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Blog - Exponential Zero

edited June 2016 in Azimuth Blog

I just reworked my latest forum post into a short blog article. It's a combination of math and biography.

John, I recall that Andrew told you how to turn on Markdown on the blog, so that we don't have to convert the markdown to html by hand. If that's not working right, just let us know, so that can continue to do the conversions.

Comments

  • 1.

    Updated the text, and added some problems.

    Comment Source:Updated the text, and added some problems.
  • 2.
    edited April 2016

    Added more problems.

    John, as I mentioned, I haven't worked through the answers to the problems (for the most part). Let me know if the answers to any will be too obvious to the readership of the blog, in which case we can rework the questions -- maybe just by identifying what is a simple exercise and what is not.

    Comment Source:Added more problems. John, as I mentioned, I haven't worked through the answers to the problems (for the most part). Let me know if the answers to any will be too obvious to the readership of the blog, in which case we can rework the questions -- maybe just by identifying what is a simple exercise and what is not.
  • 3.

    The first problem is really just about understanding exponentiation between complex numbers, which certainly is an interesting creature -- so maybe it should be flagged as an exercise?

    And feel free anyone here to work out any of the answers right in this thread.

    Comment Source:The first problem is really just about understanding exponentiation between complex numbers, which certainly is an interesting creature -- so maybe it should be flagged as an exercise? And feel free anyone here to work out any of the answers right in this thread.
  • 4.

    Hi! You say

    "So the graph of f(x) has an impressive discontinuity at zero."

    Can you draw this and include a figure? I know it's a bit silly, but I find it a bit frustrating to be told something looks impressive but not get to see it. I can easily visualize it, of course.

    Mathematicians would draw a horizontal line at y = 1 ending in an empty circle at x = 0, and then a black dot at x = 0, y = 0. Here are some similar drawings.

    If it's too much work, don't bother...

    Comment Source:Hi! You say "So the graph of f(x) has an impressive discontinuity at zero." Can you draw this and include a figure? I know it's a bit silly, but I find it a bit frustrating to be told something looks impressive but not get to see it. I can easily visualize it, of course. Mathematicians would draw a horizontal line at y = 1 ending in an empty circle at x = 0, and then a black dot at x = 0, y = 0. [Here are some similar drawings](https://www.google.com/search?q=step+discontinuity&client=ubuntu&hs=t7p&channel=fs&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwj5wuXbo5HMAhWHeSYKHZVmDqwQsAQILg&biw=1288&bih=671&dpr=0.9). If it's too much work, don't bother...
  • 5.
    edited April 2016

    I got scared for a second and thought you were asking me to draw the "wild singularity" described in the Calabi lecture. :)

    Can you visualize that one?

    I can only picture it in a qualitative way. As for the geometry of the level sets, and how they all bunch up together to a shared limit point at (0,0) -- I am not able to visualize this.

    That's a two-parameter family of level sets, each with dimension $d$, which form a nice partition of $\mathbb{C}^2$ in the neighborhood of (0,0), all of which "almost touch" at the shared limit point (0,0).

    I said dimension $d$, because that was one of my questions -- what is the dimension of the level sets, is it fractal near the singularity, and does it depend on the level set?

    Is this family of level sets "smoothly layered," or are they twisted and entangled?

    If these appear to be reasonable descriptions and questions, I would add them to the text of the blog.

    Comment Source:I got scared for a second and thought you were asking me to draw the "wild singularity" described in the Calabi lecture. :) Can you visualize that one? I can only picture it in a qualitative way. As for the geometry of the level sets, and how they all bunch up together to a shared limit point at (0,0) -- I am not able to visualize this. That's a two-parameter family of level sets, each with dimension $d$, which form a nice partition of $\mathbb{C}^2$ in the neighborhood of (0,0), all of which "almost touch" at the shared limit point (0,0). I said dimension $d$, because that was one of my questions -- what is the dimension of the level sets, is it fractal near the singularity, and does it depend on the level set? Is this family of level sets "smoothly layered," or are they twisted and entangled? If these appear to be reasonable descriptions and questions, I would add them to the text of the blog.
  • 6.
    edited April 2016

    Can you draw this and include a figure?

    Sure, I will look into making the graph that you mentioned. To be honest, I've been lazy about learning plotting packages, and am way behind the times. But it's time to upgrade!

    I'm hoping to do this in matplotlib/python, as that's becoming my new home base for scientific computing.

    Any tips, folks, on making this graph in matplotlib?

    Thanks!

    Comment Source:> Can you draw this and include a figure? Sure, I will look into making the graph that you mentioned. To be honest, I've been lazy about learning plotting packages, and am way behind the times. But it's time to upgrade! I'm hoping to do this in matplotlib/python, as that's becoming my new home base for scientific computing. Any tips, folks, on making this graph in matplotlib? Thanks!
  • 7.
    edited April 2016

    Mathematicians would draw a horizontal line at y = 1 ending in an empty circle at x = 0, and then a black dot at x = 0, y = 0.

    In the text, my starting point is the function $y = 0^x$ (not the level sets of the function $x ^ y$).

    For that, I see:

    a rightward-directed horizontal ray starting at (0,0), with an empty circle at (0,0), and then a black dot at x = 0, y = 1.

    It's infinite for $x < 0$ -- and arguably positive infinity, if you take the limit as the base goes to zero from above. That's too too hard to depict on the graph, so we'll leave it undefined.

    Comment Source:> Mathematicians would draw a horizontal line at y = 1 ending in an empty circle at x = 0, and then a black dot at x = 0, y = 0. In the text, my starting point is the function $y = 0^x$ (not the level sets of the function $x ^ y$). For that, I see: a rightward-directed horizontal ray starting at (0,0), with an empty circle at (0,0), and then a black dot at x = 0, y = 1. It's infinite for $x < 0$ -- and arguably positive infinity, if you take the limit as the base goes to zero from above. That's too too hard to depict on the graph, so we'll leave it undefined.
  • 8.
    edited April 2016

    That's the easy part.

    But there are some nuances to be worked through, due to the fact that $x ^ y$ is a multi-valued function.

    It doesn't make sense to speak of "the" limit of $x ^ y$, as $(x, y)$ approaches zero along a path, because there will be a different limit for each branch of the function.

    Comment Source:That's the easy part. But there are some nuances to be worked through, due to the fact that $x ^ y$ is a multi-valued function. It doesn't make sense to speak of "the" limit of $x ^ y$, as $(x, y)$ approaches zero along a path, because there will be a different limit for each branch of the function.
  • 9.
    edited April 2016

    In the article I wrote:

    Note the nuance that, since $ln(x)$ is multi-valued, so is the function $x^y = e^{y \ ln(x)}$. I am assuming that a level set $x^y = c$ of this function means all points $(x,y)$ where $x^y$ contains $c$.

    But this doesn't look like a useful definition, because these "level sets" no longer partition the domain of the function.

    EDIT: I removed this definition from the problems at the end, and tightened up the statement of the problems.

    Comment Source:In the article I wrote: > Note the nuance that, since $ln(x)$ is multi-valued, so is the function $x^y = e^{y \ ln(x)}$. I am assuming that a level set $x^y = c$ of this function means all points $(x,y)$ where $x^y$ contains $c$. But this doesn't look like a useful definition, because these "level sets" no longer partition the domain of the function. EDIT: I removed this definition from the problems at the end, and tightened up the statement of the problems.
  • 10.
    edited April 2016

    Educational sidenote: I am seeing that there is some beautiful geometry involved in just picturing the basic operation of raising one complex number to the power of another complex number.

    For instance, try to picture these facts geometrically:

    • $0 ^ x$ is 0 if $x$ is positive, infinite if $x$ is negative, and cannot be defined (divergent) if $x$ is imaginary. The key to this is that, viewed as a limit, $ln(0)$ is negative infinity. No matter from what angle it is approached, as complex $\epsilon$ goes to zero, $\epsilon ^ i$ keeps orbiting around the origin, with a non-zero radius.

    • For rational $q = m / n$, where $m$ and $n$ have no common factors, $x ^ q$ has exactly $n$ solutions, all with the same absolute value, with phases differing in increments of $2 \pi / n$.

    • For irrational $r$, $x ^ r$ has an infinite number of solutions, all with the same absolute value, with a countable set of phases that are dense in the interval from $-\pi$ to $\pi$.

    Comment Source:Educational sidenote: I am seeing that there is some beautiful geometry involved in just picturing the basic operation of raising one complex number to the power of another complex number. For instance, try to picture these facts geometrically: * $0 ^ x$ is 0 if $x$ is positive, infinite if $x$ is negative, and cannot be defined (divergent) if $x$ is imaginary. The key to this is that, viewed as a limit, $ln(0)$ is negative infinity. No matter from what angle it is approached, as complex $\epsilon$ goes to zero, $\epsilon ^ i$ keeps orbiting around the origin, with a non-zero radius. * For rational $q = m / n$, where $m$ and $n$ have no common factors, $x ^ q$ has exactly $n$ solutions, all with the same absolute value, with phases differing in increments of $2 \pi / n$. * For irrational $r$, $x ^ r$ has an infinite number of solutions, all with the same absolute value, with a countable set of phases that are dense in the interval from $-\pi$ to $\pi$.
  • 11.
    edited April 2016

    Now I suspect that the point he was making in the lecture holds true for any branch of $x ^ y$.

    And, as long as we restrict ourselves to one branch of $ln(x)$, and hence of $x ^ y$, everything is single-valued, and the usual definition of a level set is what is meant in the statement that all of the level sets have the origin as a limit point.

    Comment Source:Now I suspect that the point he was making in the lecture holds true for any branch of $x ^ y$. And, as long as we restrict ourselves to one branch of $ln(x)$, and hence of $x ^ y$, everything is single-valued, and the usual definition of a level set is what is meant in the statement that all of the level sets have the origin as a limit point.
  • 12.

    David wrote:

    In the text, my starting point is the function $y=0^x$ (not the level sets of the function $x^y$

    Right.

    For that, I see:

    a rightward-directed horizontal ray starting at (0,0), with an empty circle at (0,0), and then a black dot at x = 0, y = 1.

    Right. It's pathetically simple, one can draw it with any technology that can draw a line, a small circle and a dot. image Perhaps it's pointless to bother. It's just mildly frustrating to hear that something looks "impressive" and not see it.

    Never mind... I think I should go ahead and post this. Okay?

    Comment Source:David wrote: > In the text, my starting point is the function $y=0^x$ (not the level sets of the function $x^y$ Right. > For that, I see: > a rightward-directed horizontal ray starting at (0,0), with an empty circle at (0,0), and then a black dot at x = 0, y = 1. Right. It's pathetically simple, one can draw it with any technology that can draw a line, a small circle and a dot. <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif"> Perhaps it's pointless to bother. It's just mildly frustrating to hear that something looks "impressive" and not see it. Never mind... I think I should go ahead and post this. Okay?
  • 13.

    I see your point. I'd like to add the graph. I'm slammed for the next couple of days, but then will attend to it. Thanks!

    Comment Source:I see your point. I'd like to add the graph. I'm slammed for the next couple of days, but then will attend to it. Thanks!
  • 14.

    Okay, I'll wait until you add it. Monday I will post Matteo Smerlak's new post; then I'll post yours later in the week if you get it ready.

    Comment Source:Okay, I'll wait until you add it. Monday I will post Matteo Smerlak's new post; then I'll post yours later in the week if you get it ready.
  • 15.

    I added the graph, and am now developing the text somewhat further. I'll let you know when it's ready, which will be this week. Thanks.

    Comment Source:I added the graph, and am now developing the text somewhat further. I'll let you know when it's ready, which will be this week. Thanks.
  • 16.

    Okay, great! We can use a new blog article anytime now.

    Comment Source:Okay, great! We can use a new blog article anytime now.
  • 17.
    edited May 2016

    This is ready to go. I may fix a word or two here or there, but I'm letting it go.

    John, can you review it for math errors? Thanks!

    Thanks to my daughter Katy Tanzer, who showed me how to use Google Drawings to make the graph.

    Comment Source:This is ready to go. I may fix a word or two here or there, but I'm letting it go. John, can you review it for math errors? Thanks! Thanks to my daughter Katy Tanzer, who showed me how to use Google Drawings to make the graph.
  • 18.
    edited May 2016

    The graph looks nice! It's very simple but it somehow makes everything more real, the way images always do.

    I don't see any math errors. This post may get a lot of pushback from professional mathematicians, because people like them - that is, like me - don't talk about these issues in quite these ways. If I wrote a post about this subject it would sound very different. However, I don't want to turn this post into something it's not. I hope it starts some discussions that lead to a lot of people learning a lot of math.

    Can you post it, David? You have the power to post blog articles on Azimuth, and if you post it it'll come out under your name.

    You'll need to change every single equation to the Wordpress version of LaTeX, where equations are formatted like this:

    $latex z^2 = 0^{0+0} = z$

    instead of like this

    $z^2 = 0^{0+0} = z$

    That is, the word "latex" must occur directly after the left dollar sign, with no intervening space, and it must have a space after it.

    An annoying feature of old-fashioned LaTeX is that there's no instant way to do this with a global search-and-replace, because a left dollar sign looks just like a right dollar sign. Newer versions use \( instead of a left dollar sign and \) for a right dollar sign, but alas, those commands don't seem to work on the Azimuth Wiki. So, whenever I transfer a blog post from the wiki to the blog I need to suffer a bit to format the equations. I have some tricks to reduce my suffering, but not eliminate it.

    Comment Source:The graph looks nice! It's very simple but it somehow makes everything more real, the way images always do. I don't see any math errors. This post may get a lot of pushback from professional mathematicians, because people like them - that is, like me - don't talk about these issues in quite these ways. If I wrote a post about this subject it would sound very different. However, I don't want to turn this post into something it's not. I hope it starts some discussions that lead to a lot of people learning a lot of math. Can you post it, David? You have the power to post blog articles on _Azimuth_, and if you post it it'll come out under your name. You'll need to change every single equation to the Wordpress version of LaTeX, where equations are formatted like this: `$latex z^2 = 0^{0+0} = z$` instead of like this `$z^2 = 0^{0+0} = z$` That is, the word "latex" must occur directly after the left dollar sign, with no intervening space, and it must have a space after it. An annoying feature of old-fashioned LaTeX is that there's no instant way to do this with a global search-and-replace, because a left dollar sign looks just like a right dollar sign. Newer versions use `\(` instead of a left dollar sign and `\)` for a right dollar sign, but alas, those commands don't seem to work on the Azimuth Wiki. So, whenever I transfer a blog post from the wiki to the blog I need to suffer a bit to format the equations. I have some tricks to reduce my suffering, but not eliminate it.
  • 19.

    I am not sure about this, let me sleep on it.

    If this comes across as too naive, then I might want to let it go -- or you could help me with some of the professional feedback that you have in mind. I'm sure there's a way to say it that isn't too harsh, and which I could absorb into my thinking, process it, and still keep the article in my own voice.

    I can only surmise what you're hinting at, perhaps along these lines:

    • Format is a semi-biographical narrative. If it's just a matter of the literary format, then I don't care too much about it.

    • The math part has explanatory information that is too obvious/basic. E.g. I included the footnote defining the geometric interpretation of addition and multiplication of complex numbers. That will be obvious and basic to mathematicians, but probably not to all readers of the blog -- so it was an element of "outreach," which I tried to encapsulate in a footnote.

    • Maybe there is some other, perhaps, higher-level way to look at the matter, so that my questions about the structure at the origin has an obvious answer. Suppose you told me that, through the use of the yigma-zigma function, it is obvious that the behavior at the origin looks smooth. And then I though, wow, why didn't I see that. Well, I could incorporate that into the narrative. First I was wondering how the structure looked at the origin, so I raised it on the Azimuth forum. Then John led me to the yigma-zigma function, which works this way...

    In the end, the essence of it is a story about how following up on something that seems just a bit quirky can lead into deeper territory, and a little quest in that direction. I could cut down on the more technical paragraphs, or even extend them just a bit, and still keep this story line.

    Meanwhile, there are plenty of other things to be done, and I working up some other blog/wiki/forum material which is directly in line with my professional education.

    This blog was a one-off affair, and tangential to my main line.

    Comment Source:I am not sure about this, let me sleep on it. If this comes across as too naive, then I might want to let it go -- or you could help me with some of the professional feedback that you have in mind. I'm sure there's a way to say it that isn't too harsh, and which I could absorb into my thinking, process it, and still keep the article in my own voice. I can only surmise what you're hinting at, perhaps along these lines: * Format is a semi-biographical narrative. If it's just a matter of the literary format, then I don't care too much about it. * The math part has explanatory information that is too obvious/basic. E.g. I included the footnote defining the geometric interpretation of addition and multiplication of complex numbers. That will be obvious and basic to mathematicians, but probably not to all readers of the blog -- so it was an element of "outreach," which I tried to encapsulate in a footnote. * Maybe there is some other, perhaps, higher-level way to look at the matter, so that my questions about the structure at the origin has an obvious answer. Suppose you told me that, through the use of the yigma-zigma function, it is obvious that the behavior at the origin looks smooth. And then I though, wow, why didn't I see that. Well, I could incorporate that into the narrative. First I was wondering how the structure looked at the origin, so I raised it on the Azimuth forum. Then John led me to the yigma-zigma function, which works this way... In the end, the essence of it is a story about how following up on something that seems just a bit quirky can lead into deeper territory, and a little quest in that direction. I could cut down on the more technical paragraphs, or even extend them just a bit, and still keep this story line. Meanwhile, there are plenty of other things to be done, and I working up some other blog/wiki/forum material which is directly in line with my professional education. This blog was a one-off affair, and tangential to my main line.
  • 20.
    edited May 2016

    Okay, I'll be less circumspect in my remarks.

    The debate over "what's $0^0$?" is extremely famous. Every mathematician has had to grapple with it. Most of us have argued about it in our youth, often in ways that seem embarrassing in retrospect. So, raising the issue at all is likely to provoke some groans among professionals. On the other hand, amateurs start arguing about it - and they can get very heated! I think you could head off both these reactions by briefly acknowledging the tangled history of this issue. Here's a good quick overview:

    Here's a good summary of what some of those debates are like, minus the curses and insults:

    You'll notice this blog article provoked 1,120 comments!

    There's been a similar long argument about 'multi-valued functions', though fewer people are involved because it's more technical.

    I don't think you made any actual mistakes. And I don't think the "semi-biographical narrative" is bad - in fact, I really liked that! So, I think you should go ahead with this, maybe after some minor tweaks.

    Comment Source:Okay, I'll be less circumspect in my remarks. The debate over "what's $0^0$?" is extremely famous. Every mathematician has had to grapple with it. Most of us have argued about it in our youth, often in ways that seem embarrassing in retrospect. So, raising the issue at all is likely to provoke some groans among professionals. On the other hand, amateurs start arguing about it - and they can get very heated! I think you could head off both these reactions by briefly acknowledging the tangled history of this issue. Here's a good quick overview: * _Wikipedia_, [Zero to the power of zero](https://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero). Here's a good summary of what some of those debates are like, minus the curses and insults: * [Q: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree?](http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/), _Ask a Mathematician_. You'll notice this blog article provoked 1,120 comments! There's been a similar long argument about 'multi-valued functions', though fewer people are involved because it's more technical. I don't think you made any actual mistakes. And I don't think the "semi-biographical narrative" is bad - in fact, I really liked that! So, I think you should go ahead with this, maybe after some minor tweaks.
  • 21.
    edited May 2016

    Thanks! That's a good bit of perspective, which I will use to reprocess it one more time.

    Speaking of the long ruminative arguments about what $0^0$ should be, I like the statement that I came up with :), which is factual:

    But what about $0^0$? By the rules of exponents, if it is defined, it must be 1. Let $z = 0^0$. Then $z^2 = 0^{0+0} = z$, and $1/z = 0^{-0} = z$. Since $z$ equals its square, and equals its reciprocal, it must be 1.

    The other option, of course, is to leave it undefined -- that's a subjective choice.

    Comment Source:Thanks! That's a good bit of perspective, which I will use to reprocess it one more time. Speaking of the long ruminative arguments about what $0^0$ should be, I like the statement that I came up with :), which is factual: > But what about $0^0$? By the rules of exponents, _if_ it is defined, it must be 1. Let $z = 0^0$. Then $z^2 = 0^{0+0} = z$, and $1/z = 0^{-0} = z$. Since $z$ equals its square, and equals its reciprocal, it must be 1. The other option, of course, is to leave it undefined -- that's a subjective choice.
  • 22.
    edited May 2016

    You're not just assuming $0^0$ is defined; you're assuming certain rules hold, and your rule $1/0^0 = 0^{-0}$ implicitly assumes $0^0$ is nonzero: otherwise the rule doesn't make sense.

    As a pedant, I'd say something like this:

    Suppose we assume $0^x$ is defined for all $x \ge 0$.

    If we additionally assume $0^{x+y} = 0^x 0^y$ for all $x,y\ge 0$, then we must have $0^0 = 0^{0+0} = 0^0 0^0$, so $0^0$ must equal its square. This forces $0^0$ to equal either $0$ or $1$, which of course are the top two contenders in arguments about this issue.

    If we additionally assume $0^x = 0$ for $x > 0$ and $0^x$ is continuous as a function of $x$, then we must have $0^0 = 0$.

    On the other hand, if we additionally assume that $0^{-0}$ is the inverse of $0^0$, then we must have $0^0 = 1$.

    However, this is why I'm not the right guy to write a fun, entertaining post on this subject!

    Comment Source:You're not just assuming $0^0$ is defined; you're assuming certain rules hold, and your rule $1/0^0 = 0^{-0}$ implicitly assumes $0^0$ is nonzero: otherwise the rule doesn't make sense. As a pedant, I'd say something like this: > Suppose we assume $0^x$ is defined for all $x \ge 0$. > If we additionally assume $0^{x+y} = 0^x 0^y$ for all $x,y\ge 0$, then we must have $0^0 = 0^{0+0} = 0^0 0^0$, so $0^0$ must equal its square. This forces $0^0$ to equal either $0$ or $1$, which of course are the top two contenders in arguments about this issue. > If we additionally assume $0^x = 0$ for $x > 0$ and $0^x$ is continuous as a function of $x$, then we must have $0^0 = 0$. > On the other hand, if we additionally assume that $0^{-0}$ is the inverse of $0^0$, then we must have $0^0 = 1$. However, this is why I'm not the right guy to write a fun, entertaining post on this subject! <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif" alt = ""/>
  • 23.
    edited May 2016

    You're not just assuming $0^0$ is defined; you're assuming certain rules hold, and your rule $1/0^0 = 0^{-0}$ implicitly assumes $0^0$ is nonzero: otherwise the rule doesn't make sense.

    Let me restate my point, this time in a clearer form. It is hard to avoid being pedantic here, but what can we do, we now have on the table a question about what is mathematically true.

    Claim: If the general propositions of arithmetic are accepted without introducing ad-hoc exceptions, then if $0^0$ is defined, it must equal 1.

    Here is the general proposition that I have in mind:

    For all a,b, if $a^b$ is defined, then $a^{-b} = 1/a^b$.

    I am content to accept this without qualification, and to see what its logical consequences are.

    Therefore:

    For all a, if $a^0$ is defined, then $a^0 = 1/a^0$.

    Further specialization gives:

    If $0^0$ is defined, then $0^0 = 1/0^0$.

    Hence it is not 0.

    The rest of the argument is as given above.

    Comment Source:> You're not just assuming $0^0$ is defined; you're assuming certain rules hold, and your rule $1/0^0 = 0^{-0}$ implicitly assumes $0^0$ is nonzero: otherwise the rule doesn't make sense. Let me restate my point, this time in a clearer form. It is hard to avoid being pedantic here, but what can we do, we now have on the table a question about what is mathematically true. Claim: If the general propositions of arithmetic are accepted without introducing ad-hoc exceptions, then if $0^0$ is defined, it must equal 1. Here is the general proposition that I have in mind: For all a,b, if $a^b$ is defined, then $a^{-b} = 1/a^b$. I am content to accept this without qualification, and to see what its logical consequences are. Therefore: For all a, if $a^0$ is defined, then $a^0 = 1/a^0$. Further specialization gives: If $0^0$ is defined, then $0^0 = 1/0^0$. Hence it is not 0. The rest of the argument is as given above.
  • 24.
    edited May 2016

    Synoposis: there are three chief candidates for $0^0$: 1, 0 and undefined.

    Undefined is certainly a plausible option, especially given that in the complex domain, the function $a^b$ can be led to approach any limiting value whatsoever, by choosing an appropriate path to the origin.

    $0^0$ can be defined to be 0, on the grounds of making the function right-continuous.

    But:

    • Continuity from the right looks like an arbitrary assumption, especially considering that, in the complex domain, $x^y$ is wildly discontinuous.

    • This assumption entails that an exception be added to the general proposition that for all a,b, if $a^b$ is defined, then $a^{-b} = 1/a^b$.

    Comment Source:Synoposis: there are three chief candidates for $0^0$: 1, 0 and undefined. Undefined is certainly a plausible option, especially given that in the complex domain, the function $a^b$ can be led to approach any limiting value whatsoever, by choosing an appropriate path to the origin. $0^0$ _can_ be defined to be 0, on the grounds of making the function right-continuous. But: * Continuity from the right looks like an arbitrary assumption, especially considering that, in the complex domain, $x^y$ is wildly discontinuous. * This assumption entails that an exception be added to the general proposition that for all a,b, if $a^b$ is defined, then $a^{-b} = 1/a^b$.
  • 25.

    I don't mind whatever you want to do here. Post it!

    Comment Source:I don't mind whatever you want to do here. Post it!
  • 26.

    Did you just give up on trying to post this article? At this point it would take at most half an hour to copy it to the blog and hit "post".

    Comment Source:Did you just give up on trying to post this article? At this point it would take at most half an hour to copy it to the blog and hit "post".
  • 27.
    edited June 2016

    Hi John, I have not given up. I know that it seems that I have dropped off, sorry, but I ask for a bit of extra patience. I have been going through some hard times, and have not had the concentration for anything abstract, besides daily programming. I know that I could just paste it in, but the current version does not reflect the further conversations that we have had about $0^x$ -- and they belong there. So things have been slooowed down, but I'm restarting as we speak. Let me chit-chat a bit about modules and tensors on the other thread, and then finish up with $0^x$. Thanks!

    Comment Source:Hi John, I have not given up. I know that it seems that I have dropped off, sorry, but I ask for a bit of extra patience. I have been going through some hard times, and have not had the concentration for anything abstract, besides daily programming. I know that I could just paste it in, but the current version does not reflect the further conversations that we have had about $0^x$ -- and they belong there. So things have been slooowed down, but I'm restarting as we speak. Let me chit-chat a bit about modules and tensors on the other thread, and then finish up with $0^x$. Thanks!
  • 28.
    edited June 2016

    Okay, I have given this the rewrite that I intended:

    Changes:

    • New title
    • I treat it as a riddle, just showing the graph, asking the reader to guess the function. I build up to the answer.
    • Some enhancement of the literary structure
    • Give due acknowledgement that the topic of $0^0$ is a mathematical commonplace, which has engendered many ruminative debates. That notwithstanding, it is still interesting!
    • Tighten up my two cents on the longstanding question, which is that if it is defined, it is most consistently defined as 1, not 0.

    Unless I get further feedback on this, I will post it by the end of the weekend. I may may further grammar tweaks, but the content is fully settled for me.

    You see, after a leave of absence, I really am back :)

    With Best Azimuth Regards

    Comment Source:Okay, I have given this the rewrite that I intended: * [[Blog - A Quirky Function]] Changes: * New title * I treat it as a riddle, just showing the graph, asking the reader to guess the function. I build up to the answer. * Some enhancement of the literary structure * Give due acknowledgement that the topic of $0^0$ is a mathematical commonplace, which has engendered many ruminative debates. That notwithstanding, it is still interesting! * Tighten up my two cents on the longstanding question, which is that _if_ it is defined, it is most consistently defined as 1, not 0. Unless I get further feedback on this, I will post it by the end of the weekend. I may may further grammar tweaks, but the content is fully settled for me. You see, after a leave of absence, I really am back :) With Best Azimuth Regards
  • 29.

    David, I fixed a typo. It now reads "attempted to sketch its level sets"

    Comment Source:David, I fixed a typo. It now reads "attempted to sketch its level sets"
  • 30.

    I enjoyed this post. You inspire a lot of ideas. But I will wait until you've officially posted it. :) I like the exercises you included.

    Comment Source:I enjoyed this post. You inspire a lot of ideas. But I will wait until you've officially posted it. :) I like the exercises you included.
  • 31.

    Thank you Andrius.

    Comment Source:Thank you Andrius.
  • 32.

    I just posted the article to the blog: here

    Comment Source:I just posted the article to the blog: <a href="https://johncarlosbaez.wordpress.com/2016/06/25/a-quirky-function/">here</a>
  • 33.
    edited June 2016

    Congratulations, David! As I mentioned on the blog, your article produced a nice spike of readership, over 1000 per hour for a while:

    These readers came mainly from Hacker News. There was a second bump of readers, not shown here, coming from Reddit.

    It would be great if you could write such popular things more frequently, and even better if they leaned toward "saving the planet" themes.

    Comment Source:Congratulations, David! As I mentioned on the blog, your article produced a nice spike of readership, over 1000 per hour for a while: <img src = "http://math.ucr.edu/home/baez/azimuth-statistics_a_quirky_function_6-25-2016.jpg" alt = ""/> These readers came mainly from Hacker News. There was a second bump of readers, not shown here, coming from Reddit. It would be great if you could write such popular things more frequently, and even better if they leaned toward "saving the planet" themes.
  • 34.
    edited June 2016

    By the way, you changed the title of the post from "Zero to the X" to "A Quirky Function" while I wasn't paying enough attention. Is it okay if I change it back, at least on the Azimuth Blog?

    The reason is as follows. I have a very deliberately policy of making the blog articles have titles that are descriptive rather than cute or tempting. The idea is that it should be easy to find posts on a given subject. This is why I have blog articles with titles like "Vehicle-to-Grid" or "Hard X-Ray Burst" or "El Niño Project (Part 4)". It's not because I'm unimaginative and lack a sense of poetry. There's a macho utilitarian esthetic at work here, sort of like a lab where the machines have color-coded wires coming out and lying on the floor.

    It's really easy to tell what "Zero to the X" is about. "A Quirky Function", not so much. Lots of functions are quirky. Quirkiness is subjective.

    Comment Source:By the way, you changed the title of the post from "Zero to the X" to "A Quirky Function" while I wasn't paying enough attention. Is it okay if I change it back, at least on the Azimuth Blog? The reason is as follows. I have a very deliberately policy of making the blog articles have titles that are descriptive rather than cute or tempting. The idea is that it should be easy to find posts on a given subject. This is why I have blog articles with titles like "Vehicle-to-Grid" or "Hard X-Ray Burst" or "El Ni&ntilde;o Project (Part 4)". It's not because I'm unimaginative and lack a sense of poetry. There's a macho utilitarian esthetic at work here, sort of like a lab where the machines have color-coded wires coming out and lying on the floor. It's really easy to tell what "Zero to the X" is about. "A Quirky Function", not so much. Lots of functions are quirky. Quirkiness is subjective.
  • 35.

    Thanks! I'll post a link to this on my facebook page, which may pull in some readers from wider circles -- with all the pals from college, grad school, professional circles, and musicians too.

    It would be great if you could write such popular things more frequently, and even better if they leaned toward "saving the planet" themes.

    Funny you should mention this, I was just drumming some ideas up along these ideas. I will post when I have a draft.

    Comment Source:Thanks! I'll post a link to this on my facebook page, which may pull in some readers from wider circles -- with all the pals from college, grad school, professional circles, and musicians too. > It would be great if you could write such popular things more frequently, and even better if they leaned toward "saving the planet" themes. Funny you should mention this, I was just drumming some ideas up along these ideas. I will post when I have a draft.
  • 36.

    John, I totally understand why you want to have descriptive titles, but let's not go back to Zero to the X, for the following reason. I start out by challenging the reader to guess the formula for the function, and the whole thing would be spoiled if the reader already read the formula in the title.

    I will come up with something else, which is descriptive, but doesn't give it away. Perhaps: The Exponential at the Origin?

    Comment Source:John, I totally understand why you want to have descriptive titles, but let's not go back to Zero to the X, for the following reason. I start out by challenging the reader to guess the formula for the function, and the whole thing would be spoiled if the reader already read the formula in the title. I will come up with something else, which is descriptive, but doesn't give it away. Perhaps: The Exponential at the Origin?
  • 37.

    Can we go with this: "The Complex Exponential at Zero"

    That's the heart of the matter, to which Zero to the X is the lead-in. So it factually descriptive.
    And it hints at but does not give away the opening question.

    Comment Source:Can we go with this: "The Complex Exponential at Zero" That's the heart of the matter, to which Zero to the X is the lead-in. So it factually descriptive. And it hints at but does not give away the opening question.
  • 38.
    edited June 2016

    Even though we change the title, I suggest that we leave the URL as it is now. It's imperfect, but since it's already been linked to from various places, let's not break these links.

    Comment Source:Even though we change the title, I suggest that we leave the URL as it is now. It's imperfect, but since it's already been linked to from various places, let's not break these links.
  • 39.

    Change of suggested title:

    Exponential Zero

    Comment Source:Change of suggested title: Exponential Zero
  • 40.

    Done.

    Last note: thanks to my daughter Katy Tanzer, for helping me to make the graph using Google Drawings.

    Comment Source:Done. Last note: thanks to my daughter Katy Tanzer, for helping me to make the graph using Google Drawings.
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