Great video. One question - isn't the assertion that j's representation must be unique exactly the same thing we're trying to prove in the first place (i.e., that j = qk +r is unique)? I can't get past the notion that the proof asserts what it's trying to prove right there. I'm referring to what you said right at 9:00, and maybe the 15 or 20 seconds before that too. Thanks.
@rdorsey Great question! There are two different representations involved here. At 9:00 and beforehand, I use the word representation to mean "basis representation". But we ultimately are proving that j=qk+r is a unique representation. We use the uniqueness of the first to prove the uniqueness of the second. Should have made that more clear. Thanks!
Great video. One question - isn't the assertion that j's representation must be unique exactly the same thing we're trying to prove in the first place (i.e., that j = qk +r is unique)? I can't get past the notion that the proof asserts what it's trying to prove right there. I'm referring to what you said right at 9:00, and maybe the 15 or 20 seconds before that too. Thanks.
rdorsey 2 months ago
@rdorsey Great question! There are two different representations involved here. At 9:00 and beforehand, I use the word representation to mean "basis representation". But we ultimately are proving that j=qk+r is a unique representation. We use the uniqueness of the first to prove the uniqueness of the second. Should have made that more clear. Thanks!
steinmath 2 months ago