3 ms·

Yup. Frequently, in mathematics, we have some ternary relation r(a, b, c) such that fixing values for any two arguments uniquely determines a corresponding valu

by Chinjut 10y ago

Yup. Frequently, in mathematics, we have some ternary relation r(a, b, c) such that fixing values for any two arguments uniquely determines a corresponding value for the third. This encapsulates not only the relationship between a^b = c, log_a(c) = b, and b-th root of c = a, but also the relationship between a + b = c, c - a = b, and c - b = a, the relationship between a * b = c, c/a = b, and c/b = a (as well as the non-commutative version of this where we have separate notions of left- and right-division), and myriad other such ternary relations. In all of these cases, a triangular notation is apt for expressing and emphasizing the structure that any two arguments fix the third. That having been said, that is all the triangular notation does; it does not help with any further structure that may be around for that particular relation (e.g., rules like a^(b * c) = (a^b)^c, which the triangular notation does little to simplify, though perhaps further notational choices could make these also immediate and clear). Still, when that ternary structure is what one cares to emphasize (and why shouldn't it often be of interest?), one should go ahead and emphasize it.

posterboy 10y ago
> a * b = c, c/a = b, and c/b = a this readily shows that it's not about triangles, but some sort of binary tree, because / or * do not operate on the same domain. One is a special case of the other, or an extension. a bifurcation or whatchamacallit.
- Chinjut 10y ago
  Huh? I'm having a little bit of difficulty following you. The triangle just captures that there are three quantities involved (with any two determining the third). I don't know what you mean with references to bifurcations and binary trees and so on. If the mention of different domains is about the fact that you can't divide by zero, sure; in the same way, we find problems taking the logarithm of zero, or raising zero to negative powers, with uniquely pinning down roots of negative numbers, etc. For now, I am glossing over these things; let us suppose, for example, that in the ternary relation a * b = c, I intend all quantities to be drawn from some multiplicative group (thus, nonzero, and thus, with any two determining the third).
  - posterboy 10y ago
    I didn't watch the video for several reasons, so if I was off, I'm sorry. From the comments I couldn't deduce what's actually wrong with the notation, or which notation for that matter. Yes, bad HN ettiquette is to respond anyway. > If the mention of different domains is about the fact that you can't divide by zero > I intend all quantities to be drawn from some multiplicative group That's it. Still, I was trying to draw some hierarchical network, where log and root are both inverses of exp. In the same way, multiplication is a special case of addition. Although It might not have to be, if it's just my preference to look at it that way. It reminds me of the diamond dependency problem (https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_problem https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_pr...). Multiplicative groups to me look like a special case, too. The arrow diagrams look like category theory. I on the other hand just talk from intuition and my experience with the elementary functions, the order I learned in school. EDIT: Subtraction and division as well aren't associative, so do they really form a subgroup? Another problem besides needlessly complicated notation is ambiguous notation. Wikipedia lists two alternatives for multiplicative groups. One is a special case of a ring which does have a null element. Now, in c/b=a, c can be the null element, but then a would be too, so c/a=b is still undefined. I'd guess that holds for the non-commutative version as well.

by Chinjut 10y ago

posterboy 10y ago

> a * b = c, c/a = b, and c/b = a this readily shows that it's not about triangles, but some sort of binary tree, because / or * do not operate on the same domain. One is a special case of the other, or an extension. a bifurcation or whatchamacallit.

Chinjut 10y ago
Huh? I'm having a little bit of difficulty following you. The triangle just captures that there are three quantities involved (with any two determining the third). I don't know what you mean with references to bifurcations and binary trees and so on. If the mention of different domains is about the fact that you can't divide by zero, sure; in the same way, we find problems taking the logarithm of zero, or raising zero to negative powers, with uniquely pinning down roots of negative numbers, etc. For now, I am glossing over these things; let us suppose, for example, that in the ternary relation a * b = c, I intend all quantities to be drawn from some multiplicative group (thus, nonzero, and thus, with any two determining the third).
- posterboy 10y ago
  I didn't watch the video for several reasons, so if I was off, I'm sorry. From the comments I couldn't deduce what's actually wrong with the notation, or which notation for that matter. Yes, bad HN ettiquette is to respond anyway. > If the mention of different domains is about the fact that you can't divide by zero > I intend all quantities to be drawn from some multiplicative group That's it. Still, I was trying to draw some hierarchical network, where log and root are both inverses of exp. In the same way, multiplication is a special case of addition. Although It might not have to be, if it's just my preference to look at it that way. It reminds me of the diamond dependency problem (https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_problem https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_pr...). Multiplicative groups to me look like a special case, too. The arrow diagrams look like category theory. I on the other hand just talk from intuition and my experience with the elementary functions, the order I learned in school. EDIT: Subtraction and division as well aren't associative, so do they really form a subgroup? Another problem besides needlessly complicated notation is ambiguous notation. Wikipedia lists two alternatives for multiplicative groups. One is a special case of a ring which does have a null element. Now, in c/b=a, c can be the null element, but then a would be too, so c/a=b is still undefined. I'd guess that holds for the non-commutative version as well.