Now that's simple - imaginary numbers are composed of two part - a real part and an imaginary one.
You can write them as x = a + bi or simply as a pair (a,b).
Programs working with imaginary numbers only need to apply the set of rules defined for them, e.g. i2=-1.
Now if you have something like z = x + y it boils down to z = (xa + ixb) + (ya + iyb) = (xa + ya) + i(xb*yb).
Since all components are real numbers, the computer can easily deal with them. If you use the pair notation you don't even 'see' the i from the imaginary part.
Hope this helps,
Pat.
Can anyone confirm the Sqrt(-1)?
Author
Re: AABBFFLLUUGG.
So I would be correct to assume that "i" is not a variable, but a value?
So I would be correct to assume that "i" is not a variable, but a value?
Quote:Original post by Anri
Re: AABBFFLLUUGG.
So I would be correct to assume that "i" is not a variable, but a value?
Yes.
Quote:Original post by Anri
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".
You're absolutely right, and they do. Imaginary number are numbers that are the sqrt of a negative value. Let's say you wnat the sqrt(-16). Now, this is exactly the same ax (-16)^0.5, so we can break this up into 2 parts, (16^0.5)*((-1)^0.5), or alternatively sqrt(16)*sqrt(-1) = 4i. Doing things this way we can represent any imaginary number as a real number (that the computer *can* work with) multiplied by i (which to the computer is totally meaningless).
Complex numbers include a 'normal' real number with this, so a complex number can be represented as x+yi. Now, while the computer can't do jack with 'i', it *can* do anything it pleases with x and y. The fact that the i is there means there are certain rules that need to be followed to make the mathematical operations valid. For example, multipication. Say you want to find (1+2i)*(3+4i), you just multiply them together as you would normally with an unknown variable, so you get (3+10i+8i^2). However, since we know that i=sqrt(-1) then i^2=-1. So really the result is (-5+10i), which is again in a form that the computer can represent using x and y.
Now when coding this multiplication on computer, we don't try and plug in a value for i, we just apply some simple rules that get us the same results. Say you have 2 complex number on the computer, (x1=1,y1=2) and (x2=3,y2=4). To multiply them together we would calculate x3=x1*x2-y1*y2, y3=x1*y2+x2*y1.
Also, another way to help visualise complex numbers is to consider real numbers as the x axis and imaginary numbers as the y axis. This also allows you to represent a complex number as a magnitude and an angle, which is sometimes very useful (for example, in AC electricity as mentioned earlier).
Quote:Original post by joanusdmentiaQuote:Original post by Anri
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".
You're absolutely right, and they do. Imaginary number are numbers that are the sqrt of a negative value. Let's say you wnat the sqrt(-16). Now, this is exactly the same ax (-16)^0.5, so we can break this up into 2 parts, (16^0.5)*((-1)^0.5), or alternatively sqrt(16)*sqrt(-1) = 4i. Doing things this way we can represent any imaginary number as a real number (that the computer *can* work with) multiplied by i (which to the computer is totally meaningless).
Complex numbers include a 'normal' real number with this, so a complex number can be represented as x+yi. Now, while the computer can't do jack with 'i', it *can* do anything it pleases with x and y. The fact that the i is there means there are certain rules that need to be followed to make the mathematical operations valid. For example, multipication. Say you want to find (1+2i)*(3+4i), you just multiply them together as you would normally with an unknown variable, so you get (3+10i+8i^2). However, since we know that i=sqrt(-1) then i^2=-1. So really the result is (-5+10i), which is again in a form that the computer can represent using x and y.
Now when coding this multiplication on computer, we don't try and plug in a value for i, we just apply some simple rules that get us the same results. Say you have 2 complex number on the computer, (x1=1,y1=2) and (x2=3,y2=4). To multiply them together we would calculate x3=x1*x2-y1*y2, y3=x1*y2+x2*y1.
Erhh, you just repeated what I already wrote (just using more words[smile]).
Regards,
Pat.
Quote:
Original post by darookie
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).
Indeed, quantum mechanics too (don't get much more physical than that ;). My point was more that GA is fundamentally grounded in geometry which actually gives a way of visualizing i. I realize the standard Argand diagram gives a 'visual' appearance but that doesn't really explain what i is.
Quote:Original post by JuNCQuote:
Original post by darookie
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).
Indeed, quantum mechanics too (don't get much more physical than that ;). My point was more that GA is fundamentally grounded in geometry which actually gives a way of visualizing i.
Yes, I understand that.
Cheers,
Pat.
Quote:Original post by darookie
Erhh, you just repeated what I already wrote (just using more words[smile]).
Regards,
Pat.
Yeah, but mine looks more impressive.... [smile]
Seriously though, I was being lazy and just replied to his post without checking the couple below it. My bad!
Author
Hmmm. Okay, I'm starting to get the idea - i ain't a variable. Thats the hard part over with. Great.
Once I'm over this, I shall take a Holiday!
Cheers.
Once I'm over this, I shall take a Holiday!
Cheers.
Quote:Original post by Anri
Hmmm. Okay, I'm starting to get the idea - i ain't a variable. Thats the hard part over with. Great.
Once I'm over this, I shall take a Holiday!
Cheers.
Exactly. It's a symbolic representation of an impossible (yet useful) number. Well, impossible in the sense that you can't write it as x=1.234
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