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Good article - but couple of points. I'm not formally trained Mathematician, but I think this is not an analytic proof - it lacks rigour. It is more a graphica

Good article - but couple of points. I'm not formally trained Mathematician, but I think this is not an analytic proof - it lacks rigour. It is more a graphical approach that appeals to the reader's intuition. I could be wrong here though. Second, I really really thought the last graph will include y = x line and was very surprised to not see it :)

chx 3y ago
No, the graph parts are explanation. The analysis of the derivative function maybe would be better if it were rewritten with the usual limes notations, nonetheless it's correct in the crucial conclusion of the ln x / x function is monotonically increasing (0, e) and monotonically decreasing (e, inf) and as such one of every pair of equal but distinct values must be from (0,e) the other from (e,inf). There are not many integers between 0 and e: 1 and 2. At 1, ln x / x is zero and in (e,inf) this function has a strict lower boundary of 0, it is never 0. Thus the only possible solution is 2 and it is.
- arketyp 3y ago
  In general, couldn't a function be monotonically increasing in some interval and still have many points such that y = x? A curve that jiggles around the line y = x for instance.
  - chx 3y ago
    So while the original problem was x^y=y^x he have rearranged this into ln x / x = ln y / y and so now are investigating the ln x / x function. Yes, the lettering could be clearer because y at some points means one variable and later on it means the value of the function.
  - deredede 3y ago
    We are not looking for points such that y = x but for distinct points x, x' that have the same y (f(x) = f(x'))
NotYourLawyer 3y ago
It's not totally rigorous, but it wouldn't be hard to fill in the details. This is a sketch of a proof, not the proof itself.
- karthickgururaj 3y ago
  Yes, that's basically what I said and meant in the parent post. I was suggesting the article call it a "graphical" solution approach, not analytical. And if I wasn't clear - it wasn't my intent to treat this any less "useful". If anything, it is more interesting (for me) to read about solutions that give an intuitive feel, than dry analysis.
  - hgsgm 3y ago
    "analytical" means "calculus" here.
  - n4r9 3y ago
    Proofs can be graphical or diagrammatic, though. Take for example this proof of the Pythagorean Theorem: https://cdn.britannica.com/43/70143-004-CCB17706/theorem-demonstration-squares-proof-Pythagorean-b-square.jpg?s=1500x700&q=85 https://cdn.britannica.com/43/70143-004-CCB17706/theorem-dem...

chx 3y ago

No, the graph parts are explanation. The analysis of the derivative function maybe would be better if it were rewritten with the usual limes notations, nonetheless it's correct in the crucial conclusion of the ln x / x function is monotonically increasing (0, e) and monotonically decreasing (e, inf) and as such one of every pair of equal but distinct values must be from (0,e) the other from (e,inf). There are not many integers between 0 and e: 1 and 2. At 1, ln x / x is zero and in (e,inf) this function has a strict lower boundary of 0, it is never 0. Thus the only possible solution is 2 and it is.

arketyp 3y ago
In general, couldn't a function be monotonically increasing in some interval and still have many points such that y = x? A curve that jiggles around the line y = x for instance.
- chx 3y ago
  So while the original problem was x^y=y^x he have rearranged this into ln x / x = ln y / y and so now are investigating the ln x / x function. Yes, the lettering could be clearer because y at some points means one variable and later on it means the value of the function.
- deredede 3y ago
  We are not looking for points such that y = x but for distinct points x, x' that have the same y (f(x) = f(x'))

NotYourLawyer 3y ago

It's not totally rigorous, but it wouldn't be hard to fill in the details. This is a sketch of a proof, not the proof itself.

karthickgururaj 3y ago
Yes, that's basically what I said and meant in the parent post. I was suggesting the article call it a "graphical" solution approach, not analytical. And if I wasn't clear - it wasn't my intent to treat this any less "useful". If anything, it is more interesting (for me) to read about solutions that give an intuitive feel, than dry analysis.
- hgsgm 3y ago
  "analytical" means "calculus" here.
- n4r9 3y ago
  Proofs can be graphical or diagrammatic, though. Take for example this proof of the Pythagorean Theorem: https://cdn.britannica.com/43/70143-004-CCB17706/theorem-demonstration-squares-proof-Pythagorean-b-square.jpg?s=1500x700&q=85 https://cdn.britannica.com/43/70143-004-CCB17706/theorem-dem...