Well, there is a reason "they" say you can't divide by zero. It is not all made up :)
Oftentimes when you increase the amount of things you have which are called numbers you lose interesting properties (but maybe gain other ones along the way).
When you go from the complex numbers to the quaternions you lose commutativity. When you go from the quaternions to the octonions you lose associativity. If you try to apply the algebraic rules you previously learned you will end up in trouble.
When you go from the real numbers to the complex numbers you lose a "total order" compatible with the + operator. However, this is not such a big deal for the algebra involved. So an ordinary person doesn't need to worry too much about the "rules for numbers" they learned not applying when they think about complex numbers.
The real numbers are a "field", and so people learn that all numbers have an additive inverse. People also learn that multiplication (but not division) is closed. The Riemann sphere is no longer a field though. So you need to tell people: if you want to divide by zero and call it infinity, then your "numbers" will no longer obey the nice algebraic rules you are used to.
Goodness, can I not add infinity and infinity and get a definite result on the Riemann sphere? Maybe I should use the extended reals. Or if I like, the surreals; omega plus omega is quite well behaved there.
Are you sure I need to tell people what you want me to tell them?
You can add them, but there is no additive or multiplicative inverse. So you can’t do the following:
Infinity +5 = Infinity
Therefore,
Infinity - Infinity + 5 = Infinity - Infinity
5=0
So if you have an equation like
x+y=x + z
You cannot conclude y=z because your ordinary algebra doesn’t work. I guess my point is the following:
The word “can’t” is rarely absolute*. You “can” do anything you want mathematically, but you might end up with a useless system. If you want to divide infinity by zero you end up with a different system than you might have expected, so you “can” do it, but you “can’t” do it and keep the nice properties of numbers you are used to.
*You even can have inconsistent logical systems and just accept the principle of explosion, but few would consider systems like this useful.
No additive or multiplicative inverse? There's no additive or multiplicative inverse to the naturals! The natural numbers aren't a field, a Banach space, a Hilbert space, or anything else you seem to think we want our numbers to do.
Meanwhile, on the surreals "Infinity + 5 = infinity" is wrong, and while it's fine on the Riemann or extended set, "infinity - infinity" doesn't return a defined quantity there; it's undefined there, so you *never* end up with "5=0" as you're trying to claim. If you think I'm leaving out some set that has infinity in it like the hyperreals (or, hey, some other set you personally defined), you can tell me all about it, but frankly It really seems you don't grasp the concept of infinity.
I think I must have unknowingly said something rude in one of my replies so far, because I do not know why you replied in kind. I have not really disagreed with anything you said in your original post so far or any of your comments. I was making (what I thought) was a friendly comment about the consequences of some different sets in the manner of "here is another way of looking at it." ( I do have multiple degrees in relevant fields, by the way, if that means anything to you.)
What am I misunderstanding? I know that infinity - infinity doesn’t return a defined quantity, that was my point. And I know that infinity + 5 = infinity is wrong on the surreals, I was talking only about the Riemann sphere.
I guess I think the word "can't" has different shades of meaning. If you tell me that you can't subtract a number x which is bigger than a number y; i.e. y-x, then I would say yes that is true in the natural numbers. But it is not true in the integers. If you told me that every number can be written as the ratio of two natural numbers, I would say that is true if by "numbers" you mean the rationals. If you tell me you can't divide by zero, I would say that is true in the real or complex numbers. And etc.
Most people don't know that you can extend your number systems so that you can divide by zero. That is what your original post was about. It was good, and I liked it. A while back as I initially learned all of this stuff my response was to gradually lose the feeling that the word "number" had any ultimate meaning. Any set we create which has properties we like can be called "numbers" in a family resemblance. (https://en.wikipedia.org/wiki/Family_resemblance)
So when I get questions from family members about whether or not imaginary numbers are "real" or if they are actually "numbers" I typically ask them in turn if the real numbers are "real" and I ask them what makes something a number.
Thank you for being patient. After carefully rereading your comments, I suspect that I owe you an apology! I don't typically use the word "you" that way - I objected particularly to your statement "you need to tell people: if you want to divide by zero and call it infinity, then your 'numbers' will no longer obey the nice algebraic rules you are used to." That's why I asked you "Are you sure I need to tell people what you want me to tell them?"
I am sorry that I was so keen to see a specific response from you. It sounds like you never really meant any more than "One ought to be clear: if we want to divide by zero and call it infinity, then our numbers will no longer obey the algebraic rules people are used to."
Let me add that I've actually been writing a post about the idea you expressed very straghtforwardly here: There are no real / correct / ultimate / true numbers. I want to be careful before just taking it for granted, but yet if that's correct it might prove to be very, very useful. I appreciate your bringing this idea up, and I hope I didn't cause too much offense with my previous remarks.
No worries! We are communicating by text over the internet, and I am sure I communicated in a subpar manner. I have been really enjoying your blog.
To take the idea a step further you could even call matrices “numbers” if you wanted to. It would be a strange, and I haven’t been around anybody who does that, but if that convention had taken off… so what?
C is just R^2 with a special multiplication. Alternatively, it is just a subring of the 2x2 matrixes looking like x + iy -> {{x,-y}, {y,x}}.
Or if you wanted to call three dimensional vectors numbers you could do so (with multiplication being the cross product).
I like Wittgenstein’s idea of the “family resemblance” a lot here, and in general that idea has influenced a lot of my thinking about issues like this.
I never noticed this comment! The absurdities you point out don't exist once you realize that 0/0 is still undefined, just as ∞/∞ is undefined. For example, step 5 of this proof... https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero ...has not just division by zero, but 0 divided by 0. So, we don't have 2 = 1, just undefined = undefined.
I know you don't think I'm that bright, but at the very least, be aware that I've thought about this for many years while you were doing more useful things!
But then, what do you mean by _reciprocal_ and _division_? According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1. Therefore, 0/0 = 0 × 1/0 = 0 × ∞ = 1, rather than being undefined.
I suppose anyone sufficiently interested already knows this much better than I do, but there’s a reason the definition of a field (<https://en.m.wikipedia.org/wiki/Field_(mathematics)>) does not require a reciprocal for the identity element of addition; namely, that fields with such a reciprocal are not very interesting or useful.
Let’s suppose such element exists and call it ∞. Then, 0 × ∞ = 1. In any field, 0 is an absorbing element, so it’s also true that 0 × ∞ = 0. Therefore, 1 = 0. Now, for any element a in the field, 1 × a = a, and 0 × a = 0 = 1, so a = 0 = 1; that is, the field has but one element, which, of course, must be the identity element of both addition and multiplication, and its own reciprocal. That’s why what Eric Brown called “absurdities” are actually true in this system.
I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
> According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1.
No; if that were true, the usual definitions would also imply that 1×0 = 0 means 0/0 = 1, and now we are dividing by zero. The usual way of carrying out math in grade school holds that multiplication by 0 has no legal inverse, in contradiction to that kind of reasoning.
Although there are other ways of handling it, the "divide by 0" function can be treated as non-invertible in the same way that grade school math treats "multiply by 0" as invertible, because it is many-to-one. That is, if any x divided by 0 = complex infinity, then complex infinity times 0 = any x, or 0 × ∞ = undefined. This is nothing special; according to conventional mathematics which avoids surreals and transfinites, many-to-one functions are often treated as non-invertible.
Granted, sometimes many-to-oneedness can be overcome. Say in the case of e^z, with complex number z, where the inverse function is many-valued log(z) where periodicity requires making a branch cut (e.g. https://functions.wolfram.com/ElementaryFunctions/Log/visualizations/5/ ). This kind of overcoming of objections is common, because "you can't do that" is often a barrier to getting a useful result.
Ultimately, I didn't invent the Riemann Sphere or the hyperreals ( https://en.wikipedia.org/wiki/Hyperreal_number ). Arguments that these are "absurdities" are analogous to claims the square root of negative 1 result in "absurdities." OK, but physicists are going to drive right past those objections with the Dirac Delta function and the Schroedinger Equation, and show our entire reality is "absurd" with the discovery of Quantum Mechanics.
> I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
By that line of reasoning I may as well just welcome you to Things to Read, and remind you that I am in charge. I don't really care if you use my terms and concepts, though; I'm just going to keep pointing out that "Annie Ant is on a pilgrimage to the Holy Land, she's 8 miles away, and travels at 1/4 mile per hour, how long does it take her to get there" is a reasonable question and has a reasonable answer, and "If Annie is squashed flat by an apple cart and her speed is reduced to 0, how long does it take her to get there" is just as reasonable, and has just as reasonable an answer.
There might be something wrong with me, but I just don't care. If they say dividing things by zero is impossible, I say fine. They make the rules! If they say dividing things by zero equals infinity, I say fine too. It makes sense, after all, because dividing something by 0.000001 results in a very big number.
That exercise with nines was difficult to understand. It gave me the impression that there are people who actually like mathematical concepts, the same way that I actually like words. People who like to learn new mathematical concepts for the sake of it. That is strange to me.
Before I moved here to the midlatitudes, I didn't care about climate, at all. Sure, CO2 --> global warming --> something something coral reefs die, and that bothered me, but the actual workings of the Earth could not be of less interest.
Then a time came when I had to look around and consider where to move. I started poking and poking, and found this hideous map, Köppen's map of dizzying complexity, where blue means hot, and also cold:
As before, I had zero interest. But this way, Köppen's way, was the standard way of discussing climates across Wikipedia. And because I was considering places to live for long periods, I grit my teeth for a few hours, and learned the meaning of all the little letter codes to try to make sense of things, and determine where good climates could be found. And then - then, like a ray shining forth from the clouds, I Found This:
Legible, comprehensible, and so, so lovely. The reds, the greens, the blues, the shining whites! Trewartha's glorious map called to me, and I listened; what began as hours stretched into years. I gathered data, wrangled with climate officianados, and finally produced my own map, and my own classification system, about which no one cares.
// that's hilarious, I went to all this effort to make sense but in that moment pander to the lowest common denominator as described in that normal distribution above, all while targeting a pet peeve of yours. Hindsight is a painful logic. I guess any mathematical law or expectation works the same way. Parallel lines defined as never meeting would be another.
// takeaway, there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
Well, there is a reason "they" say you can't divide by zero. It is not all made up :)
Oftentimes when you increase the amount of things you have which are called numbers you lose interesting properties (but maybe gain other ones along the way).
When you go from the complex numbers to the quaternions you lose commutativity. When you go from the quaternions to the octonions you lose associativity. If you try to apply the algebraic rules you previously learned you will end up in trouble.
When you go from the real numbers to the complex numbers you lose a "total order" compatible with the + operator. However, this is not such a big deal for the algebra involved. So an ordinary person doesn't need to worry too much about the "rules for numbers" they learned not applying when they think about complex numbers.
The real numbers are a "field", and so people learn that all numbers have an additive inverse. People also learn that multiplication (but not division) is closed. The Riemann sphere is no longer a field though. So you need to tell people: if you want to divide by zero and call it infinity, then your "numbers" will no longer obey the nice algebraic rules you are used to.
Goodness, can I not add infinity and infinity and get a definite result on the Riemann sphere? Maybe I should use the extended reals. Or if I like, the surreals; omega plus omega is quite well behaved there.
Are you sure I need to tell people what you want me to tell them?
You can add them, but there is no additive or multiplicative inverse. So you can’t do the following:
Infinity +5 = Infinity
Therefore,
Infinity - Infinity + 5 = Infinity - Infinity
5=0
So if you have an equation like
x+y=x + z
You cannot conclude y=z because your ordinary algebra doesn’t work. I guess my point is the following:
The word “can’t” is rarely absolute*. You “can” do anything you want mathematically, but you might end up with a useless system. If you want to divide infinity by zero you end up with a different system than you might have expected, so you “can” do it, but you “can’t” do it and keep the nice properties of numbers you are used to.
*You even can have inconsistent logical systems and just accept the principle of explosion, but few would consider systems like this useful.
No additive or multiplicative inverse? There's no additive or multiplicative inverse to the naturals! The natural numbers aren't a field, a Banach space, a Hilbert space, or anything else you seem to think we want our numbers to do.
Meanwhile, on the surreals "Infinity + 5 = infinity" is wrong, and while it's fine on the Riemann or extended set, "infinity - infinity" doesn't return a defined quantity there; it's undefined there, so you *never* end up with "5=0" as you're trying to claim. If you think I'm leaving out some set that has infinity in it like the hyperreals (or, hey, some other set you personally defined), you can tell me all about it, but frankly It really seems you don't grasp the concept of infinity.
I think I must have unknowingly said something rude in one of my replies so far, because I do not know why you replied in kind. I have not really disagreed with anything you said in your original post so far or any of your comments. I was making (what I thought) was a friendly comment about the consequences of some different sets in the manner of "here is another way of looking at it." ( I do have multiple degrees in relevant fields, by the way, if that means anything to you.)
What am I misunderstanding? I know that infinity - infinity doesn’t return a defined quantity, that was my point. And I know that infinity + 5 = infinity is wrong on the surreals, I was talking only about the Riemann sphere.
I guess I think the word "can't" has different shades of meaning. If you tell me that you can't subtract a number x which is bigger than a number y; i.e. y-x, then I would say yes that is true in the natural numbers. But it is not true in the integers. If you told me that every number can be written as the ratio of two natural numbers, I would say that is true if by "numbers" you mean the rationals. If you tell me you can't divide by zero, I would say that is true in the real or complex numbers. And etc.
Most people don't know that you can extend your number systems so that you can divide by zero. That is what your original post was about. It was good, and I liked it. A while back as I initially learned all of this stuff my response was to gradually lose the feeling that the word "number" had any ultimate meaning. Any set we create which has properties we like can be called "numbers" in a family resemblance. (https://en.wikipedia.org/wiki/Family_resemblance)
So when I get questions from family members about whether or not imaginary numbers are "real" or if they are actually "numbers" I typically ask them in turn if the real numbers are "real" and I ask them what makes something a number.
Thank you for being patient. After carefully rereading your comments, I suspect that I owe you an apology! I don't typically use the word "you" that way - I objected particularly to your statement "you need to tell people: if you want to divide by zero and call it infinity, then your 'numbers' will no longer obey the nice algebraic rules you are used to." That's why I asked you "Are you sure I need to tell people what you want me to tell them?"
I am sorry that I was so keen to see a specific response from you. It sounds like you never really meant any more than "One ought to be clear: if we want to divide by zero and call it infinity, then our numbers will no longer obey the algebraic rules people are used to."
Let me add that I've actually been writing a post about the idea you expressed very straghtforwardly here: There are no real / correct / ultimate / true numbers. I want to be careful before just taking it for granted, but yet if that's correct it might prove to be very, very useful. I appreciate your bringing this idea up, and I hope I didn't cause too much offense with my previous remarks.
No worries! We are communicating by text over the internet, and I am sure I communicated in a subpar manner. I have been really enjoying your blog.
To take the idea a step further you could even call matrices “numbers” if you wanted to. It would be a strange, and I haven’t been around anybody who does that, but if that convention had taken off… so what?
C is just R^2 with a special multiplication. Alternatively, it is just a subring of the 2x2 matrixes looking like x + iy -> {{x,-y}, {y,x}}.
Or if you wanted to call three dimensional vectors numbers you could do so (with multiplication being the cross product).
I like Wittgenstein’s idea of the “family resemblance” a lot here, and in general that idea has influenced a lot of my thinking about issues like this.
I look forward to reading your article!
My problem with dividing by zero is that it lets you conclude that 2 = 1, and other absurdities.
There are tons of algebraic reductions that smuggle in a division by zero along the way that end up concluding that 2 = 1.
I never noticed this comment! The absurdities you point out don't exist once you realize that 0/0 is still undefined, just as ∞/∞ is undefined. For example, step 5 of this proof... https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero ...has not just division by zero, but 0 divided by 0. So, we don't have 2 = 1, just undefined = undefined.
I know you don't think I'm that bright, but at the very least, be aware that I've thought about this for many years while you were doing more useful things!
But then, what do you mean by _reciprocal_ and _division_? According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1. Therefore, 0/0 = 0 × 1/0 = 0 × ∞ = 1, rather than being undefined.
I suppose anyone sufficiently interested already knows this much better than I do, but there’s a reason the definition of a field (<https://en.m.wikipedia.org/wiki/Field_(mathematics)>) does not require a reciprocal for the identity element of addition; namely, that fields with such a reciprocal are not very interesting or useful.
Let’s suppose such element exists and call it ∞. Then, 0 × ∞ = 1. In any field, 0 is an absorbing element, so it’s also true that 0 × ∞ = 0. Therefore, 1 = 0. Now, for any element a in the field, 1 × a = a, and 0 × a = 0 = 1, so a = 0 = 1; that is, the field has but one element, which, of course, must be the identity element of both addition and multiplication, and its own reciprocal. That’s why what Eric Brown called “absurdities” are actually true in this system.
I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
> According to the usual definitions, 1/0 = ∞ means 0 × ∞ = 1.
No; if that were true, the usual definitions would also imply that 1×0 = 0 means 0/0 = 1, and now we are dividing by zero. The usual way of carrying out math in grade school holds that multiplication by 0 has no legal inverse, in contradiction to that kind of reasoning.
Although there are other ways of handling it, the "divide by 0" function can be treated as non-invertible in the same way that grade school math treats "multiply by 0" as invertible, because it is many-to-one. That is, if any x divided by 0 = complex infinity, then complex infinity times 0 = any x, or 0 × ∞ = undefined. This is nothing special; according to conventional mathematics which avoids surreals and transfinites, many-to-one functions are often treated as non-invertible.
Granted, sometimes many-to-oneedness can be overcome. Say in the case of e^z, with complex number z, where the inverse function is many-valued log(z) where periodicity requires making a branch cut (e.g. https://functions.wolfram.com/ElementaryFunctions/Log/visualizations/5/ ). This kind of overcoming of objections is common, because "you can't do that" is often a barrier to getting a useful result.
Ultimately, I didn't invent the Riemann Sphere or the hyperreals ( https://en.wikipedia.org/wiki/Hyperreal_number ). Arguments that these are "absurdities" are analogous to claims the square root of negative 1 result in "absurdities." OK, but physicists are going to drive right past those objections with the Dirac Delta function and the Schroedinger Equation, and show our entire reality is "absurd" with the discovery of Quantum Mechanics.
> I think the real lesson in any human interaction is, “I am in charge, so use my terms and concepts or you’ll suffer”, said or implied by whoever is actually in charge.
By that line of reasoning I may as well just welcome you to Things to Read, and remind you that I am in charge. I don't really care if you use my terms and concepts, though; I'm just going to keep pointing out that "Annie Ant is on a pilgrimage to the Holy Land, she's 8 miles away, and travels at 1/4 mile per hour, how long does it take her to get there" is a reasonable question and has a reasonable answer, and "If Annie is squashed flat by an apple cart and her speed is reduced to 0, how long does it take her to get there" is just as reasonable, and has just as reasonable an answer.
> I am in charge
Duly noted.
Sorry if you suffered.
There might be something wrong with me, but I just don't care. If they say dividing things by zero is impossible, I say fine. They make the rules! If they say dividing things by zero equals infinity, I say fine too. It makes sense, after all, because dividing something by 0.000001 results in a very big number.
That exercise with nines was difficult to understand. It gave me the impression that there are people who actually like mathematical concepts, the same way that I actually like words. People who like to learn new mathematical concepts for the sake of it. That is strange to me.
I like both.
I liked the mathematical concepts more in school, where I was fed them. That made them seem more relevant.
Interest is a strange thing.
Before I moved here to the midlatitudes, I didn't care about climate, at all. Sure, CO2 --> global warming --> something something coral reefs die, and that bothered me, but the actual workings of the Earth could not be of less interest.
Then a time came when I had to look around and consider where to move. I started poking and poking, and found this hideous map, Köppen's map of dizzying complexity, where blue means hot, and also cold:
https://upload.wikimedia.org/wikipedia/commons/d/d5/K%C3%B6ppen-Geiger_Climate_Classification_Map.png
As before, I had zero interest. But this way, Köppen's way, was the standard way of discussing climates across Wikipedia. And because I was considering places to live for long periods, I grit my teeth for a few hours, and learned the meaning of all the little letter codes to try to make sense of things, and determine where good climates could be found. And then - then, like a ray shining forth from the clouds, I Found This:
https://en.wikipedia.org/wiki/K%C3%B6ppen_climate_classification#/media/File:K%C3%B6ppen-Geiger_Climate_Classification_Map.png
Legible, comprehensible, and so, so lovely. The reds, the greens, the blues, the shining whites! Trewartha's glorious map called to me, and I listened; what began as hours stretched into years. I gathered data, wrangled with climate officianados, and finally produced my own map, and my own classification system, about which no one cares.
Interest is a strange thing.
// I find support for my "logic is a hindsight", in Wolfram's words on entropy & thermodynamics at https://www.youtube.com/watch?v=dkpDjd2nHgo
// that's hilarious, I went to all this effort to make sense but in that moment pander to the lowest common denominator as described in that normal distribution above, all while targeting a pet peeve of yours. Hindsight is a painful logic. I guess any mathematical law or expectation works the same way. Parallel lines defined as never meeting would be another.
// takeaway, there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
Worry not, dear samorzewski. If you hadn't commented, I wouldn't have even connected this to your post on ratios at https://whyweshould.substack.com/p/the-janus-ratio
When I read it, I had a vague sense you might just be talking about division by zero somewhat metaphorically, or for the sake of making a point.
> there is always hope for narcissist "to world", it will just take longer than the expected lifespan of the universe.
LOL