You mentioned that you disliked the way math is taught in the USA, particularly the simulation of steps done by tutors. I completely agree with you, and I’ve taken inspiration from your perspective to create my own channel for the Arab world to address this very issue. It seems we share the same concern, and I truly appreciate your insights. Thank you for recommending those three books-they’ve been incredibly helpful. I wish you all the best with your channel and your endeavors.
Thank you very much! That is an excellent idea, and you have my support. The more we can spread around the world the foundations of proper math, the better the world will be to equip our young people to think PROPERLY about things. I hope your channel goes far as well!
I disliked mathematics, not for its subject matter, but because no one explained why things worked as they did. This questioning intensified with physics, leading me to drop it, pursue intermediate mathematics, and focus on science and English. Years later, as a lawyer, I remain curious about mathematics and its logic.
@13:28 The whole numbers include zero. Since you can't have a denominator of zero, you rather should have said that n must be a positive integer. Regarding @24:00 There is a big difference between polite correction and abuse.
Yes, I agree it would have been clearer to specify n must be a positive integer! For sure! And yes, you are right about the abuse, but I am used to it all. Been "controversial" for a long time, so I am used to both!
Mr Cromwell, I need your opinion. I have been studying Calculus for some time now with Stewarts book. I went through the first 5 Chapters, so differentiation and integration with Riemann's method, moreover I just started the rules on Integration. So I can say I have a relative strong foundation. I can use differentiation, I understand the main concept behind it, why we are doing it and so on, but sonething is missing. It is a shapeless feeling, I can not put my finger on it. Maybe I miss deepness or something. Rigour maybe... Stewarts book is not bad at all, considering I use it as a mechanical engineering student, but instead of just seeing how things work out I truly want to feel it deeply! I hope you understand my point here. Can you recommend me calculus books that have a kind of different "magic" approach? Currently I found Apostols Calculus which seemst to be a better choice but I hope you can recommend me some other literature here. And if possible, it would be very kind If you also could state the type of teaching method those books work with. I mean who do you recommend one over another? What type of person? Thank you very much! Congratulation on the 4k+ sub!!!🎉😊
I am sorry, I was not able to respond to you earlier! Fear not; you're experiencing the natural frustration of someone ready to transcend the mechanical into the metaphysical. Stewart is a practical workhorse of a book: it gets you from point A to point B but rarely invites you to ponder the scenery. If you're craving rigor and magic, here’s your itinerary: Apostol’s Calculus - A superb choice if you're a fan of mathematical dual citizenship: theory and application. Apostol doesn’t just teach you how to differentiate; he’ll make you prove why differentiation deserves to exist. Be ready for axioms and integrals from day one-no shortcuts here. Recommended for anyone who likes structured thinking and doesn’t mind pausing to sip the pure nectar of mathematical proofs. Spivak’s Calculus - Think of this as calculus written by a poet who happens to also be a sadist (and I mean that as a compliment). Spivak’s book is rigorous and has a charm that will either break you or make you fall in love with epsilon-delta proofs. Ideal for those who appreciate quality and are willing to work hard for their epiphanies. Courant’s Differential and Integral Calculus - The lovechild of mathematical rigor and applied sensibility. Courant dives deeper than Stewart but still remembers you’re human. Great for an engineering mindset with a penchant for rigor. Kline’s Calculus: An Intuitive and Physical Approach - If Apostol and Spivak feel like they’re delivering calculus sermons from on high, Kline will come down to earth and walk you through with vivid, intuitive examples. A gentle but insightful companion. Choose your poison based on your personality: Apostol if you want rigor served with a dash of formality. Spivak if you're after rigor with a pinch of whimsy. Courant if you want to bridge engineering and theoretical perspectives. Kline if you’re a lover of intuition who doesn’t want to lose sight of reality. Remember, deep understanding isn’t about jumping into a new book-it’s about taking the time to wrestle with the ideas. Let these authors guide you, but don’t expect them to solve the ‘shapeless feeling’ for you. That’s the part of the journey you get to carve out yourself.
@citytutoring Mr. Cromwell, Your ability to amaze me from time to time is beyond compare. Not just a dozens of books you list but you put them into a "tale". Into perspective. That is rellay helpful when choosing the rigtht book(s). Moreover, you your vocabulary is superb. A true poet.😀 Thank you very much, you really helped me with this one! By the way, let me ask you one more question. As I looked into some of the books you mentiond, tipically Apostol and Spivak, in order to understand the deepness of the topics, It is some proof writing skills are required. Do you agree with me? So do you recommend to start with the basics of proof writing in order to get the proofs mentiond in these books?
@@petihundeuhun8950 Thank you for your kind and generous words. They are like a balm to my soul amidst so many that try to attack mathematics. Regarding your question: you’re absolutely right. Apostol and Spivak don’t just hand you answers; they expect you to waltz through proofs like a mathematician at a formal ball. Proof-writing skills are the key to understanding their reasoning. If proofs seem intimidating at first, think of learning them as sharpening your detective skills-spotting assumptions, following logical breadcrumbs, and arriving triumphantly at 'QED.' I'd suggest starting with a gentle primer on proofs, like "How to Prove It" by Velleman. It's like a training montage for your brain before diving into the heavyweight champions like Apostol and Spivak. So yes, master the art of proofs first. Once you do, Apostol and Spivak’s pages will feel much better. Happy proving-may your theorems always be true and your proofs ever elegant!
I disagree with your analysis of the right hand side. You are interpreting it as ((-4)^(1/2))^2=(2i)^2=-4 But the expression as written is (-4^(1/2))^2, namely there are no brackets around the -4. The actual value is therefore (-4^(1/2))^2=(-1 * 4^(1/2))^2=(-1 * 2)^2=(-2)^2=4. So, in fact the two sides are equal.
I am a little confused about the Use of the bracket. I told my Students, that the minus before a number ist only Affected by the Exponent, If ITS written with a bracket. In this Case i would say that (-4)^(1/2) is equal to 2i and -4^(1/2) is equal to -2 because the minus is Not touched by the ^(1/2). Maybe this is a Thing of Definition or so....
For example: -3^2 =-9 whereas (-3)^2 =9 Some calculators and computational software may fail to interpret such problems correctly, but this is a long-established convention for the use of these symbols. As you correctly told your students, the exponent only affects the number or letter immediately before it, not the minus sign, unless the number and minus sign are surrounded by parentheses (or brackets, as some English speakers call them).
Thank you for your comment -- The brackets are necessary to clarify the intended order of operations. Without them, the interpretation changes due to how exponents are applied. Without parentheses, you could end up with a different calculation order: (-4^(1/2)) means -(4^(1/2)), which is the negative square root of 4, or -2. Squaring that gives 4. [(-4)^2]^(1/2) clearly means you first square -4 to get 16 and then take the square root of 16, which equals 4. So, the two expressions are not the same-the parentheses are crucial for defining the order of operations. the brackets are like traffic lights-they make sure everything stops in the right order! Without them, the operations would go off in the wrong direction. It’s all about making sure the right operations happen first. Think of it as the difference between 'squaring the negative' and 'taking the square root of the square. Also, when you have more than one parenthesis, the brackets help to keep things orderly for multiple parenthesis. Things can get very messy! When you have a negative real number as a radicand, there are cases for which [a^r1]^r^2 may NOT equal [a^r2]^r1 for a < 0. The brackets are absolutely necessary to show this proof, hence the example that many students have gotten wrong.
How is the teaching "wrong"? You have not provided any reason at all for your comment. The video clearly states why they are not equal. And if you argue that they are, I want a mathematical explanation other than "the teaching is wrong".
@@novalistutoring9603 This is what I have said as well, so not sure why you are misinterpreting. I specifically said, quoting myself (I hate to do this), but I must: "4 is not equal to -4, and the result is not independent of the order of operations when the base is a NEGATIVE real number and the root is even. And yes, -3^2 means negative 9, and (-3)^2 is positive 9. But when you have a negative real number as a radicand, there are cases for which [a^r1]^r^2 may NOT equal [a^r2]^r1 for a < 0. The brackets are absolutely necessary to show this, hence the example that many students have gotten wrong.
Wow, your channel is really growing. Congratulations.
Thank you! Yes, it is shocking to me...never would have expected this from a Math channel!
You mentioned that you disliked the way math is taught in the USA, particularly the simulation of steps done by tutors. I completely agree with you, and I’ve taken inspiration from your perspective to create my own channel for the Arab world to address this very issue. It seems we share the same concern, and I truly appreciate your insights.
Thank you for recommending those three books-they’ve been incredibly helpful. I wish you all the best with your channel and your endeavors.
Thank you very much! That is an excellent idea, and you have my support. The more we can spread around the world the foundations of proper math, the better the world will be to equip our young people to think PROPERLY about things. I hope your channel goes far as well!
@@citytutoring and the world is welcoming the approach, thaanks dear
I disliked mathematics, not for its subject matter, but because no one explained why things worked as they did. This questioning intensified with physics, leading me to drop it, pursue intermediate mathematics, and focus on science and English. Years later, as a lawyer, I remain curious about mathematics and its logic.
@13:28 The whole numbers include zero. Since you can't have a denominator of zero, you rather should have said that n must be a positive integer.
Regarding @24:00 There is a big difference between polite correction and abuse.
Yes, I agree it would have been clearer to specify n must be a positive integer! For sure! And yes, you are right about the abuse, but I am used to it all. Been "controversial" for a long time, so I am used to both!
Yeah unfortunately.
Mr Cromwell, I need your opinion. I have been studying Calculus for some time now with Stewarts book. I went through the first 5 Chapters, so differentiation and integration with Riemann's method, moreover I just started the rules on Integration. So I can say I have a relative strong foundation. I can use differentiation, I understand the main concept behind it, why we are doing it and so on, but sonething is missing. It is a shapeless feeling, I can not put my finger on it. Maybe I miss deepness or something. Rigour maybe... Stewarts book is not bad at all, considering I use it as a mechanical engineering student, but instead of just seeing how things work out I truly want to feel it deeply! I hope you understand my point here. Can you recommend me calculus books that have a kind of different "magic" approach? Currently I found Apostols Calculus which seemst to be a better choice but I hope you can recommend me some other literature here. And if possible, it would be very kind If you also could state the type of teaching method those books work with. I mean who do you recommend one over another? What type of person?
Thank you very much!
Congratulation on the 4k+ sub!!!🎉😊
I am sorry, I was not able to respond to you earlier! Fear not; you're experiencing the natural frustration of someone ready to transcend the mechanical into the metaphysical. Stewart is a practical workhorse of a book: it gets you from point A to point B but rarely invites you to ponder the scenery. If you're craving rigor and magic, here’s your itinerary:
Apostol’s Calculus - A superb choice if you're a fan of mathematical dual citizenship: theory and application. Apostol doesn’t just teach you how to differentiate; he’ll make you prove why differentiation deserves to exist. Be ready for axioms and integrals from day one-no shortcuts here. Recommended for anyone who likes structured thinking and doesn’t mind pausing to sip the pure nectar of mathematical proofs.
Spivak’s Calculus - Think of this as calculus written by a poet who happens to also be a sadist (and I mean that as a compliment). Spivak’s book is rigorous and has a charm that will either break you or make you fall in love with epsilon-delta proofs. Ideal for those who appreciate quality and are willing to work hard for their epiphanies.
Courant’s Differential and Integral Calculus - The lovechild of mathematical rigor and applied sensibility. Courant dives deeper than Stewart but still remembers you’re human. Great for an engineering mindset with a penchant for rigor.
Kline’s Calculus: An Intuitive and Physical Approach - If Apostol and Spivak feel like they’re delivering calculus sermons from on high, Kline will come down to earth and walk you through with vivid, intuitive examples. A gentle but insightful companion.
Choose your poison based on your personality:
Apostol if you want rigor served with a dash of formality.
Spivak if you're after rigor with a pinch of whimsy.
Courant if you want to bridge engineering and theoretical perspectives.
Kline if you’re a lover of intuition who doesn’t want to lose sight of reality.
Remember, deep understanding isn’t about jumping into a new book-it’s about taking the time to wrestle with the ideas. Let these authors guide you, but don’t expect them to solve the ‘shapeless feeling’ for you. That’s the part of the journey you get to carve out yourself.
@citytutoring Mr. Cromwell, Your ability to amaze me from time to time is beyond compare. Not just a dozens of books you list but you put them into a "tale". Into perspective. That is rellay helpful when choosing the rigtht book(s). Moreover, you your vocabulary is superb. A true poet.😀 Thank you very much, you really helped me with this one! By the way, let me ask you one more question. As I looked into some of the books you mentiond, tipically Apostol and Spivak, in order to understand the deepness of the topics, It is some proof writing skills are required. Do you agree with me? So do you recommend to start with the basics of proof writing in order to get the proofs mentiond in these books?
@@petihundeuhun8950 Thank you for your kind and generous words. They are like a balm to my soul amidst so many that try to attack mathematics. Regarding your question: you’re absolutely right. Apostol and Spivak don’t just hand you answers; they expect you to waltz through proofs like a mathematician at a formal ball. Proof-writing skills are the key to understanding their reasoning.
If proofs seem intimidating at first, think of learning them as sharpening your detective skills-spotting assumptions, following logical breadcrumbs, and arriving triumphantly at 'QED.' I'd suggest starting with a gentle primer on proofs, like "How to Prove It" by Velleman. It's like a training montage for your brain before diving into the heavyweight champions like Apostol and Spivak.
So yes, master the art of proofs first. Once you do, Apostol and Spivak’s pages will feel much better. Happy proving-may your theorems always be true and your proofs ever elegant!
@@citytutoring Thank you! By the way, your channel is skyrocketing. Congratulations for the subs!!!
I disagree with your analysis of the right hand side.
You are interpreting it as ((-4)^(1/2))^2=(2i)^2=-4
But the expression as written is (-4^(1/2))^2, namely there are no brackets around the -4.
The actual value is therefore (-4^(1/2))^2=(-1 * 4^(1/2))^2=(-1 * 2)^2=(-2)^2=4.
So, in fact the two sides are equal.
I am a little confused about the Use of the bracket. I told my Students, that the minus before a number ist only Affected by the Exponent, If ITS written with a bracket. In this Case i would say that (-4)^(1/2) is equal to 2i and -4^(1/2) is equal to -2 because the minus is Not touched by the ^(1/2). Maybe this is a Thing of Definition or so....
You are correct. It is a matter of defining the meaning of mathematical symbols. And the teaching in the video is wrong.
For example: -3^2 =-9 whereas (-3)^2 =9
Some calculators and computational software may fail to interpret such problems correctly, but this is a long-established convention for the use of these symbols. As you correctly told your students, the exponent only affects the number or letter immediately before it, not the minus sign, unless the number and minus sign are surrounded by parentheses (or brackets, as some English speakers call them).
Thank you for your comment -- The brackets are necessary to clarify the intended order of operations. Without them, the interpretation changes due to how exponents are applied. Without parentheses, you could end up with a different calculation order:
(-4^(1/2)) means -(4^(1/2)), which is the negative square root of 4, or -2. Squaring that gives 4. [(-4)^2]^(1/2) clearly means you first square -4 to get 16 and then take the square root of 16, which equals 4. So, the two expressions are not the same-the parentheses are crucial for defining the order of operations. the brackets are like traffic lights-they make sure everything stops in the right order! Without them, the operations would go off in the wrong direction. It’s all about making sure the right operations happen first. Think of it as the difference between 'squaring the negative' and 'taking the square root of the square. Also, when you have more than one parenthesis, the brackets help to keep things orderly for multiple parenthesis. Things can get very messy! When you have a negative real number as a radicand, there are cases for which [a^r1]^r^2 may NOT equal [a^r2]^r1 for a < 0. The brackets are absolutely necessary to show this proof, hence the example that many students have gotten wrong.
How is the teaching "wrong"? You have not provided any reason at all for your comment. The video clearly states why they are not equal. And if you argue that they are, I want a mathematical explanation other than "the teaching is wrong".
@@novalistutoring9603 This is what I have said as well, so not sure why you are misinterpreting. I specifically said, quoting myself (I hate to do this), but I must: "4 is not equal to -4, and the result is not independent of the order of operations when the base is a NEGATIVE real number and the root is even. And yes, -3^2 means negative 9, and (-3)^2 is positive 9. But when you have a negative real number as a radicand, there are cases for which [a^r1]^r^2 may NOT equal [a^r2]^r1 for a < 0. The brackets are absolutely necessary to show this, hence the example that many students have gotten wrong.
Proofs make everything needlessly complicated and inefficient.
That's why separate courses are required for proofs.