13:27 This example involves the square root of 7. But the minimum polynomial will be the same if 7 is replaced by 3 or 10, or *anything*. Is that OK, professor?
Can you explain why *all* constructible numbers must lie in K0,K1,K2,...? I understand why all of the numbers that belong to this list must be constructible, but I do not understand why all constructible numbers *must* belong K0,K1,K2,... . In other words, in the constructible numebr theorem, I understand the "if" but i do not understand why it must be an "if and only if"
I concur with Akroker - You certainly did explain how to construct square roots - and how they generate throughout the Ki, but you never even stated (no less proved) that square roots were the only root constructions that could be accomplished.
Ahh I miss these videos! And great one by the way :) I do think now that I've experienced how some math professors teach at usc that you spoil your students with how well explained and easily accessible the content of your lectures are (especially since you use these videos). I have had to get use to fumbling and a lack of interaction especially with this one math professor. But anyways I was wondering..what is a definition for a direct limit and what in another example of one?
you seem to arbitrarily take Q (certainly constructible) and iteratively add square roots. this gives you your set K. so your set K is certainly constructible. by why not more? why not start instead of Q start with a set that also includes some cube roots? or why only add square roots to Q, why not cube roots?
The power of two is duality -- constructable numbers are dual! Real is dual to imaginary -- complex numbers are dual. The master is dual to the apprentice -- the rule of two, Darth Bane, Sith Lord. "Always two there are" -- Yoda.
Where's the note?? I rediscovered the error too, please add the note so I can discover something useful. But thanks for the video, it is very well done otherwise!!. . ..
The geometric mean! I teach it to my sophomores but never realized that it has implications in constructible numbers. Brilliant explanation!
The more one studies disparate math subjects, the more one realizes it is just the same bowl of pudding.
How have your classes been going since COVID?
One of the greatest curiosities in my life has solved and I am very happy now.
I think you're right, Cori. (Since the constant term should be the product of the roots 4 ± sqrt(5), which is 11.) I'll add a note.
nyc prof
Thank you. I really appreciate you for transforming the mythology into a logos.
13:27 This example involves the square root of 7.
But the minimum polynomial will be the same if 7 is replaced by 3 or 10, or *anything*. Is that OK, professor?
9:18 It's cool to know there's more algebra beyond this, and I'm sure it's irrationally great!
Another very good video! However, I caught another mistake. (4 + √5)(4 - √5) = 11, not 9.
how are you able to draw unit segment (since you can't measure it)?
Step 1: draw a fixed segment of arbitrary length. Step 2: there is no step 2. 😉
Matthew Salomone
Lol you clown
Can you explain why *all* constructible numbers must lie in K0,K1,K2,...? I understand why all of the numbers that belong to this list must be constructible, but I do not understand why all constructible numbers *must* belong K0,K1,K2,... . In other words, in the constructible numebr theorem, I understand the "if" but i do not understand why it must be an "if and only if"
Do you talk about integral domains anywhere?
I concur with Akroker - You certainly did explain how to construct square roots - and how they generate throughout the Ki, but you never even stated (no less proved) that square roots were the only root constructions that could be accomplished.
Ahh I miss these videos! And great one by the way :) I do think now that I've experienced how some math professors teach at usc that you spoil your students with how well explained and easily accessible the content of your lectures are (especially since you use these videos). I have had to get use to fumbling and a lack of interaction especially with this one math professor. But anyways I was wondering..what is a definition for a direct limit and what in another example of one?
you seem to arbitrarily take Q (certainly constructible) and iteratively add square roots. this gives you your set K. so your set K is certainly constructible. by why not more? why not start instead of Q start with a set that also includes some cube roots? or why only add square roots to Q, why not cube roots?
The power of two is duality -- constructable numbers are dual!
Real is dual to imaginary -- complex numbers are dual.
The master is dual to the apprentice -- the rule of two, Darth Bane, Sith Lord.
"Always two there are" -- Yoda.
Ah geometric means theorem
Where's the note?? I rediscovered the error too, please add the note so I can discover something useful. But thanks for the video, it is very well done otherwise!!. . ..