Quote:Original post by JimPrice Oh yeah, and wait till you get to Quaternions...
And just to let you know what you're in for with them, rumor has it they were invented by a drunk irishman as he was heading home from the pub (across a bridge, no less) [grin]
"Voilà! In view, a humble vaudevillian veteran, cast vicariously as both victim and villain by the vicissitudes of Fate. This visage, no mere veneer of vanity, is a vestige of the vox populi, now vacant, vanished. However, this valorous visitation of a bygone vexation stands vivified, and has vowed to vanquish these venal and virulent vermin vanguarding vice and vouchsafing the violently vicious and voracious violation of volition. The only verdict is vengeance; a vendetta held as a votive, not in vain, for the value and veracity of such shall one day vindicate the vigilant and the virtuous. Verily, this vichyssoise of verbiage veers most verbose, so let me simply add that it's my very good honor to meet you and you may call me V.".....V
Thanks for the encouraging words - it can feel like the world caving in around you when you can't understand something!
The funny thing, is - I can understand the Quarternion side of it, but its just this little piece of the puzzle that eludes me. Still, I shall keep at it. All tough nuts crack in the end!
After much sweat, blood and suger - I think I have not only quaternions, but the imaginary number sussed! I shall tell you my findings...
Just before dinner, I asked my father( he is really good at maths ) what the imaginary number is. He gave me a rather mystic idea - this what he said...
"Imagine I have this sphere in my hands and inside it is the square root of -1. With this value, I can now work out many other things about it, including the surface of this ball, the curve etc, etc..."
...I thought he was full of baloney just like the rest! I still didn't know what the square root of -1 was...grrrrr!
Get this - I was looking for a value. But it wasn't a value as such I was looking for...
I looked back at the form of a quaternion...
P = Pa + Pb*i + Pc*j + Pd*k
...in which I wondered how this was applied to our whole 3D pipeline. It goes something like this...
P = Pa + x*i + y*j + z*k
...so the x,y,z is pretty much our 3D values in question, and they are multiplied by their imaginary unit to make our "imaginary part", and those units are i,j,k( notice they resemble x,y,z ).
Now lets look at something else...
-4 = 4*-1
...and...
4 = 4*1
...and...
0 = 4*0
...I don't know about you - but I see a pattern here! Lets go back to our beloved quaternion and seperate "x*i" from the equation...
x*0 = 0
x*1 = x
x*-1 = -x
...and know let us look at an example identity matrix(its the only thing I can compare this to )...
1 0 0 0 1 0 0 0 1
...hmmm, interesting. And in a certain Tricks book, I found the author using this concept...
i = <1,0,0> j = <0,1,0> k = <0,0,1>
...At this stage, I'm a wolf that has caught my prey's scent. Lets try to unravel more of the puzzle( I'm hoping this is to the benefit of others stuggling with imaginary numbers! )...
i = <-1,0,0> j = <0,-1,0> k = <0,0,-1>
...AHA! So, if I make i = -1, I'm basically instructing the following...
P = Pa + x*-1 + y*j + z*k
...I think I am ready to spell it out now.
The imaginary number is an "instruction" more than it is a value. The value is simply what kind of instruction I want to have performed.
If I give the imaginary unit the value of 0, you will get...
x*0 = 0
...and as such, the value of 0 adds nothing to the final quaternion sum. In other words - "I do not want to perform an operation on x!". However...
x*1 = x
...I'm saying that I want x to be positive! And thus...
x*-1 = -x
...I would be saying that I want x to be negative. This whole thing also says that I want/don't want x to be included in the overall equation.
And thats it for now. I feel as thought I have caught on to the idea of what the use of an imaginary number is.
This also applies to the quaternion components, although in those cases they are vectors and not scalars.
So
q = w + x * i + y * j + z * k
i2 = -1 j2 = -1 k2 = -1
i * j = - j * i = k j * k = - k * j = i k * i = - i * k = j
These basis vectors have these special properties which leads to the whole quaternion algebra and calculus.
This is an abstract definition and trying to find physical meaning just isn't important. The question 'what *is* i' is almost meaningless (your father obviously gets it) - it is a definition which allows for solviing a whole range of neat problems which we couldn't before.
For instance, try and solve:
x2 + 2 = 0
This is a perfectly reasonable quadratic equation, very easily stated but it cannot be solved without some value for sqrt(-1). You basically have two choices,
a) Just say that the equation is undefined (i.e. there is no value of x for which the equation holds)
b) Extend the number system you pick x from.
People wrestled with this for a long time before 'inventing' i to stand for sqrt(-1). Unfortunately to properly grok it you really need to have some experience in abstract algebra systems, where you'll see that everything is just definitions and asking 'what *really* is this' becomes meaningless at best (paradoxical at worst ;).
If you want some 'physical' meaning to attribute to these concepts, look at 'Geometric Algebra' which unifies quaternions and complex numbers (and a huge number of other things).
Quote:Original post by JuNC If you want some 'physical' meaning to attribute to these concepts, look at 'Geometric Algebra' which unifies quaternions and complex numbers (and a huge number of other things).
Physical meaning of imaginary numbers can easily be seen if you deal with AC voltage calculations (ok maybe that's more of a practical application[smile]).
I'm now at the stage where I'm Gary Oldman from The Fifth Element - "HAHAHAHAHHAHAHAHAHAAAaaaamwmwmwmwmwmmw - they're not in here!"
Okay, you know how this is possible, but I see it in this way...the computer MUST have a value to work with. You simply cannot ask the computer x*i if i = ?. You cannot give it a "question mark".
That value is i. i is as much a number as 5 or 12, it is just not real. Working with complex numbers, you have i DEFINED as root -1, in much the same way that when you learn numbers you have 1 DEFINED as being a singular quantity or the first counting number. i is just another number.
