Don't know if anyone would see this, but I just feel like leaving some messages and hopefully someone can be benfited from it. When I was in elementry school, I always wondered the deep-down knowledge of certain thing and that of-course includes math. However, I had always being considered as a bad student who couldn't get a good grade or even fail tests in elementry, especially in math. It could be because of the eduaction system here in Taiwan, but anyways, when I was a child and which was when I was being considered as a stupid student. I always questioned about something that the teacher even couln't repondse. For example, why is 1+1=2, why does pi have this 3.14... value, why we couldn't divide by zero. and mostly why would say that reason. It wasn't getting any better until I got out of the system and finally have the chance to answer all of thoses questions everything seems so clear and I realized it's not because I am stupid or something, it's because the system pretty ban us to think and always wants us to memorize. These videos in the channel made me feel incrdiable or even more, and it helps me to purse my dream to get a master degree in cybersecurity in the nearly future. Thank you, and I love math.
The first axiom of a ring you gave was that it's a set which is an abelian group under addition. The last axiom you gave that makes a ring commutative says that the multiplication operation must be commutative as well. This makes me think... Could we just call a commutative ring a set which is an abelian group under both addition AND multiplication? Wikipedia describes a commutative ring like this: "the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition". Where does my definition fail? I want to say it's because of the fact that we don't require each element of the ring to have a multiplicative inverse. Is this correct? If each element did have a multiplicative inverse, we could call the set a field, correct? Just trying to get some things straightened out before moving on. Thanks in advance!
Correct. Its not a group under multiplication because not every element will have a multiplicative inverse. Even a field isn’t a group under multiplication because 0 doesn’t have an inverse and cannot have an inverse if you want distributivity.
People's mileages vary, but ring theory is hard and counter-intuitive, and it's very useful to have some videos that take it steadily and address every point that could cause confusion. Imho these videos do just that.
Don't know if anyone would see this, but I just feel like leaving some messages and hopefully someone can be benfited from it.
When I was in elementry school, I always wondered the deep-down knowledge of certain thing and that of-course includes math.
However, I had always being considered as a bad student who couldn't get a good grade or even fail tests in elementry, especially in math.
It could be because of the eduaction system here in Taiwan, but anyways, when I was a child and which was when I was being considered as a stupid student.
I always questioned about something that the teacher even couln't repondse. For example, why is 1+1=2, why does pi have this 3.14... value, why we couldn't divide by zero. and mostly why would say that reason.
It wasn't getting any better until I got out of the system and finally have the chance to answer all of thoses questions everything seems so clear and I realized it's not because I am stupid or something, it's because the system pretty ban us to think and always wants us to memorize.
These videos in the channel made me feel incrdiable or even more, and it helps me to purse my dream to get a master degree in cybersecurity in the nearly future.
Thank you, and I love math.
All of these videos are amazing. They follow your own style which is excellent.
Excellent explanations. You set off lots of lightbulbs in my head.
The first axiom of a ring you gave was that it's a set which is an abelian group under addition. The last axiom you gave that makes a ring commutative says that the multiplication operation must be commutative as well. This makes me think...
Could we just call a commutative ring a set which is an abelian group under both addition AND multiplication?
Wikipedia describes a commutative ring like this: "the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition". Where does my definition fail? I want to say it's because of the fact that we don't require each element of the ring to have a multiplicative inverse. Is this correct? If each element did have a multiplicative inverse, we could call the set a field, correct?
Just trying to get some things straightened out before moving on. Thanks in advance!
Correct. Its not a group under multiplication because not every element will have a multiplicative inverse.
Even a field isn’t a group under multiplication because 0 doesn’t have an inverse and cannot have an inverse if you want distributivity.
@@elliotnicholson5117That makes complete sense. Thanks a ton, your videos are great.
Your videos are really helpful but you do tend to over-explain stuff a bit. And that is a wastage of precious time. But really good explanations.
2X speed is your friend.
No he doesn’t, his videos are amazing no over explaining here.
People's mileages vary, but ring theory is hard and counter-intuitive, and it's very useful to have some videos that take it steadily and address every point that could cause confusion. Imho these videos do just that.
no he doesnt, put x2 speed and use clicks.