Split Thread Musings on Infinity

Argumemnon said:
No, it's not a fact. It's your unsupported opinion, but I can see why that would confuse you.



Now you're just posting nonsense in order to make yourself feel smart. Even Dave's admitted that in the general case, I am correct. He's just unable to admit that the special case doesn't apply to our conversation.
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Congratulation, you get the final word. On the downside, you missed an opportunity to learn.
 
jt512 said:
Congratulation, you get the final word. On the downside, you missed an opportunity to learn.
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Oh, quite the contrary. I learned two things: that there is a special case in which that claim might be true but that doesn't satisfy every arithmetic rule; and that some people are so unable to admit to being wrong that they will pretend that this special case supercedes the general case.

This isn't a matter of "winning". It's a matter of what the correct answer is. I guess this forum has gone so far off the deep end that we can't agree on basic arithmetic anymore. You'd be telling me that 2+2=5 in a special, incomplete theory means that my arguing that 2+2=4 is incorrect.
 
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Argumemnon said:
Even Dave's admitted that in the general case, I am correct.
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I admitted no such thing; you don't even appear to have understood that. The idea that simple arithmetic is the general and complex analysis the special case is purely your construct, and has nothing to do with actual mathematics.

Argumemnon said:
He's just unable to admit that the special case doesn't apply to our conversation.
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We're talking about how to handle division by infinity. Are you arguing that a simplified form of mathematics that forbids that operation applies better to it than a more developed form that allows it? If so, you're obviously arguing nonsense.

Enough of this, anyway. I have no more interest in the products of your ignorance, and neither should anyone else.

Dave
 
Dave Rogers said:
I admitted no such thing; you don't even appear to have understood that. The idea that simple arithmetic is the general and complex analysis the special case is purely your construct, and has nothing to do with actual mathematics.
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Then clarify it for me. The very article you quoted says that it doesn't "satisfy every rule of arithmetic."

We're talking about how to handle division by infinity. Are you arguing that a simplified form of mathematics that forbids that operation applies better to it than a more developed form that allows it?
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It's not "simplified". It's normal arithmetics. A wheel is not a simplified turbine, either.

Enough of this, anyway. I have no more interest in the products of your ignorance, and neither should anyone else.
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Another poster who feels the need to boast his superiority, and yet quotes wikipedia just like everyone else. :dqueen
 
Argumemnon said:
Expertise? What are your credentials? I'll remind you that Dave quoted wikipedia.
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Oh, for Christ's sake. I quoted Wikipedia because it happened to be right. My credentials are a doctorate in physics, and no I'm not going to provide evidence because it's irrelevant.

Let me have one more go at explaining this to you. As I said, if we're limited to simple arithmetic then you're right; 1/0 is a forbidden operation and has no value. But the question was what happens when you divide 4 by infinity, and simple arithmetic can't answer that either because no operations involving infinity as anything other than a limit are defined in simple arithmetic either. To answer questions involving infinity, we need to resort to higher mathematics such as complex analysis, which allow a reduced set of operations involving infinity - in other words, more operations than simple arithmetic - so that we can address these questions. Complex analysis is not, as you believed from reading Wikipedia, a special case of simple arithmetic; in fact it's the other way round, simple arithmetic is a simplified set of operations that only work for real, finite numbers, and is effectively a special case of complex analysis. And using complex analysis, we can define infinity as equal to 1/0, giving it the same properties that we understand infinity to possess, and determine that 0/0 and 0 times infinity are both indeterminate. Those two results are critically important to differential and integral calculus respectively, because without them each wouldn't work. I can explain that in more detail if you're interested. But it's important to understand, as I said above, that complex analysis isn't a limited special case of simple arithmetic; simple arithmetic is a limited special case of complex analysis, and when we discuss infinity we've stepped outside those limits.

Dave
 
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Dave Rogers said:
Oh, for Christ's sake.
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Calm down. I didn't mention the wikipedia thing as a slight. I simply mentioned it to show that both you and I were using the very same source in our arguments.

Let me have one more go at explaining this to you. As I said, if we're limited to simple arithmetic then you're right
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Isn't that what we were discussing? I mean, I'm sure string theory offers interesting solutions to some stuff that newtonian physics struggles with, but since we're rarely talking about such complex theories, why use them as a counter-argument in a discussion such as this one?

But the question was what happens when you divide 4 by infinity, and simple arithmetic can't answer that either because no operations involving infinity as anything other than a limit are defined in simple arithmetic either.
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Which is exactly what I was talking about: arithmetics. If you had simply said that, in some higher mathematics, there are solutions, and we agreed that it doesn't change the general use of these equations, there would be no disagreement.

Dave
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I appreciate the longer explanation, Dave. I really do. I just don't think the higher maths you brought into the discussion are relevant to my original comment, which was nothing more than a joke anyway. Advanced maths wasn't required to understand it, especially as, like Wiki points out, they don't always apply.
 
Argumemnon said:
I appreciate the longer explanation, Dave. I really do. I just don't think the higher maths you brought into the discussion are relevant to my original comment, which was nothing more than a joke anyway.
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Well then, I think we can agree that lim(x->∞) 4/x is zero, and that a finite amount divided by a number increasing without upper bound decreases in quantity without lower bound. So shall we go back to watching Jabba divide his finite level of knowledge and understanding between a sequence of posts that seems to be increasing without an upper bound, and observe the inevitable consequences?

Dave
 
Argumemnon said:
. Your say-so does not determine reality.
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You're right it doesn't. You have the right to remain ignorant, despite the help of me and others. So, enjoy your ignorance.

Dave has patiently laid everything out for you here, including how you misinterpreted the wikipedia article. Now you really have no excuse to persist in your ignorance.
 
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jt512 said:
You're right it doesn't. You have the right to remain ignorant, despite the help of me and others. So, enjoy your ignorance.

Dave has patiently laid everything out for you here, including how you misinterpreted the wikipedia article. Now you really have no excuse to persist in your ignorance.
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Since Dave and I have come to an agreement, including popcorn and soda, you're starting to look a bit silly, here. Pat yourself on the back all you want.
 
Argumemnon said:
Since Dave and I have come to an agreement, including popcorn and soda, you're starting to look a bit silly, here. Pat yourself on the back all you want.
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You haven't come to an agreement. You're still insisting that Dave's "higher math" doesn't apply to your argument, and on top of that you're playing the "I was just joking anyway" card. So you now claim that you were right, but just joking. What does that even mean?
 
jt512 said:
You haven't come to an agreement. You're still insisting that Dave's "higher math" doesn't apply to your argument, and on top of that you're playing the "I was just joking anyway" card. So you now claim that you were right, but just joking. What does that even mean?
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Easy: that I was making a joke but that it was based on correct arithmetics. Why is that hard to understand? And yes, we did come to an agreement. Why is that so hard for you to understand? And if you doubt that it's a joke, look back at the original post:

Argumemnon said:
abaddon said:
Dividing by infinity makes any claim valid, if one is let away with it.
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Doesn't dividing by infinity yield a number approaching zero?
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It should be obvious since you can't make a claim valid by dividing by infinity. But of course, as usual people who don't get the joke think that explaining the joke is some sort of lie. :rolleyes:

Now can we get back to the soda and popcorn?
 
Argumemnon said:
Easy: that I was making a joke but that it was based on correct arithmetics. Why is that hard to understand? And yes, we did come to an agreement. Why is that so hard for you to understand? And if you doubt that it's a joke, look back at the original post:



It should be obvious since you can't make a claim valid by dividing by infinity. But of course, as usual people who don't get the joke think that explaining the joke is some sort of lie. :rolleyes:

Now can we get back to the soda and popcorn?
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Ugh. Fine. [SNIP] Have the last word:

Edited by kmortis: 
Removed to comply with Rule 0
 
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The reason why dividing by infinity is wrong is more fundamental than already suggested in this thread - infinity is a mathematical concept, not a number.
See Infinity is NOT a number
If you treat infinity as a number, you fundamentally break everything that makes arithmetic work. For example, the most basic definition of numbers that I know of is Peano arithmetic. Peano arithmetic is a set of axioms that defines how the natural numbers work. It’s the set of axioms that are typically used as the fundamental basis of a formal definition of numbers. One of the Peano axioms says that for every natural number, there is exactly one natural number that is its successor; and every natural number except zero is the successor of exactly one natural number.
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jt512 said:
In fact, division of a real number by infinity is defined to be 0 in broad areas of mathematics. Look at any advanced calculus text.
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I have never seen a calculus text that says that. The only thing you will find is that infinity is a limit.
 
This is why I went into electrical engineering. We get to skip all the hard math and just say "yep, we can divide by infinity, woohoo!"
 
John Jones said:
You cannot divide by infinity that way. Notice that if we wrote 1/∞ =0
, we could multiply by infinity on both sides and obtain 1=∞⋅0
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No, you can divide by infinity, that's fine. It's sort of like a function. The reverse doesn't work because not all functions have an inverse.

But the logic is wrong anyways, because there is in fact a limited number of possible selves (at least, if you're confining your set to humans). That's just basic thermodynamics. The number is incomprehensibly big, but it isn't actually infinity.
 
There is no real number corresponding to infinity, and there is no complex number corresponding to infinity.

Whenever someone speaks of dividing by infinity, therefore, we immediately know they are not talking about the field of real numbers or the field of complex numbers.

There are mathematical systems that include some notion of infinity, but those are not the familiar systems of real or complex numbers.

Whenever someone argues about dividing by infinity, regardless of their position, without explaining what they take to be the domain of the division operation or the definition of the division operation, they are talking nonsense.
 
jt512 said:
Jesus, after reading through that crap, I'm led to believe that I'm the only person on the internet who ever studied real analysis.
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Are you referring to the crap that the Infinity is NOT a number addresses, e.g. divining by zero, jt512? That is really basic math rather than real analysis. If you look further through the blog you will see even more deluded math being addressed.
Otherwise the points the blog raises are basic mathematics
  • Infinity is a concept, not a number
  • Treating infinity breaks foundations of number theory (Peano axioms)
  • Rational numbers, real numbers, complex numbers, etc. form a field.
    Treating infinity as a number breaks this and thus most of our existing mathematical proofs.
Real analysis does have the useful concept of an extended real number line where two "elements" of +∞ and –∞ are added to the line. It is important to note that these are not real numbers.
 
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Reality Check said:
  • Infinity is a concept, not a real or complex number
  • Treating infinity as a real or complex number breaks foundations of number theory (Peano axioms)
  • Rational numbers, real numbers, complex numbers, etc. form a field.
    Treating infinity as a number breaks this and thus most of our existing mathematical proofs, if those proofs are predicated on the field axioms for those number fields.
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FTFY
 
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mijopaalmc said:
FTFY
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Not really needed - the "most" part is that most of our existing mathematical proofs are based on numbers being fields. Which anyone like jt512 who claims to know real analysis will know. And there are other kinds of numbers than real or complex.
  • Infinity is a concept, not a real, complex, rational, etc. number
  • Treating infinity as a natural number breaks foundations of number theory (Peano axioms).
  • Rational numbers, real numbers, complex numbers, etc. form a field.
    Treating infinity as a number breaks this and thus most of our existing mathematical proofs which based on numbers being fields
 
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Reality Check said:
Not really needed - the "most" part is that most of our existing mathematical proofs are based on numbers being fields. Which anyone like jt512 who claims to know real analysis will know.
And there are other kinds of numbers than real or complex.
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Except that that's not even close to true. Most of the proof in real and complex analysis are extensible to compactified versions of the real or complex plane that include infinity (see: affinely extended real number line, projectively extended real line, and extended complex plane).

P.S. Repeating yourself doesn't make you right.
 
W.D.Clinger said:
There is no real number corresponding to infinity, and there is no complex number corresponding to infinity.

Whenever someone speaks of dividing by infinity, therefore, we immediately know they are not talking about the field of real numbers or the field of complex numbers.

There are mathematical systems that include some notion of infinity, but those are not the familiar systems of real or complex numbers.

Whenever someone argues about dividing by infinity, regardless of their position, without explaining what they take to be the domain of the division operation or the definition of the division operation, they are talking nonsense.
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(bold added)

What he said. Times infinity ;)

Mathematicians have studied, and refined, the concepts of "infinity" for well over a century. Their published results are, today, not the least bit controversial, or all that difficult to understand (in general).

Perhaps this is a chance for us to have an informative discussion of what, in mathematics, a "field" is? And what are the things ("numbers") in the "real number field" and the "complex number field"? (Can you plant sunflowers in such fields? :D)

Also, what are the sorts of "mathematical systems that include some notion of infinity"?

Oh, and what is a "domain", as in "the domain of the division operation"?

Also, isn't "the definition of the division operation" intuitively obvious? What needs explaining?
 
mijopaalmc said:
Most of the proof in real and complex analysis are extensible to compactified versions of the real or complex plane that include infinity.
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Not supporting your assertion does not make your assertion right.

As I wrote:
Reality Check said:
Real analysis does have the useful concept of an extended real number line where two "elements" of +∞ and –∞ are added to the line. It is important to note that these are not real numbers.
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That is the "affinely extended real number line": Extended real number line

Links to the projectively extended real line and Riemann sphere (great coverage in Penrose's The Road to Reality book) are not evidence for your assertion.

N.B. The real line extensions do not include infinity as an actual number. Infinity is till the concept of "unboundedness", e.g. real numbers that are increasing and unbounded. But by adding a point "at infinity", dividing a non-zero number by zero gains meaning as infinity at the cost of no longer being a field. Thus all proofs that depend on there being a field fail.
Not sure about a Riemann sphere though.
 
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psionl0 said:
I have never seen a calculus text that says that. The only thing you will find is that infinity is a limit.
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Try Principles of Mathematical Analysis by Rudin, for one example among many.
 
jt512 said:
Try Principles of Mathematical Analysis by Rudin, for one example among many.
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I have the 3rd edition available - no sign on skimming it of "division of a real number by infinity is defined to be 0" - can you give the page number?
ETA: You may be confused by the description of the extended real number system where there is a convention only in that context that x/+infinity = x/-infinity = 0.
 
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For the record, I'm with you Reality Check. I think you are correct.

But I'm supposed to believe you read jt512's reply, have this edition of the text at the ready, grabbed it, skimmed it, AND typed your response, all in 7 minutes? That's quite a trick!
 
DuvalHMFIC said:
For the record, I'm with you Reality Check. I think you are correct.

But I'm supposed to believe you read jt512's reply, have this edition of the text at the ready, grabbed it, skimmed it, AND typed your response, all in 7 minutes? That's quite a trick!
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Yes, I also find it hard to believe that RC spent that much time on it. ;)

Although it is possible to talk about "countably infinite" and "uncountably infinite" (suggesting that there are degrees of infinity), treating infinity as anything other than a limit is bound to get you tied up in knots sooner or later.
 
JeanTate asked some excellent questions. I'm sure he knows the answers to his questions, but I agree the level of discourse in this thread might be improved by mentioning a few mathematical facts.

JeanTate said:
(bold added)

What he said. Times infinity ;)

Mathematicians have studied, and refined, the concepts of "infinity" for well over a century. Their published results are, today, not the least bit controversial, or all that difficult to understand (in general).

Perhaps this is a chance for us to have an informative discussion of what, in mathematics, a "field" is? And what are the things ("numbers") in the "real number field" and the "complex number field"? (Can you plant sunflowers in such fields? :D)
Click to expand...
In mathematics, a field is a set of things (which we call numbers) together with two binary operations (which we call addition and subtraction multiplication) that obey algebraic laws. Those algebraic laws say the numbers form a commutative group under addition, the numbers excluding the additive identity (which we call zero) form a commutative group under multiplication, and the addition and multiplication operations are connected by a distribution law.

Example: The non-negative integers less than 5, {0, 1, 2, 3, 4}, form a field with respect to the usual definitions of addition and multiplication modulo 5.

Non-example: The non-negative integers less than 4, {0, 1, 2, 3}, do not form a field with respect to the usual definitions of addition and multiplication modulo 4 because 2 times 2 mod 4 is zero, which is not a non-zero element of the set.

JeanTate said:
Also, what are the sorts of "mathematical systems that include some notion of infinity"?
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There are many such systems. RealityCheck cited one example: the real projective line, which is not a field but is a one-dimensional manifold, which basically means pieces of it look like pieces of the real number line so long as you don't look at the big picture.

JeanTate said:
Oh, and what is a "domain", as in "the domain of the division operation"?
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The domain of the division operator is the set of ordered pairs of numbers on which the division operator is defined. For a field, the domain of the division operator consists of all ordered pairs of the field's numbers whose second element is nonzero.

JeanTate said:
Also, isn't "the definition of the division operation" intuitively obvious? What needs explaining?
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For a field, the division operator is a derived operation, defined using the field's addition and multiplication operations. The addition operation has a unique identity, which we call zero. The division operation is then defined as follows: For any pair of numbers <x,y> for which y is nonzero, x/y is defined as the unique number z such that x equals y multiplied by z.

If that were intuitively obvious to everyone posting in this thread, many of the things that have been said would not have been said. Some, for example, have said infinity is defined to be 1/0, but the ordered pair <1,0> is excluded from the domain of the division operation. The reason for that exclusion is that there is no unique z such that 1 equals 0 multiplied by z.

I suppose some would claim 1 equals 0 multiplied by infinity, but that claim founders when, applying the algebraic laws that hold for all fields, you use the claimed 1=0*∞ to prove 2=1 as follows:

2 = 2*1 = 2*(0*∞)=(2*0)*∞=0*∞ = 1​

That's why infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers. That's why infinity cannot be regarded as a complex number.
 
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W.D.Clinger said:
JeanTate asked some excellent questions. I'm sure he knows the answers to his questions, but I agree the level of discourse in this thread might be improved by mentioning a few mathematical facts.


In mathematics, a field is a set of things (which we call numbers) together with two binary operations (which we call addition and subtraction) that obey algebraic laws. Those algebraic laws say the numbers form a commutative group under addition, the numbers excluding the additive identity (which we call zero) form a commutative group under multiplication, and the addition and multiplication operations are connected by a distribution law.

Example: The non-negative integers less than 5, {0, 1, 2, 3, 4}, form a field with respect to the usual definitions of addition and multiplication modulo 5.

Non-example: The non-negative integers less than 4, {0, 1, 2, 3}, do not form a field with respect to the usual definitions of addition and multiplication modulo 4 because 2 times 2 mod 4 is zero, which is not a non-zero element of the set.


There are many such systems. RealityCheck cited one example: the real projective line, which is not a field but is a one-dimensional manifold, which basically means pieces of it look like pieces of the real number line so long as you don't look at the big picture.


The domain of the division operator is the set of ordered pairs of numbers on which the division operator is defined. For a field, the domain of the division operator consists of all ordered pairs of the field's numbers whose second element is nonzero.


For a field, the division operator is a derived operation, defined using the field's addition and multiplication operations. The addition operation has a unique identity, which we call zero. The division operation is then defined as follows: For any pair of numbers <x,y> for which y is nonzero, x/y is defined as the unique number z such that x equals y multiplied by z.

If that were intuitively obvious to everyone posting in this thread, many of the things that have been said would not have been said. Some, for example, have said infinity is defined to be 1/0, but the ordered pair <1,0> is excluded from the domain of the division operation. The reason for that exclusion is that there is no unique z such that 1 equals 0 multiplied by z.

I suppose some would claim 1 equals 0 multiplied by infinity, but that claim founders when, applying the algebraic laws that hold for all fields, you use the claimed 1=0*∞ to prove 2=1 as follows:

2 = 2*1 = 2*(0*∞)=(2*0)*∞=0*∞ = 1​

That's why infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers. That's why infinity cannot be regarded as a complex number.
Click to expand...

Just a nitpick: This should be multiplication, not subtraction.

Oh, it's always a joy to read your posts on math. You may make some typos :p but as long as I can follow, I find them correct and very well explained. Your style reminds me a bit of that of my math hero Klaus Jänich. You don't by chance understand some german?
 
W.D.Clinger said:
The domain of the division operator is the set of ordered pairs of numbers on which the division operator is defined. For a field, the domain of the division operator consists of all ordered pairs of the field's numbers whose second element is nonzero.
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Isn't the set of ordered pairs the division operation itself?

As far as I understand it, a set relation is a set of ordered pairs, whereas the domain of that relation is the set of all the first members in each ordered pair.
 
mijopaalmc said:
W.D.Clinger said:
The domain of the division operator is the set of ordered pairs of numbers on which the division operator is defined. For a field, the domain of the division operator consists of all ordered pairs of the field's numbers whose second element is nonzero.
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Isn't the set of ordered pairs the division operation itself?

As far as I understand it, a set relation is a set of ordered pairs, whereas the domain of that relation is the set of all the first members in each ordered pair.
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A binary relation R can indeed be identified with the set of ordered pairs <x,z> for which xRz, and a function f is a special kind of binary relation so f can be identified with its graph, which is the set of ordered pairs <x,z> for which z=f(x).

In this case, the addition and multiplication operations are binary, meaning they take two arguments, which we can treat as an ordered pair, so the graph of + (for example) is the set of ordered pairs <<x,y>,z> for which z=x+y. The domain of + is then the set of ordered pairs <x,y> for which <<x,y>,z> belongs to the graph of +.

Similarly for division.
 
W.D.Clinger said:
JeanTate asked some excellent questions. I'm sure he knows the answers to his questions, but I agree the level of discourse in this thread might be improved by mentioning a few mathematical facts.


In mathematics, a field is a set of things (which we call numbers) together with two binary operations (which we call addition and subtraction multiplication) that obey algebraic laws. Those algebraic laws say the numbers form a commutative group under addition, the numbers excluding the additive identity (which we call zero) form a commutative group under multiplication, and the addition and multiplication operations are connected by a distribution law.

Example: The non-negative integers less than 5, {0, 1, 2, 3, 4}, form a field with respect to the usual definitions of addition and multiplication modulo 5.

Non-example: The non-negative integers less than 4, {0, 1, 2, 3}, do not form a field with respect to the usual definitions of addition and multiplication modulo 4 because 2 times 2 mod 4 is zero, which is not a non-zero element of the set.


There are many such systems. RealityCheck cited one example: the real projective line, which is not a field but is a one-dimensional manifold, which basically means pieces of it look like pieces of the real number line so long as you don't look at the big picture.


The domain of the division operator is the set of ordered pairs of numbers on which the division operator is defined. For a field, the domain of the division operator consists of all ordered pairs of the field's numbers whose second element is nonzero.


For a field, the division operator is a derived operation, defined using the field's addition and multiplication operations. The addition operation has a unique identity, which we call zero. The division operation is then defined as follows: For any pair of numbers <x,y> for which y is nonzero, x/y is defined as the unique number z such that x equals y multiplied by z.

If that were intuitively obvious to everyone posting in this thread, many of the things that have been said would not have been said. Some, for example, have said infinity is defined to be 1/0, but the ordered pair <1,0> is excluded from the domain of the division operation. The reason for that exclusion is that there is no unique z such that 1 equals 0 multiplied by z.

I suppose some would claim 1 equals 0 multiplied by infinity, but that claim founders when, applying the algebraic laws that hold for all fields, you use the claimed 1=0*∞ to prove 2=1 as follows:

2 = 2*1 = 2*(0*∞)=(2*0)*∞=0*∞ = 1​

That's why infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers. That's why infinity cannot be regarded as a complex number.
Click to expand...


Just sittin' back LMAO, watching people make the same bad arguments over and over again.
 
jt512 said:
Just sittin' back LMAO, watching people make the same bad arguments over and over again.
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LMAO at the denial of mathematics showing in that remark, jt512.
W.D.Clinger makes the basic mathematical point that "infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers". Real numbers are a field as described in his post. Treating infinity as a real number makes real numbers not a field.
 
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Reality Check said:
LMAO at the denial of mathematics showing in that remark, jt512.
W.D.Clinger makes the basic mathematical point that "infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers". Real numbers are a field as described in his post. Treating infinity as a real number makes real numbers not a field.
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Do real and complex analysis depend on the underlying algebraic structure being a field?
 
Reality Check said:
W.D.Clinger makes the basic mathematical point that "infinity cannot be regarded as a real number without breaking the fundamental algebraic laws that hold for real numbers". Real numbers are a field as described in his post. Treating infinity as a real number makes real numbers not a field.
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My two cents worth.

This goes waaayyy back for me, but I recall my prof discussing zero and infinity thusly:
whereas zero is definable, infinity is not.
Zero is absence of quantity, infinity is not similarily definable.
One can take any non-negative number and subtract 1. Do it enough times and the result will be zero. The number of times one must substract 1 is known and specific but with infinity that does not hold.
 
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