## |i|

For the discussion of math. Duh.

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cjdrum
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### |i|

I was thinking about maths for some reason imaginary numbers just the other day, while doing Absolute Values in class.

Now, i being sqrt(-1) and imaginary, how do you take the absolute value of it? Is it possible, or does the universe implode when somebody figures it out?

jestingrabbit
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### Re: |i|

cjdrum wrote:I was thinking about maths for some reason imaginary numbers just the other day, while doing Absolute Values in class.

Now, i being sqrt(-1) and imaginary, how do you take the absolute value of it? Is it possible, or does the universe implode when somebody figures it out?

In the set of complex numbers, we don't talk about absolute value, we talk about modulus. Its defined as [imath]|a + ib|= \sqrt{a^2 + b^2}[/imath], so |i| = 1.

edit: fixed for missing a square root.
Last edited by jestingrabbit on Sat Feb 19, 2011 12:24 am UTC, edited 1 time in total.
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Talith
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### Re: |i|

If you've studied the complex plane yet (some people call it the argand plane), then the modulus of a complex number is the distance away from 0 that the number is (distance in the standard euclidean sense that you're probably used to). This is just a more geometrical way of looking at what Jestingrabbit's already said and it really helped me picture this stuff a lot better when I was first learning it.

Meem1029
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### Re: |i|

Just another comment on a connection with that that really helped me get both complex modulus and absolute value: You can think of absolute value in the real numbers as the distance from 0. It's the same in the complex plane.
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Eebster the Great
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### Re: |i|

jestingrabbit wrote:In the set of complex numbers, we don't talk about absolute value, we talk about modulus.

Well, absolute value, complex modulus, and Euclidean norm are all closely related concepts. I have heard the modulus called the "absolute value."

the tree
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### Re: |i|

Meem1029 wrote: You can think of absolute value in the real numbers as the distance from 0. It's the same in the complex plane.
I always like to go for "how 'big' a number is", which if you're thinking of complex numbers as vectors then that works quite neatly.

silverhammermba
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### Re: |i|

the tree wrote:
Meem1029 wrote: You can think of absolute value in the real numbers as the distance from 0. It's the same in the complex plane.
I always like to go for "how 'big' a number is", which if you're thinking of complex numbers as vectors then that works quite neatly.

Except that the imaginary numbers are not linearly ordered like the real numbers are - so "bigness" is not intuitively defined.

Which is bigger: 2358 or 7809? The answer is intuitive and obvious.

Which is bigger: 4525+682i or 3826-2643i? Not so obvious, and it isn't intuitive what the question is even asking.

Thinking about | | as being "distance from 0" is more general and correct.

z4lis
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### Re: |i|

silverhammermba wrote:Which is bigger: 2358 or 7809? The answer is intuitive and obvious.

Which is bigger: 4525+682i or 3826-2643i? Not so obvious, and it isn't intuitive what the question is even asking.

Thinking about | | as being "distance from 0" is more general and correct.

Eh, if you were talking to me about complex numbers and mentioned that you wished to "let z get large", I would assume you meant "let |z| go to infinity". Your example also seems to play off the fact that the idea that choosing the larger out of 4525+682i or 3826-2643i isn't intuitively obvious, but that's not because there's not a good notion of size. Rather, it's because computing it isn't easy to do mentally, while we have nice tricks to look at the decimal representations of two reals and figure out which goes on which side of the inequality. At any rate, It feels like we're just arguing semantics here, and I would hate to think that's what I'm doing.
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Eebster the Great
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### Re: |i|

On the other hand, if you have ℝ x I in the order topology, 2 < i, while if you have I x ℝ in the order topology, i < 2. So it isn't always clear what "bigger" means.

But if you take the quotient ℂ / ~ where ℂ is the complex numbers in the topology where r1 eiθ1 > r2 eiθ2 if r1 > r2 or r1 = r2 and θ1 > θ2, and ~ is defined such that z1 ~ z2 if |z1| = |z2|, then that quotient space will give you something like what you want.

I'm pretty sure that's just homeomorphic to ℝ, though.

jestingrabbit
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### Re: |i|

z4lis wrote:Eh, if you were talking to me about complex numbers and mentioned that you wished to "let z get large", I would assume you meant "let |z| go to infinity".

That's fine once you've decided that the size of a complex number is its modulus, rather than the absolute value of its real part (as we might sensibly do with dual numbers) or something stranger still (in the split complex numbers for instance).

You're basically saying that you've included "modulus = size" in your intuition, but its not obvious to someone who's just coming across this stuff that that is how to go, especially when they haven't seen the definition of modulus yet.
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Eebster the Great
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### Re: |i|

^For an intuitive notion of "size" that does not match "modulus," consider the 1-norm |z| = Re(z) + Im(z).

the tree
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### Re: |i|

silverhammermba wrote:
the tree wrote:
Meem1029 wrote: You can think of absolute value in the real numbers as the distance from 0. It's the same in the complex plane.
I always like to go for "how 'big' a number is", which if you're thinking of complex numbers as vectors then that works quite neatly.

Except that the imaginary numbers are not linearly ordered like the real numbers are - so "bigness" is not intuitively defined.
Like I said, it works if you're willing to see complex numbers as vectors. So when a+ib is represented as (a,b)T, then the modulus is quite clearly the same as the length of that vector.

I remember at A-level I was really irritated by some algorithm that in the textbook required the "smallest negative number" which I eventually scribbled out and replaced with "most negative" because intuitively to me, what it said was more or less the opposite to what it meant.

Kurushimi
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### Re: |i|

silverhammermba wrote:Except that the imaginary numbers are not linearly ordered like the real numbers are - so "bigness" is not intuitively defined.

Which is bigger: 2358 or 7809? The answer is intuitive and obvious.

Which is bigger: 4525+682i or 3826-2643i? Not so obvious, and it isn't intuitive what the question is even asking.

Thinking about | | as being "distance from 0" is more general and correct.

I think you're mixing up size and order. Just because you can't order them like real numbers doesn't mean you can't associate a size with it. Two different numbers can have the same size. Complex numbers have size like houses have size. Houses have size, but I also can't think of a meaningful way to order houses.

Eebster the Great
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### Re: |i|

Kurushimi wrote:I think you're mixing up size and order. Just because you can't order them like real numbers doesn't mean you can't associate a size with it. Two different numbers can have the same size. Complex numbers have size like houses have size. Houses have size, but I also can't think of a meaningful way to order houses.

Wherever there are things of different sizes, you can have a semiorder of size. That's pretty obvious. You have a > b iff a is bigger than b. You might not have a total order, admittedly, but it's silly to say the two aren't related.

Yakk
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### Re: |i|

A common generalization of |.| is a norm. A common generalization of a norm is a metric.

You can think of |x| as being distance(x, zero) -- and distance(x,y) == |x-y|.

Now what |i| is is pretty obvious.

Often norms are denoted as || x || instead of one set of bars, as absolute value has a certain primogeniture. Sometimes you do things like || x ||t to denote one kind of norm.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Kurushimi
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### Re: |i|

Eebster the Great wrote:
Kurushimi wrote:I think you're mixing up size and order. Just because you can't order them like real numbers doesn't mean you can't associate a size with it. Two different numbers can have the same size. Complex numbers have size like houses have size. Houses have size, but I also can't think of a meaningful way to order houses.

Wherever there are things of different sizes, you can have a semiorder of size. That's pretty obvious. You have a > b iff a is bigger than b. You might not have a total order, admittedly, but it's silly to say the two aren't related.

Well, of course there can be a partial order. When I and, I think, silverhammermamba said "order" we meant "total order". I was just saying you can still have bigness without total order.

gorcee
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### Re: |i|

Yakk wrote:A common generalization of |.| is a norm. A common generalization of a norm is a metric.

You can think of |x| as being distance(x, zero) -- and distance(x,y) == |x-y|.

Now what |i| is is pretty obvious.

Often norms are denoted as || x || instead of one set of bars, as absolute value has a certain primogeniture. Sometimes you do things like || x ||t to denote one kind of norm.

There is, unfortunately, somewhat of a conflation of notation and nomenclature based on the varying levels and branches of mathematics in which they appear.

For complex numbers, if we want to talk about the "size" of a complex number, perhaps it is easiest to avoid norms and vertical bars altogether, and instead just write [imath]\left(z\bar{z}\right)^{1/2}[/imath].

Eebster the Great
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### Re: |i|

Kurushimi wrote:
Eebster the Great wrote:
Kurushimi wrote:I think you're mixing up size and order. Just because you can't order them like real numbers doesn't mean you can't associate a size with it. Two different numbers can have the same size. Complex numbers have size like houses have size. Houses have size, but I also can't think of a meaningful way to order houses.

Wherever there are things of different sizes, you can have a semiorder of size. That's pretty obvious. You have a > b iff a is bigger than b. You might not have a total order, admittedly, but it's silly to say the two aren't related.

Well, of course there can be a partial order. When I and, I think, silverhammermamba said "order" we meant "total order". I was just saying you can still have bigness without total order.

Actually you don't, in general, have a partial order, since if a ≥ b and b ≥ a, it does not follow that a = b (they could be different houses of the same size).

Though I guess it depends on how you precisely define '≥'.

silverhammermba
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### Re: |i|

z4lis wrote:Eh, if you were talking to me about complex numbers and mentioned that you wished to "let z get large", I would assume you meant "let |z| go to infinity".

Once again, the meaning of that is not intuitively clear: "let |z| go to infinity" is not well-defined with regards to z itself. On the complex plane there are an infinite number of ways in which z can be set to a limit such that |z| approaches infinity. The only way saying "let |z| go to infinity" is well defined is if you're only dealing with |z| and not z itself. But then [imath]|z| \in \mathbf{R}[/imath], where "bigness" is intuitively defined!

Kurushimi wrote:Well, of course there can be a partial order. When I and, I think, silverhammermamba said "order" we meant "total order". I was just saying you can still have bigness without total order.

Oops! You are correct. The complex numbers are not totally ordered by | |, unlike the real numbers. Obviously they are both linearly ordered...

antonfire
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### Re: |i|

The real numbers are not totally ordered by |.| either. Compare 1 and -1.
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### Re: |i|

antonfire wrote:The real numbers are not totally ordered by |.| either. Compare 1 and -1.

What do you mean by "totally ordered by |.|?" The real numbers obviously are totally ordered, as are the nonnegative numbers. And the set of absolute values of real numbers is simply the set of nonnegative numbers.

Do you mean that the relation ~ where a ~ b iff |a| > |b| is not a total order on the set of reals? Because that seems obvious.

antonfire
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### Re: |i|

Yes, it is pretty obvious.
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z4lis
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### Re: |i|

silverhammermba wrote:"let |z| go to infinity" is not well-defined with regards to z itself. On the complex plane there are an infinite number of ways in which z can be set to a limit such that |z| approaches infinity. The only way saying "let |z| go to infinity" is well defined is if you're only dealing with |z| and not z itself. B

There are plenty of estimates for which the path you take to infinity doesn't really matter, as long as z is outside some bounded set. If we're talking about the Riemann sphere, then letting z go to infinity is unambiguous. At any rate, my statement is really:

If you want to let "z get large", I'm going to assume that you mean "for any bounded set S, z eventually leaves S", which is stated easily as "let |z| go to infinity." If the direction z takes as it runs away matters, then you wouldn't be using phrases like "let z get large" without talking about the paths it takes anyway.
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silverhammermba
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### Re: |i|

antonfire wrote:The real numbers are not totally ordered by |.| either. Compare 1 and -1.

Crap again. I was thinking the nonnegative real numbers, in which case everything I said actually makes sense. "Let |z| go to infinity" isn't even well-defined if z is real. z would need to be nonnegative or nonpositive.

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### Re: |i|

antonfire wrote:Yes, it is pretty obvious.

Not really.

z4lis
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### Re: |i|

silverhammermba wrote: "Let |z| go to infinity" isn't even well-defined if z is real. z would need to be nonnegative or nonpositive.

I provided a perfectly acceptable definition. "|z_n| goes to infinity as n goes to infinity" means exactly "given any bounded set S, there is some N for which n larger than N implies z_n does not lie in S".

If the LHS is true and you give me a bounded set, I can choose some N for which z_n lies outside the bounded set for n larger than N.
If the RHS is true and you give me a fixed C, then I'll let my bounded set be the ball of radius C around zero. The N for which z_n falls outside of that bounded set will by the N I hand you to make |z_n| > C for all n larger than N.
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silverhammermba
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### Re: |i|

z4lis wrote:I provided a perfectly acceptable definition. "|z_n| goes to infinity as n goes to infinity" means exactly "given any bounded set S, there is some N for which n larger than N implies z_n does not lie in S".

If the LHS is true and you give me a bounded set, I can choose some N for which z_n lies outside the bounded set for n larger than N.
If the RHS is true and you give me a fixed C, then I'll let my bounded set be the ball of radius C around zero. The N for which z_n falls outside of that bounded set will by the N I hand you to make |z_n| > C for all n larger than N.

Yes, and I totally agree with that. But that's only sufficient in a certain context. If the context in which you are working relies on the specific behavior of z as it goes off to some infinity, then saying "Let |z| go to infinity" might not tell you enough information. A trivial example:
$\lim_{|z|\rightarrow \infty} \text{Re}(z)$

My point is that if you're in a context where only |z| matters (and so saying "Let |z| go to infinity" gives you enough information), then you might as well be working in the nonnegative real numbers where "bigness" is intuitively defined.

achan1058
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### Re: |i|

silverhammermba wrote:My point is that if you're in a context where only |z| matters (and so saying "Let |z| go to infinity" gives you enough information), then you might as well be working in the nonnegative real numbers where "bigness" is intuitively defined.
How so? I distinctively remember a bunch of complex analysis theorems that uses "Let |z| go to infinity". (where you usually want f(z)->0, for purposes of integration)

Cranica
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### Re: |i|

Eebster the Great wrote:^For an intuitive notion of "size" that does not match "modulus," consider the 1-norm |z| = Re(z) + Im(z).

This isn't any better than just taking the magnitude, though - it's not even a partial order, since |a+bi| = |b+ai| but a+bi != b+ai

How so? I distinctively remember a bunch of complex analysis theorems that uses "Let |z| go to infinity". (where you usually want f(z)->0, for purposes of integration)

In some contexts I imagine this is enough, namely when you only care about the magnitude (or at least that it's enough to prove certain behaviors). For instance, it makes sense to say that ||1/z|| goes to 0 as ||z|| increases in any direction. But it certainly isn't general, given the Re(z) example above, and this example at least is only using magnitudes, which are themselves (totally-ordered) reals.

Meem1029
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### Re: |i|

I'm not 100% sure, but I seem to remember that for all the proofs where we had to let ||z|| -> infinity we generally did something with considering an arbitrary point on or outside a circle of radius R, and then let R approach infinity. Or something along those lines.
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Eebster the Great
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### Re: |i|

Cranica wrote:
Eebster the Great wrote:^For an intuitive notion of "size" that does not match "modulus," consider the 1-norm |z| = Re(z) + Im(z).

This isn't any better than just taking the magnitude, though - it's not even a partial order, since |a+bi| = |b+ai| but a+bi != b+ai

It is actually much worse which is why it is almost never used. My point was merely that there can be more than one idea of "size" for the complex numbers.

Cranica
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### Re: |i|

Meem1029 wrote:I'm not 100% sure, but I seem to remember that for all the proofs where we had to let ||z|| -> infinity we generally did something with considering an arbitrary point on or outside a circle of radius R, and then let R approach infinity. Or something along those lines.

"outside a circle of radius R" would simply be |z| > r, so your proofs would rely only on the magnitude and not the direction.

achan1058
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### Re: |i|

Cranica wrote:
Meem1029 wrote:I'm not 100% sure, but I seem to remember that for all the proofs where we had to let ||z|| -> infinity we generally did something with considering an arbitrary point on or outside a circle of radius R, and then let R approach infinity. Or something along those lines.

"outside a circle of radius R" would simply be |z| > r, so your proofs would rely only on the magnitude and not the direction.
But you are still doing the rest of your algebra with complex numbers, including multiplication, functions, etc. It might be true that only the magnitude really matters, but turning everything inside the proof into magnitude will probably fail. For example, if you are taking a contour integral of a curve outside that circle of radius R.

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### Re: |i|

I often catch myself calling |a+bi| the "absolute value" of a+bi, and I don't think I've ever been misunderstood by saying that (though I have been corrected, which implies that the listener knew exactly what I meant).

It's a bad habit, but not seriously so. I sometimes suspect we should just drop the terms "absolute value" and "modulus" and use the word norm from the moment the || notation is introduced. (Sure, modulus isn't the only possible norm on the complex numbers; but the sort of students who will be exposed to abstract algebra are not the sort of students who will be confused by terminology.)
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### Re: |i|

I'm still unsure why anyone would object to calling |a+bi| the "absolute value" of a+bi.