6 ms·

Although really the first two already have a unified notation, namely x^y and x^(1/y). There are some minor differences in usage between x^(1/y) and the yth roo

Although really the first two already have a unified notation, namely x^y and x^(1/y). There are some minor differences in usage between x^(1/y) and the yth root, but those come down to the fact that the yth root isn't uniquely determined. And there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base.

Jenz 6y ago
> You don't need the ones with a different base. To add on that, we need only one logarithm in the sense that all other then follows. For any base b, we have log(x; b) = log(x)/log(b)
- 6gvONxR4sf7o 6y ago
  I think to complete the point, it’s also worth pointing out that for any bases b, c and d: log(x; b) = log(x; c)/log(b; c) = log(x; d)/log(b; d) The second equality is why you can ignore c and d, and just pretend there's one logarithm when you do log(x; b) = log(x)/log(b)
qsort 6y ago
Yeah, their assumption that there's a ternary relation between those immediately breaks down when you consider sets other than the real numbers. And you don't even need to go that far, roots as used on the real numbers don't make sense in C. The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...) I guess a different notation could have some meaning pedagogically, math notation is incredibly inconsistent at times, but there really is no "deeper truth" here.
- contravariant 6y ago
  Exponentiation does have some scenarios where it can be defined without an (obvious) exponential function, and roots may not be uniquely defined (or rather they are almost never unique), which means you need to be a bit careful, and which means that in theory the 3rd root could differ from the definition of x^(1/3). However in the cases where you need to be careful most of stuff you'd use the more general notation for wouldn't be applicable anyway, you'd have a high chance of writing down an expression that has no unique value, or can't even be evaluated.
- jessriedel 6y ago
  > The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...) Why do you think this? It doesn't seem to fit historically or formally. 3^2 is a more elementary object than anything built with exp, and there's often no natural notion of exp(A) of a function A even when finite powers like A^3 are defined. exp(A) is defined with a power series that may not converge.
  - qsort 6y ago
    Premise: Forgive the sloppiness, I have some math background but I'm not a mathematician, any correction is more than welcome. I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function. Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.
    - jessriedel 6y ago
      > the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed. > When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function. > Since the OP was about a notation propos... My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
- naringas 6y ago
  > but there really is no "deeper truth" here. I disagree, I do think there's a deeper truth. It seems to me that you have already internalized and understood it very well; that is the goal. would you at least agree that there's beauty here? and this notation does make it more apparent; or as you say "could have some meaning pedagogically"
  - qsort 6y ago
    The thing is that ultimately notation isn't that important, what matters are the formal concepts at play (and to develop an intuition on how to manipulate them, if you're learning). If this can help students "see the light", then sure, but I'm not entirely sold on the idea that the notation is actually the hardest part. This is similar (with a different twist) to the various ideas going around that we should stop using base 10 and think in hexadecimal. It might be better in some absolute sense, but it's not something it's worth spending any energy into. I'm just not sure I see much from this, but obviously I'm not the "intended target", so please take this with a pinch of salt.
    - deleted 6y ago
      [deleted]
    - alisonkisk 6y ago
      Notation is extremely important for humans. That's why math notation is used instead of the extremely wordy sentences that were used centuries ago.
- kortex 6y ago
  IMHO, the "deeper truth" is addition is translation and multiplication is rotation/stretching (at least when it comes to rings and fields). It involves getting a geometric understanding of e to the pi i, and 3blue1brown explains it better than I could: https://www.youtube.com/watch?v=F_0yfvm0UoU https://www.youtube.com/watch?v=F_0yfvm0UoU [6.28 min] https://www.youtube.com/watch?v=v0YEaeIClKY https://www.youtube.com/watch?v=v0YEaeIClKY [3.14 min] https://www.youtube.com/watch?v=mvmuCPvRoWQ https://www.youtube.com/watch?v=mvmuCPvRoWQ [25 min]
tgv 6y ago
For log, you could also write x ^-1 y if you want uniformity, although it looks rather ugly and confusing to me.
- contravariant 6y ago
  Yeah, let's not. The main function of the logarithm is to transform multiplication into addition, so log(x y) = log(x) + log(y) and log(x^k) = k log(x) are uniform and consistent with that idea. Also exponentiation is a way more fundamental property than the logarithm so it's weird to place its notation on the same footing.
  - alisonkisk 6y ago
    Exponentiation to logarithm is like addition to subtraction and multiplication to division, which are all pretty much on the same footing.
rocqua 6y ago
Yes, the current notation is sufficiently expressive. That is not what is being discussed here. The question is, can we find a notation that is much better at teaching the intuitive relationship between these 3 operations. Because notation isn't just about formalizing. Notation is about enabling better understanding. And for a large part, good notation should help teaching. Especially something like exponents, roots, and logarithms which are taught to many. People who happen to not get this probably stop doing math, and we need more people who can do math it seems.
bonzini 6y ago
> there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base. That's not true. The natural logarithm is the only one whose derivative is 1/x but the identity you wrote is true with any base. People had been using logarithmic tables to do multiplication centuries before Napier came up with "e".
- contravariant 6y ago
  True, although the fact that all of them are pretty much equivalent does mean that there's no real reason to think of it as a logarithm with 'base' anything. Also to the best of my knowledge there's no logarithm tables that predate Napier (1550-1617), and when you try to calculate those tables you naturally stumble upon the natural logarithm (or at least a very close approximation), as the easiest way to create a logarithm table is to start with the powers of something like 1.00000001.
  - bonzini 6y ago
    Right, I am not sure why I thought that the slide rule predated logarithms.
quietbritishjim 6y ago
> And there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base. Maybe I misunderstood your comment but it really sounds like you're saying that only the natural logarithm has this property, but in fact it's true with every base: log(x^y; b) = log(x^y)/log(b) = y log(x) / log(b) = y log(x; b)
- OJFord 6y ago
  The comma is important. Or replace 'which' with 'and it'.
  - quietbritishjim 6y ago
    Rejigging their text slightly, what they said was "There's only one useful logarithm. It has this property. It's the only useful logarithm." Yes I noticed their use of a comma (and "which" rather than "that"), in fact that's why I added the disclaimer at the start of my comment about possibly misunderstanding theirs. But sandwiching a mention of that property in the middle of making the same point twice only makes sense if it's a supporting argument.

Jenz 6y ago

> You don't need the ones with a different base. To add on that, we need only one logarithm in the sense that all other then follows. For any base b, we have log(x; b) = log(x)/log(b)

6gvONxR4sf7o 6y ago
I think to complete the point, it’s also worth pointing out that for any bases b, c and d: log(x; b) = log(x; c)/log(b; c) = log(x; d)/log(b; d) The second equality is why you can ignore c and d, and just pretend there's one logarithm when you do log(x; b) = log(x)/log(b)

qsort 6y ago

Yeah, their assumption that there's a ternary relation between those immediately breaks down when you consider sets other than the real numbers. And you don't even need to go that far, roots as used on the real numbers don't make sense in C. The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...) I guess a different notation could have some meaning pedagogically, math notation is incredibly inconsistent at times, but there really is no "deeper truth" here.

contravariant 6y ago
Exponentiation does have some scenarios where it can be defined without an (obvious) exponential function, and roots may not be uniquely defined (or rather they are almost never unique), which means you need to be a bit careful, and which means that in theory the 3rd root could differ from the definition of x^(1/3). However in the cases where you need to be careful most of stuff you'd use the more general notation for wouldn't be applicable anyway, you'd have a high chance of writing down an expression that has no unique value, or can't even be evaluated.
jessriedel 6y ago
> The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...) Why do you think this? It doesn't seem to fit historically or formally. 3^2 is a more elementary object than anything built with exp, and there's often no natural notion of exp(A) of a function A even when finite powers like A^3 are defined. exp(A) is defined with a power series that may not converge.
- qsort 6y ago
  Premise: Forgive the sloppiness, I have some math background but I'm not a mathematician, any correction is more than welcome. I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function. Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.
  - jessriedel 6y ago
    > the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed. > When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function. > Since the OP was about a notation propos... My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
naringas 6y ago
> but there really is no "deeper truth" here. I disagree, I do think there's a deeper truth. It seems to me that you have already internalized and understood it very well; that is the goal. would you at least agree that there's beauty here? and this notation does make it more apparent; or as you say "could have some meaning pedagogically"
- qsort 6y ago
  The thing is that ultimately notation isn't that important, what matters are the formal concepts at play (and to develop an intuition on how to manipulate them, if you're learning). If this can help students "see the light", then sure, but I'm not entirely sold on the idea that the notation is actually the hardest part. This is similar (with a different twist) to the various ideas going around that we should stop using base 10 and think in hexadecimal. It might be better in some absolute sense, but it's not something it's worth spending any energy into. I'm just not sure I see much from this, but obviously I'm not the "intended target", so please take this with a pinch of salt.
  - deleted 6y ago
    [deleted]
  - alisonkisk 6y ago
    Notation is extremely important for humans. That's why math notation is used instead of the extremely wordy sentences that were used centuries ago.
kortex 6y ago
IMHO, the "deeper truth" is addition is translation and multiplication is rotation/stretching (at least when it comes to rings and fields). It involves getting a geometric understanding of e to the pi i, and 3blue1brown explains it better than I could: https://www.youtube.com/watch?v=F_0yfvm0UoU https://www.youtube.com/watch?v=F_0yfvm0UoU [6.28 min] https://www.youtube.com/watch?v=v0YEaeIClKY https://www.youtube.com/watch?v=v0YEaeIClKY [3.14 min] https://www.youtube.com/watch?v=mvmuCPvRoWQ https://www.youtube.com/watch?v=mvmuCPvRoWQ [25 min]

tgv 6y ago

For log, you could also write x ^-1 y if you want uniformity, although it looks rather ugly and confusing to me.

contravariant 6y ago
Yeah, let's not. The main function of the logarithm is to transform multiplication into addition, so log(x y) = log(x) + log(y) and log(x^k) = k log(x) are uniform and consistent with that idea. Also exponentiation is a way more fundamental property than the logarithm so it's weird to place its notation on the same footing.
- alisonkisk 6y ago
  Exponentiation to logarithm is like addition to subtraction and multiplication to division, which are all pretty much on the same footing.

rocqua 6y ago

Yes, the current notation is sufficiently expressive. That is not what is being discussed here. The question is, can we find a notation that is much better at teaching the intuitive relationship between these 3 operations. Because notation isn't just about formalizing. Notation is about enabling better understanding. And for a large part, good notation should help teaching. Especially something like exponents, roots, and logarithms which are taught to many. People who happen to not get this probably stop doing math, and we need more people who can do math it seems.

bonzini 6y ago

> there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base. That's not true. The natural logarithm is the only one whose derivative is 1/x but the identity you wrote is true with any base. People had been using logarithmic tables to do multiplication centuries before Napier came up with "e".

contravariant 6y ago
True, although the fact that all of them are pretty much equivalent does mean that there's no real reason to think of it as a logarithm with 'base' anything. Also to the best of my knowledge there's no logarithm tables that predate Napier (1550-1617), and when you try to calculate those tables you naturally stumble upon the natural logarithm (or at least a very close approximation), as the easiest way to create a logarithm table is to start with the powers of something like 1.00000001.
- bonzini 6y ago
  Right, I am not sure why I thought that the slide rule predated logarithms.

quietbritishjim 6y ago

> And there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base. Maybe I misunderstood your comment but it really sounds like you're saying that only the natural logarithm has this property, but in fact it's true with every base: log(x^y; b) = log(x^y)/log(b) = y log(x) / log(b) = y log(x; b)

OJFord 6y ago
The comma is important. Or replace 'which' with 'and it'.
- quietbritishjim 6y ago
  Rejigging their text slightly, what they said was "There's only one useful logarithm. It has this property. It's the only useful logarithm." Yes I noticed their use of a comma (and "which" rather than "that"), in fact that's why I added the disclaimer at the start of my comment about possibly misunderstanding theirs. But sandwiching a mention of that property in the middle of making the same point twice only makes sense if it's a supporting argument.