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 Posted: Mon Jul 23, 2007 11:15 pm
Soy un hombre muy honrado,
que me gusta lo mejor was reading through a book and found out that the absolute value function for a complex number in the form z=x+iy is defined as |z|=√(x²+y²), and the only reason I could think of for this definition is that it's the best way to ensure |z| comes out positive. Am i rite!?!/1?! Las mujeres no me faltan,
ni el dinero ni el amor
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Posted: Tue Jul 24, 2007 12:12 am
What we want is actually for the absolute value (or norm) of an object to have specific properties:
1: |w+z| =< |w|+|z| triangle inequality
2: |w*z| = |w|*|z| multiplicativity
3: |z| is real for all z and |z|>0 for all nonzero z.
I'm pretty sure that these determine a unique form for the absolute value of a complex number.
You'll note that the norm of x+iy is identical to the length of the vector (x, y). Since the absolute value of a number is supposed to correspond to the size of the number, this identity makes sense.
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Posted: Tue Jul 24, 2007 7:00 pm
Yes, you are right.
Finding the square root of a number squared is a pretty sure way of making sure that it is a positive number, as squaring a number always makes it positive.
Which reminds me, my friends goal in life is to find something for which the absolute value is negative.
Somehow I have a feeling that is a goal that will never be fulfilled.
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Posted: Tue Jul 24, 2007 7:31 pm
Dewdew Yes, you are right. Finding the square root of a number squared is a pretty sure way of making sure that it is a positive number, as squaring a number always makes it positive. Which reminds me, my friends goal in life is to find something for which the absolute value is negative. Somehow I have a feeling that is a goal that will never be fulfilled. It depends. How willing is this friend to abuse the idea of the "absolute value"? I mean, absolute values are, most of the time, defined as functions from the domain space to some subset of the positive real numbers, so the absolute value of anything would end up being positive. However, for every domain space the absolute value function is different. For example, for real numbers the absolute value can be calculated as the square root of the square of the number, i.e.  However, if you took, say, i and stuck it in there, we'd get which isn't even a real number. Thus, the absolute value of a complex number z = x+iy is defined as where It can be easily proven that is real and non-negative if x and y are real numbers, so this new formula does give us an absolute value. Therefore, it is theoretically possible to find some number that stuck into some absolute value function would return a negative number. But that's really being both abusive of the notation and missing the point entirely.
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Posted: Tue Aug 07, 2007 8:59 am
Layra-chan What we want is actually for the absolute value (or norm) of an object to have specific properties: 1: |w+z| =< |w|+|z| triangle inequality 2: |w*z| = |w|*|z| multiplicativity 3: |z| is real for all z and |z|>0 for all nonzero z. I'm pretty sure that these determine a unique form for the absolute value of a complex number. You'll note that the norm of x+iy is identical to the length of the vector (x, y). Since the absolute value of a number is supposed to correspond to the size of the number, this identity makes sense. Those three properties define norms, yes, but I'm not sure what you mean by determining a unique form for the absolute value. If we think of absolute value as distance, then the distance between two points in a vector space depends on the norm you give it. We define the modulus of a complex number the way we do to match the Euclidean norm for the plane. You could use any other kind of norm and get a different space. For example, all of these are also norms on the complex numbers a+bi: max( |a| , |b| ) |a| + |b| pth-root ( |a|^p + |b|^p ) In any nontrivial metric space, there are always multiple norms to choose from. The fact that the Euclidean norm is standard or more common does not mean it is unique.
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Posted: Wed Aug 08, 2007 10:40 pm
The Mathematician Layra-chan What we want is actually for the absolute value (or norm) of an object to have specific properties: 1: |w+z| =< |w|+|z| triangle inequality 2: |w*z| = |w|*|z| multiplicativity 3: |z| is real for all z and |z|>0 for all nonzero z. I'm pretty sure that these determine a unique form for the absolute value of a complex number. You'll note that the norm of x+iy is identical to the length of the vector (x, y). Since the absolute value of a number is supposed to correspond to the size of the number, this identity makes sense. Those three properties define norms, yes, but I'm not sure what you mean by determining a unique form for the absolute value. If we think of absolute value as distance, then the distance between two points in a vector space depends on the norm you give it. We define the modulus of a complex number the way we do to match the Euclidean norm for the plane. You could use any other kind of norm and get a different space. For example, all of these are also norms on the complex numbers a+bi: max( |a| , |b| ) |a| + |b| pth-root ( |a|^p + |b|^p ) In any nontrivial metric space, there are always multiple norms to choose from. The fact that the Euclidean norm is standard or more common does not mean it is unique. What I meant is that there is only one multiplicative norm for the complex numbers. While there are other functions that could be used as the norm for the complex numbers, none of them are fully multiplicative, which is one of the more important properties of the Euclidean norm.
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Posted: Thu Aug 09, 2007 4:04 pm
Layra-chan The Mathematician Layra-chan What we want is actually for the absolute value (or norm) of an object to have specific properties: 1: |w+z| =< |w|+|z| triangle inequality 2: |w*z| = |w|*|z| multiplicativity 3: |z| is real for all z and |z|>0 for all nonzero z. I'm pretty sure that these determine a unique form for the absolute value of a complex number. You'll note that the norm of x+iy is identical to the length of the vector (x, y). Since the absolute value of a number is supposed to correspond to the size of the number, this identity makes sense. Those three properties define norms, yes, but I'm not sure what you mean by determining a unique form for the absolute value. If we think of absolute value as distance, then the distance between two points in a vector space depends on the norm you give it. We define the modulus of a complex number the way we do to match the Euclidean norm for the plane. You could use any other kind of norm and get a different space. For example, all of these are also norms on the complex numbers a+bi: max( |a| , |b| ) |a| + |b| pth-root ( |a|^p + |b|^p ) In any nontrivial metric space, there are always multiple norms to choose from. The fact that the Euclidean norm is standard or more common does not mean it is unique. What I meant is that there is only one multiplicative norm for the complex numbers. While there are other functions that could be used as the norm for the complex numbers, none of them are fully multiplicative, which is one of the more important properties of the Euclidean norm. You're mincing definitions of norm and absolute value, which you seem to think are the same. An absolute value of a vector space is sometimes a term for multiplicative norm, but is usually a term reserved for real numbers or other scalars. That's why I was correcting you. The actual definition of a norm is that it scales positively ( ||a * z|| = |a| ||z|| where a is a scalar) and that the norm of the zero vector is the zero scalar. Your three points give a norm, but not all norms. You've gone so far as to assume that the norm is a metric, with real scalars and all! If you wanted to say that it determines a unique multiplicative norm, you should have said so, but you didn't, because you mixed up the terminology. I was generous about your imprecise characterization of a norm in my last reply, but you're the one who was using a far too restrictive definition.
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Posted: Fri Aug 10, 2007 1:32 am
There is classically one form that is singled out as "the absolute value" of a complex number, and that is |x+iy| = sqrt{(x+iy)(x-iy)}. This happens to be a norm for the complex numbers, and it happens to look like Euclidean length. This does not mean that it can be replaced by any other distance function, or any other norm; it has been singled out, and statements regarding it need not reflect norms or metrics in general.
I am used to thinking of the Euclidean complex number norm as being different from a vector norm, since I came across the complex norm from a number-theoretic perspective.
Since I think of the complex numbers as a field, or perhaps even as a 1-dimensional complex vector space, to me the distinction between the absolute value for a complex number and the norm of a complex number is completely due to context.
I never stated at any point that there were not any other norms, although if you do take the complex numbers to be a 1-dimensional vector space over the complex number field, you end up with a norm that is multiplicative since your scalars are your vectors.
What I stated is that we wanted a norm with the three properties.
Since sqrt{(x+iy)(x-iy)} is in fact the standard definition of the absolute value of the field of complex numbers, I still feel that what I have said so far is correct both in terminology and in fact; there is only one multiplicative norm for the complex numbers, and the properties I listed determine a unique form for the absolute value.
In slight digression, in terms of the definition of norm that I am familiar with, all norms do induce metrics, being that norms obey the triangle inequality. Unless this is also being too restrictive.
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Posted: Sat Aug 18, 2007 12:06 pm
No, I'm pretty sure all norms induce metrics. Rather, I know that all normed vector spaces are also metric spaces. For other algebraic objects, other definitions of norms prevail, and I doubt that all of them make metrics.
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