There are a few technical points I noticed while watching this video. 1: It is odd that you did not mention the fact that 0 is often taken to be a natural number. 2: You say that irrational numbers "often cannot be described exactly by a fraction" (presumably a fraction of integers). The use of the word "often" implies that some irrational numbers can be defined as a fraction, which is clearly not true. 3: Defining the complex numbers as the algebraic closure of the real numbers is valid, but you have not justified why the real numbers should have an algebraic closure. All fields do, but that fact is probably more challenging to prove than to simply define C some other way and to prove that C is an alg. closed field. 4, and most importantly, you say that i is taken to be the positive root of x^2 + 1. This is not true, in particular the complex field cannot be extended into an ordered field regardless of how you choose the ordering, as such "positive" is not a sensible notion in the complex numbers. Rather i is chosen arbitrarily between the two roots. As a matter of fact, the map C->C that sends a+bi to a-bi is an isomorphism with respect to the field structure.
It's great to see your considerations! I appreciate you letting me know. 1. Yes, some mathematicians take 0 to be natural, others as an integer. Though, I hope my decision did not detract from your viewing of the video. 2. There are some exceptions to the idea that "no irrational number can be described exactly by a fraction", for instance phi is exactly equal to (1+√5)/2, and it is irrational. Hence why I made this nuanced point. 3. I understand, perhaps I could have gone into more depth here. However, the aim of this video was to introduce new people to the idea of the complex set, "Discovering the Complex Set", not to definitively define it or any other set. 4. I hope that the distinction between positive and negative i as the roots of P(x) was not misleading, I think it was a fair way of explaining this new concept to a learning audience: to define i through a digestible and authentic method. Thankyou for your concern.
I don’t usually comment but I’m taking a real analysis course next semester and I feel this series is gonna be a life saver 🙏🏻🙏🏻
Thankyou for dropping this comment, It's super nice to hear that! I will try to roll out the next episodes every week or two.
There are a few technical points I noticed while watching this video.
1: It is odd that you did not mention the fact that 0 is often taken to be a natural number.
2: You say that irrational numbers "often cannot be described exactly by a fraction" (presumably a fraction of integers). The use of the word "often" implies that some irrational numbers can be defined as a fraction, which is clearly not true.
3: Defining the complex numbers as the algebraic closure of the real numbers is valid, but you have not justified why the real numbers should have an algebraic closure. All fields do, but that fact is probably more challenging to prove than to simply define C some other way and to prove that C is an alg. closed field.
4, and most importantly, you say that i is taken to be the positive root of x^2 + 1. This is not true, in particular the complex field cannot be extended into an ordered field regardless of how you choose the ordering, as such "positive" is not a sensible notion in the complex numbers. Rather i is chosen arbitrarily between the two roots. As a matter of fact, the map C->C that sends a+bi to a-bi is an isomorphism with respect to the field structure.
It's great to see your considerations! I appreciate you letting me know.
1. Yes, some mathematicians take 0 to be natural, others as an integer. Though, I hope my decision did not detract from your viewing of the video.
2. There are some exceptions to the idea that "no irrational number can be described exactly by a fraction", for instance phi is exactly equal to (1+√5)/2, and it is irrational. Hence why I made this nuanced point.
3. I understand, perhaps I could have gone into more depth here. However, the aim of this video was to introduce new people to the idea of the complex set, "Discovering the Complex Set", not to definitively define it or any other set.
4. I hope that the distinction between positive and negative i as the roots of P(x) was not misleading, I think it was a fair way of explaining this new concept to a learning audience: to define i through a digestible and authentic method. Thankyou for your concern.