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Geek Culture / Complex numbers and greater than comparisons

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Neuro Fuzzy
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Posted: 18th Nov 2009 03:39
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So... i got into a little argument about my math teacher over whether you can say that one complex number is greater than another. I'm arguing that you can, and my teacher is arguing that it's impossible.

i=sqrt(-1)

Lets say you have this:
a+bi<=c

How to prove that a+bi<=c? Get rid of the i (take the eyes out first!)

bi<=c-a
i<=(c-a)/b
-1<=((c-a)/b)^2

so if -1 is less than (c-a)^2 over b^2, then the imaginary number is less than c.

Comparing two imaginary numbers; a+bi<=c+di, works out to this:
((a-c)/(d-b))^2<=-1

The thing is... i can't really see a reason why complex numbers should have special exceptions just because they're freaks , they're just another thing you can insert for x...

I've heard arguments that you can't compare them because the points lie on an complex plane, and comparing two imaginary numbers is like comparing (123,51) and (154,345), you can't really say which is greater. But a complex number is really just one number. You can't say x=(123,52) (at the same time), but you can say x=a+bi.

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demons breath
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Posted: 18th Nov 2009 04:16 Edited at: 18th Nov 2009 04:17
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You can't square something to find out if it's greater than something else.

For example

3>-10
3^2=9
(-10)^2=100
9<100
Therefore, by the logic you applied,
3<-10
Which is obviously not right

You can compare the modulus of two imaginary numbers though. That also works by a principle of squaring things and suchlike, but it doesn't show what is greater necessarily.

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Neuro Fuzzy
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Posted: 18th Nov 2009 04:24 Edited at: 18th Nov 2009 04:28
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Oh yeah... huh... i'm being retared...

Ignoring the invalidity of what i was doing, i shoulda noticed at
((a-c)/(d-b))^2<=-1

That can only be true if one of those values is imaginary, which would be able to be simplified in the first place, and would make this whole thing pointless.

okay, lesson learnt. a+bi is not less than four, 3 is in fact greater than negative ten, and 100 is not less than 9

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demons breath
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Posted: 18th Nov 2009 04:27
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Quote: "100 is not less than 9"

This point is especially important. NEVER FORGET IT!

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Neuro Fuzzy
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Posted: 18th Nov 2009 04:40
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oh wait, you can square an inequality, it's just that flipping the sign when multiplying by a negative is still a rule, and there is the plus or minus thing w/ the square root.

Ie
3>-10
3*3>-10*-10 (multiplying by -10 on one side)
9<100
+-3<+-10

possibilities:
-3>-10 ~~~~~~~~~ 3<10 ~~~~~~~~~~ 3>-10 ~~~~~~~~~~~ -3>10
of which 3 are true


Another way of phrasing my above equation:

(a-c)^2<=-(d-b)^2

which means (excluding imaginary numbers for a,b,c, and d) that a-c=0 and d-b=0, so a=c and d=b, so a+bi=c+di... so i guess it's really impossible to tell if one complex number is more than another...

I mean... i could use distance from the origin on the complex plain, but meh.

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Peter H
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Posted: 18th Nov 2009 06:24
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Poor imaginary numbers... really they are just misunderstood. They should only be called complex numbers

if you think about it negative numbers are "imaginary" too... i mean, nobody has "negative five apples". Sure you can "owe" someone five apples, but really that's an abstract concept. Which is all that complex numbers are.

so maybe negative numbers and "imaginary" numbers should both just be called "complex"

One man, one lawnmower, plenty of angry groundhogs.
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demons breath
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Posted: 18th Nov 2009 15:27
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Quote: "Poor imaginary numbers... really they are just misunderstood. They should only be called complex numbers "

But complex numbers and imaginary numbers are different things.
Imaginary numbers are multiples of i (root -1) and complex numbers are an element of the real plus an imaginary number, usually written a+bi (where a and be are real numbers)
Therefore a complex number includes an imaginary number, but they are not the same.

Quote: "if you think about it negative numbers are "imaginary" too"

Unfortunately not, if you look at my bank balance - I wish I could imagine that away...

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Seppuku Arts
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Posted: 18th Nov 2009 15:34
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Hear hear to imaginary bank balances! Next time my bank charges me for a bounce, I'll throw that into the hat.

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Libervurto
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Posted: 18th Nov 2009 19:17
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All numbers are abstract; what is the meaning of five? How are five cows the same as five eggs?

Are "imaginary"/complex numbers infinities?
If I have an infinite number of cows and an infinite number of eggs, but eggs are laid faster than cows born, do I have more eggs or cows?

All things in the universe can be explained in terms of eggs and cows.

"With games, we create these elaborate worlds in our minds, and the computer is there to do the bookkeeping." - Will Wright
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Diggsey
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Posted: 18th Nov 2009 19:45
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@Neuro Fuzzy
You can say one complex number is greater than another, if you take 'greater' to mean that the magnitude is greater. (Which is what is usually meant when dealing with vector quantities)


New DBPro IDE + integrated debugger!
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Libervurto
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Posted: 18th Nov 2009 19:54
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Quote: "You can say one complex number is greater than another, if you take 'greater' to mean that the magnitude is greater."

So is the magnitude of eggs greater than cows?

"With games, we create these elaborate worlds in our minds, and the computer is there to do the bookkeeping." - Will Wright
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Green Gandalf
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Posted: 18th Nov 2009 21:44
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Depends what you mean by "magnitude" of course. If you mean the intensity of the colour yellow then the answer might be yes - unless you're looking at a brown cow in bright sunlight.
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Diggsey
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Posted: 19th Nov 2009 00:20
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@OBese87
In mathematical terms, it's called the modulus. The modulus is the distance from the origin of the complex plane in which the complex number lies.


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Sigh
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Posted: 19th Nov 2009 00:23
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Quote: "If you mean the intensity of the colour yellow then the answer might be yes - unless you're looking at a brown cow in bright sunlight"

Bright enough to bake the cow perhaps? Mmmm, BBQ beef

Keep your Hope & Change to yourself, I choose Liberty! Stop by for a chat! [IXE]Nateholio on irc.maxgaming.net:6667 #GarageGames
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Neuro Fuzzy
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Posted: 19th Nov 2009 01:04 Edited at: 19th Nov 2009 01:10
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Quote: "Are "imaginary"/complex numbers infinities?"


i - an imaginary number - is the square root of negative one. Most of the time an imaginary number refers to ai, or a*sqrt(-1), where a is a real number.

A complex number is a real number plus an imaginary number.

i=sqrt(-1)

imaginary number:
10i
-4i
3/2i
xi

complex number:
a+bi
2+20i

etc.

Quote: "Depends what you mean by "magnitude" of course. If you mean the intensity of the colour yellow then the answer might be yes - unless you're looking at a brown cow in bright sunlight."


vector magnitude's the distance from the origin. sqrt(a^2+b^2) in a+bi. the problem with that is that there are a circle of solutions for any given distance. I was looking for just one point.

Quote: "So is the magnitude of eggs greater than cows?"


I'll just give an example of solving a problem using the definition, magnitude=sqrt(a^2+b^2)

lets say you have 3x+4i eggs, and 3+xi cows. In imagination land, there is a restriction on how many (how much?) cows and eggs you have; the magnitude of the eggs has to be less than twice that of the cows' magnitude. That gives us this inequality:

sqrt(9x^2+16)<2*sqrt(9+x^2)
so...
9x^2+16<36+4x^2
5x^2<20
x^2<4
x<2
x>-2

So x is inbetween 2 and -2, and all the farm enforcement officers in imagination land lived happily ever after. The end.

...or is it?

Quote: "All things in the universe can be explained in terms of eggs and cows."


measuring acceleration in eggs per square cow?
[href=http://www.google.com/search?q=number+of+horns+on+a+unicorn+acre+in+tea+spoons+per+light+year]
Also, did you know that there are 7.76750034 × 10^24 horns on a unicorn acre in US teaspoons per light year?[/href]

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