7 ms·

It is not. a and b are not symmetric in this equation, you can't just swap them.

by supernewton 2y ago

henry2023 2y ago
You can swap them without loss of generality (WLOG).
- davrosthedalek 2y ago
  No, this is not correct. WLOG means: I assume one of the possible cases, but the proof works the same way for other cases. But that's not true here. The proof, as shown, only works for a>b>0, it does not work (without extra work or explanation) for a<b. The proof for a<b is similar, but not the same. [And it certainly does not show it for a,b element of C]
  - edflsafoiewq 2y ago
    WLOG just means the other cases follow from the one case. There is no implication about how hard it is to get to the other cases, although generally it is easy and you don't bother spelling it out exactly.
olddustytrail 2y ago
Of course you can. What do you mean?
- davrosthedalek 2y ago
  3^2-2^2 =!= 2^2-3^2. (You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)
  - olddustytrail 2y ago
    Did you think that I meant you can switch them on one side of the equation but not the other? That's not what anyone is saying.
    - davrosthedalek 2y ago
      No, of course not.
      - olddustytrail 2y ago
        But that's literally what you just did in your example.
        davrosthedalek 2y ago
        I did not show the right side at all, so I am not sure how you can make that statement. The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric. Both left and right side are anti-symmetric.
  - wcrossbow 2y ago
    This is not what I meant. What is being proved is: a^2-b^2 - (a+b)(a-b) = 0. If you swap a and b you end up with a sign switch on the lhs which is inconsequential.
    - davrosthedalek 2y ago
      That is not what the proof proves. The proof proves the equivalence how it was originally stated, and assumes for that b<a. Your rewriting is of course true for all a,b and might be used in an algebraic proof. But this transformation is not at all shown in the geometric proof.
- Scarblac 2y ago
  The answer is going to be negative regardless of the names, so this geometric proof won't work.

It is not. a and b are not symmetric in this equation, you can't just swap them.

henry2023 2y ago

You can swap them without loss of generality (WLOG).

davrosthedalek 2y ago
No, this is not correct. WLOG means: I assume one of the possible cases, but the proof works the same way for other cases. But that's not true here. The proof, as shown, only works for a>b>0, it does not work (without extra work or explanation) for a<b. The proof for a<b is similar, but not the same. [And it certainly does not show it for a,b element of C]
- edflsafoiewq 2y ago
  WLOG just means the other cases follow from the one case. There is no implication about how hard it is to get to the other cases, although generally it is easy and you don't bother spelling it out exactly.

olddustytrail 2y ago

Of course you can. What do you mean?

davrosthedalek 2y ago
3^2-2^2 =!= 2^2-3^2. (You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)
- olddustytrail 2y ago
  Did you think that I meant you can switch them on one side of the equation but not the other? That's not what anyone is saying.
  - davrosthedalek 2y ago
    No, of course not.
    - olddustytrail 2y ago
      But that's literally what you just did in your example.
      - davrosthedalek 2y ago
        I did not show the right side at all, so I am not sure how you can make that statement. The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric. Both left and right side are anti-symmetric.
- wcrossbow 2y ago
  This is not what I meant. What is being proved is: a^2-b^2 - (a+b)(a-b) = 0. If you swap a and b you end up with a sign switch on the lhs which is inconsequential.
  - davrosthedalek 2y ago
    That is not what the proof proves. The proof proves the equivalence how it was originally stated, and assumes for that b<a. Your rewriting is of course true for all a,b and might be used in an algebraic proof. But this transformation is not at all shown in the geometric proof.
Scarblac 2y ago
The answer is going to be negative regardless of the names, so this geometric proof won't work.