when the imaginary part of the numbers is zero. So really, traditional non-complex math is actually complex math where the imaginary part is zero, and ** reduces to *.
In other words, all numbers are complex, and so there is no dimensionality problem.
Since it is just a notation system and Real numbers are just a subset of imaginary numbers, then technically I cannot see a solution to x^2+1=0. I just see the same problem resolved in a different notation system.
I understand why x=2,x=1 are complex. I understand why x=0 is real which is a subset of complex.
That is not to say I do not understand what you are saying, I do. I understand that these tools can be used to create new dimensions to formulate theories like String Theory and other multi-dimensional theories. This is why I am interested in this topic. I want to see how they developed this multi-dimensional system.
See my first comment, which included the full evaluation. The key is that i^2 is the same as i**i. i^2 is NOT the same as i*i. When you use the correct multiplication operation, you will get -1 every time.
Going back to the video, the step that failed was saying that sqrt(-1)sqrt(-1) = sqrt((-1)(-1)). You can't do that. The reason is that it reall means sqrt(-1)**sqrt(-1) = sqrt((-1)*-1)), which mixes up ** and *. It's not valid math. You can not derive 1 by starting with i^2 for that reason.
Aaron said: "...technically I cannot see a solution to x^2+1=0. I just see the same problem resolved in a different notation system."
Yes, a+ib is just a notation. But when solving x^2+1=0, we're not just "in a different notation system" but doing the same things. Instead, we have entirely new rules, which is beyond a mere change of notation.
So I argue that it does have a solution. Let's try.
rst, let's just assume we're dealing with real numbers. Then:
x^2 + 1 = 0 => x^2 = -1 => x = sqrt(-1)
And we're screwed. sqrt(-1) isn't a real number, so the equation has no (real) solution. If we didn't know anything about complex numbers, we'd just throw our hands up in the air at this point and say that algebra has holes.
But the mathematician looks at this and says: hmmm... if the solution to an equation involving real numbers isn't itself real... what is it? Is there a broader class of numbers? Have we been wrong in thinking the reals were all there is up until now? After all, it wasn't that long ago that the nobody thought the reals existed. And before that, fractions. And before that, negative numbers. And before that, zero.
So we use the algebraic axioms and derive complex numbers to conform to them, and thus have a new tool. So let's try again, and use ALL numbers, rather than just real numbers. We were given this:
So now we see that (-1 + 0i) = (0 + 1i)^2, and thus sqrt(-1 + 0i) = (0 + 1i), and thus, going back to our calculation:
=> x = (0 + 1i).
And that is a complex solution of what we now realize was a complex equation.
Key takeaway: Real numbers ARE complex numbers. It's just that they're not ALL of them.
I realize I'm repeating myself a little, but when you say "i^2=-1 still resolves to 1" and "I cannot see a solution to x^2+1=0", I can't help it. i^2 never resolves to 1, and x^2+1=0 has a solution. Hopefully, this comment helps clarify that.
For some reason I was thinking about your comment about "3 strange asymptotes', which I didn't understand, and realized it's pretty easy to plot this function.
So here it is: http://tinyurl.com/2dh4d9g
So you had x^2 + 1 = 0 => x^2 = sqrt(-1) as your equation, but the 1 could have been anything, so this plots all complex solutions x = sqrt(y).
The horizontal axis plots values of y, and the vertical axis the resulting complex value of x. The blue plot is the real part of the solution, and the red line is the imaginary part of the complex answer.
The second graph actually draws the same plot in complex space. So the vertical axis is the real part of the solution, and the depth axis is the imaginary part of the solution, and the horizontal axis is the value of y. That graph is a little hard to see, but basically the negative portion of the graph is flat on the imaginary plane, and the positive portion of the graph is flat on the real plane.
What's cool here is that you could think of another set of reals that exactly mirror the reals we know about. There are just as many of them, they follow the same rules, the two sets are mutually exclusive, any function in one could be expressed in the other, and they're both subsets of complex numbers. What you see here are those two sets of "reals", the ones we're familiar with, and their mirror image.
Rather remarkable.
In any case, I bring this up because you mentioned "3 strange asymptotes" on the real line. Note that there are infinite non-solutions as opposed to 3, just as there are infinite solutions. And in the complex space, as might be clear from either plot, ever point along the horizontal axis is associated with a solution... there are no holes.
So, the video makes a very basic mistake. Mathematicians do not define multiplication of complex numbers in the same way as multiplication of real numbers.
ReplyDeleteFirst, some definitions. Let me define ** to mean multiplication of complex numbers, in the same way that * is multiplication of "normal" real numbers, and let a complex number be defined as a + bi, where a is the real part, and i is the imaginary part.
With complex numbers, we define ** to mean (a + bi)**(c + di) -> (a*c - b*d) + (b*c + a*d)i. That shows how to convert a ** operation into a set of simpler * operations.
So complex numbers are more than just numbers, part of their definition includes how the basic arithmatical operations interact with them, and ** is the defined multiplication operation.
So back to the video, we can more fully describe i by the complex number 0 + 1i, since the real part is zero. Then we have:
-1 = i^2 = i**i = sqrt(-1)**sqrt(-1) = (0 + 1i)**(0 + 1i) = (0*0 - 1*1) + (1*0 + 0*1)i = -1 + 0i = -1
Note that when we are using the complex ** operation, and not the normal * operation. Why? Because that is the only multiplication operation defined between complex numbers. i*i is not valid. And so we apply the above rule about how to evaluate **, and proceed to get -1, which is consistent.
The mistake made in the video is clear: they tried to multiply two complex numbers using the wrong rules of multiplication. Instead of using the multiplication operation defined for complex numbers (**), the used the operation defined for real numbers (*). Mathematicians would be happy to agree that that produces the wrong result.
So why is ** defined the way it is, and why is that "correct"?
Because we say so :)
That's actually true, in a way. We created them, and we're free to define how they work. We could have defined complex numbers to work an any number of ways, but the way described above is the one way that actually conforms to the algebraic laws that real numbers also conform to, so it's the one that is actually useful. The other possible definitions are certainly possible, but why would anyone use them if they don't have useful properties from which one could derive useful results?
And in fact, complex numbers are a little less arbitrarily defined than I make them out to be. It turns out you can derive the "right" implementation by starting with the algebraic laws that apply to real numbers, and use them to determine how complex numbers should behave in order to conform to those rules. A consistent definition would follow naturally from that process, and that's how we ended up with the particular definition of complex numbers we use today, and is how the ** operation was derived.
Hope this helps.
So, the video makes a very basic mistake. Mathematicians do not define multiplication of complex numbers in the same way as multiplication of real numbers.
ReplyDeleteFirst, some definitions. Let me define ** to mean multiplication of complex numbers, in the same way that * is multiplication of "normal" real numbers, and let a complex number be defined as a + bi, where a is the real part, and i is the imaginary part.
With complex numbers, we define ** to mean (a + bi)**(c + di) -> (a*c - b*d) + (b*c + a*d)i. That shows how to convert a ** operation into a set of simpler * operations.
So complex numbers are more than just numbers, part of their definition includes how the basic arithmatical operations interact with them, and ** is the defined multiplication operation.
So back to the video, we can more fully describe i by the complex number 0 + 1i, since the real part is zero. Then we have:
-1 = i^2 = i**i = sqrt(-1)**sqrt(-1) = (0 + 1i)**(0 + 1i) = (0*0 - 1*1) + (1*0 + 0*1)i = -1 + 0i = -1
Note that when we are using the complex ** operation, and not the normal * operation. Why? Because that is the only multiplication operation defined between complex numbers. i*i is not valid. And so we apply the above rule about how to evaluate **, and proceed to get -1, which is consistent.
The mistake made in the video is clear: they tried to multiply two complex numbers using the wrong rules of multiplication. Instead of using the multiplication operation defined for complex numbers (**), the used the operation defined for real numbers (*). Mathematicians would be happy to agree that that produces the wrong result.
So why is ** defined the way it is, and why is that "correct"?
ReplyDeleteBecause we say so :)
That's actually true, in a way. We created them, and we're free to define how they work. We could have defined complex numbers to work an any number of ways, but the way described above is the one way that actually conforms to the algebraic laws that real numbers also conform to, so it's the one that is actually useful. The other possible definitions are certainly possible, but why would anyone use them if they don't have useful properties from which one could derive useful results?
And in fact, complex numbers are a little less arbitrarily defined than I make them out to be. It turns out you can derive the "right" implementation by starting with the algebraic laws that apply to real numbers, and use them to determine how complex numbers should behave in order to conform to those rules. A consistent definition would follow naturally from that process, and that's how we ended up with the particular definition of complex numbers we use today, and is how the ** operation was derived.
Hope this helps.
I thought I would do a little complex number theory while I recover from a reboot.
ReplyDeleteI now understand how the ** is derived.
This problem stems from the real problem,
x^2 + 1 = 0
x^2 = -1
x = sqrt(-1)
Using complex multiplication, I get a scalar + a vector or a+bi. Now I have a 2-dimensional problem. In some disciplines in particular, electrical engineering, where i is a symbol for current[1].
The reason I am playing with this is because I see this problem occurring in only one place in 1 dimensional Real number line. It is an asymptote. It produces a different result than the divide by zero asymptote. The result of this operation produces another number line with its own functions and sets.
I'll be back.
[1]http://en.wikipedia.org/wiki/Complex_number
It doesn't occur in just one place. x^2 + 2 = 0 is also imaginary, where x = sqrt(-2).
ReplyDeleteFirst, a + bi is just a notation. It's not a mixed-dimensional formula... it's just how a single complex number is written by convention. The "a" part is a single-dimensional quantity, and the "bi" part is a single-dimensional quantity.
And as to x^2 + 1 = 0, we're not mixing 2-dimensional complex numbers and 1-dimensional real numbers in some invalid way. Instead, the reals are to be considered complex numbers themselves, so x^2 + 1 = 0 involves only complex numbers. x = 0 + 1i, 2 = 2 + 0i, 1 = 1 + 0i, and 0 = 0 + 0i.
One can then think of all of the traditional equations involving only real numbers as mere simplifications of the same equations using complex numbers.
The key then is that you don't "mix" complex numbers and real numbers. You instead think of real numbers AS complex numbers. It just so happens that ** reduces to * when the imaginary part of the numbers is zero. So really, traditional non-complex math is actually complex math where the imaginary part is zero, and ** reduces to *.
In other words, all numbers are complex, and so there is no dimensionality problem.
First, Thanks for the conversation.
ReplyDeleteSince it is just a notation system and Real numbers are just a subset of imaginary numbers, then technically I cannot see a solution to x^2+1=0. I just see the same problem resolved in a different notation system.
I understand why x=2,x=1 are complex. I understand why x=0 is real which is a subset of complex.
That is not to say I do not understand what you are saying, I do. I understand that these tools can be used to create new dimensions to formulate theories like String Theory and other multi-dimensional theories. This is why I am interested in this topic. I want to see how they developed this multi-dimensional system.
i^2=-1 still resolves to 1.
I have to stop now! Ill be back.
Thanks
Aaron
Aaron said: "i^2=-1 still resolves to 1."
ReplyDeleteSee my first comment, which included the full evaluation. The key is that i^2 is the same as i**i. i^2 is NOT the same as i*i. When you use the correct multiplication operation, you will get -1 every time.
Going back to the video, the step that failed was saying that sqrt(-1)sqrt(-1) = sqrt((-1)(-1)). You can't do that. The reason is that it reall means sqrt(-1)**sqrt(-1) = sqrt((-1)*-1)), which mixes up ** and *. It's not valid math. You can not derive 1 by starting with i^2 for that reason.
Aaron said: "...technically I cannot see a solution to x^2+1=0. I just see the same problem resolved in a different notation system."
Yes, a+ib is just a notation. But when solving x^2+1=0, we're not just "in a different notation system" but doing the same things. Instead, we have entirely new rules, which is beyond a mere change of notation.
So I argue that it does have a solution. Let's try.
rst, let's just assume we're dealing with real numbers. Then:
ReplyDeletex^2 + 1 = 0
=> x^2 = -1
=> x = sqrt(-1)
And we're screwed. sqrt(-1) isn't a real number, so the equation has no (real) solution. If we didn't know anything about complex numbers, we'd just throw our hands up in the air at this point and say that algebra has holes.
But the mathematician looks at this and says: hmmm... if the solution to an equation involving real numbers isn't itself real... what is it? Is there a broader class of numbers? Have we been wrong in thinking the reals were all there is up until now? After all, it wasn't that long ago that the nobody thought the reals existed. And before that, fractions. And before that, negative numbers. And before that, zero.
So we use the algebraic axioms and derive complex numbers to conform to them, and thus have a new tool. So let's try again, and use ALL numbers, rather than just real numbers. We were given this:
x^2 + 1 = 0
=> x * x + 1 = 0
First, we generalize to complex numbers, to get:
ReplyDelete=> x ** x + (1 + 0i) = (0 + 0i)
where we note that we didn't just change notation... we introduced the ** operation as the generalization of *. We continue:
=> x ** x = (0 + 0i) - (1 + 0i)
=> x ** x = ((0 - 1) + (0 - 0)i) = (-1 + 0i)
=> x = sqrt(-1 + 0i)
The last line can reduced by observing that:
(0 + 1i) ** (0 + 1i) = (0*0 - 1*1) + (0*1 - 1*0)i = -1 + 0i
So now we see that (-1 + 0i) = (0 + 1i)^2, and thus sqrt(-1 + 0i) = (0 + 1i), and thus, going back to our calculation:
=> x = (0 + 1i).
And that is a complex solution of what we now realize was a complex equation.
Key takeaway: Real numbers ARE complex numbers. It's just that they're not ALL of them.
I realize I'm repeating myself a little, but when you say "i^2=-1 still resolves to 1" and "I cannot see a solution to x^2+1=0", I can't help it. i^2 never resolves to 1, and x^2+1=0 has a solution. Hopefully, this comment helps clarify that.
Thanks for listening to me. Keep blogging.
Thanks again.
ReplyDeleteI am going to take the night off.
Aaron
Hi, I think I fixed the posting. Thanks again for the conversation. I am recovering from a difficult seizure.
ReplyDeleteI know I have more questions about this, but I doubt I can formulate the questions.
I am just going to sit and think about the 3 strange asymptotes on the Real Number line.
Aaron
For some reason I was thinking about your comment about "3 strange asymptotes', which I didn't understand, and realized it's pretty easy to plot this function.
ReplyDeleteSo here it is: http://tinyurl.com/2dh4d9g
So you had x^2 + 1 = 0 => x^2 = sqrt(-1) as your equation, but the 1 could have been anything, so this plots all complex solutions x = sqrt(y).
The horizontal axis plots values of y, and the vertical axis the resulting complex value of x. The blue plot is the real part of the solution, and the red line is the imaginary part of the complex answer.
The second graph actually draws the same plot in complex space. So the vertical axis is the real part of the solution, and the depth axis is the imaginary part of the solution, and the horizontal axis is the value of y. That graph is a little hard to see, but basically the negative portion of the graph is flat on the imaginary plane, and the positive portion of the graph is flat on the real plane.
What's cool here is that you could think of another set of reals that exactly mirror the reals we know about. There are just as many of them, they follow the same rules, the two sets are mutually exclusive, any function in one could be expressed in the other, and they're both subsets of complex numbers. What you see here are those two sets of "reals", the ones we're familiar with, and their mirror image.
Rather remarkable.
In any case, I bring this up because you mentioned "3 strange asymptotes" on the real line. Note that there are infinite non-solutions as opposed to 3, just as there are infinite solutions. And in the complex space, as might be clear from either plot, ever point along the horizontal axis is associated with a solution... there are no holes.
Oops... my last attempt at rendering it cut off part of the right sides of the graphs. No biggie... you can still see enough of it.
ReplyDeleteI had a pretty horrible seizure today. Ended up in the ER.
ReplyDeleteI intend to discuss the asymptotes in my next new posting.
Dear Anon,
ReplyDeleteI am working with Google to fix some of these posting issues. They are very helpful. I should see a fix soon.
Thanks
Aaron
Interesting!
ReplyDelete