when doing equations, think of '=' as a binary operator, exactly like '+' etc. distributivity, commutativity, etc all hold. (2x = y) -> -- not equal! we multiply times 1/2 and distribute it to both sides (1/2)(2x = y) = ((1/2)2x = (1/2)y) the main thing is to use "=" as a binary operator, and use "->" when you take an equality and modify it. this makes a lot of sense when you start to try to turn this into code. division is special. when you divide by something, you need a side-condition that it's not 0. (a = 2b) -> (a = 2b)/b, b != 0 = (a/b = 2(b/b)), b != 0 = (a/b = 2), b != 0 notice that for two things to be equal, the side conditions need to be equal as well. i wish that Lean would allow me to do things like this, but it treats '=' as more than a binary operator. it would make equation solving easier to do things like "step 2, divided by z, and z != 0".
15:26 ive been waiting to hear those words for so long now. i started enjoying math because after failing my adv algebra class, i was more convinced it had to do with the teacher or the lessons it self. So i started doing pure math on my own and learning it at my own pace, im doing complex analysis and stuff because its so cool, words dont really describe just how amazing math really is. I dont really like the way my teachers teach because their WAY of teaching is so formulaic, predictable, and doesn't really give students an intuitive understanding. I feel bad for the students who wont ever get the chance to see how mindblowing the subject really is. Its all a big memory game that we must play until we graduate. They treat it more like a job. It js a job sure, but for instance: about two or three days ago in precalculus we were reviewing logs and exponentiation equations. I shared a log property to the teacher where if you bring the log into the denominator, you can switch the base and the input. I thought it was cool but my teacher immediately went on the offensive because she assumed i was questioning her education. She says she has a masters degree in math and all that. As someone who didnt mean any offense when I said that, I was caught off guard because she made me look bad infront of the class. I feel like since she is a teacher, she should be encouraging my learning and taking advice from us instead of turning it down and dismissing my comment as some rude insult. They just dont seem to share the interest in math which is also ironic because they're teaching the subject. Ive only had two teachers who have an interest in what i was doing. Those are the teachers we need. One of them even gave me his old calculus book which i study from time to time. My other math teacher is very knowledgeable and knows many people so whenever i see him, i learn something new through our insightful exchanges.
@@ZDTF I got into complex analysis when wanting to solve the basel problem, the approach with complex analysis is very unique. One of the things you learn in complex analysis ties in with line integrals and vector calculus. It is very useful in solving nonelementary improper integrals and infinite sums. You have to prove the contour goes to zero which at first sounds kind of boring but with vector calculus, it was really easy and not tedious. There is also a very rewarding feeling when using fancy notation such as a circle in between an integral sign. Time flies. Now that I think about it, you learn so much from learning one thing in maths.
@@ZDTF I mean it took me like a few months to learn it. Maybe learn how to evaluate infinite sums. The digamma function and basel problem were my starting points. Have like a goal and learn what you need to solve it. Theres a lot of good videos including visual ones and calculators.learn what interests you. Learn it visually and numerically.
For your first point it is probably clearer to say: a*b=0 implies a (inclusive) or b = 0. The way you wrote the "Fundamental Truth" would imply that for example in the case 6x+6=0 that 6=0 becuase it is a common factor which is a contradiction as 6 is not equal to 0.
I did not pursue a PhD paper, as I did the Masters program in Pure Mathematics at University of London, and then afterwards decided to concentrate on researching on Abstract Algebra as well as Number Theory.
I am just so relieved that i can vent about this in a comment section with someone that shares the same problem because i dont really feel like i can last the rest of the year in that class where learning seems to be discouraged.
Thank you for sharing your experience! It's inspiring to hear about your journey into math on your own terms. Your story shows just how powerful self-directed learning can be-especially when traditional classroom settings don’t always nurture that spark of curiosity. You’re absolutely right: real learning happens when students get to explore concepts, ask questions, and connect with ideas in a way that resonates with them personally. I will let you in on something: unfortunately, most math "teachers" actually hate mathematics. I say this as someone who has taught Math for teachers planning to become teachers at the college level. Many teachers specifically prepare for one area of Math, and learn a very "recipe" like approach, so when you throw them off guard, they get upset. It’s unfortunate that some teachers can feel defensive rather than supportive when students bring fresh perspectives to the table. Mathematics should be a process where questions and ideas are welcomed-not dismissed. The fact that you’re diving into complex analysis and finding joy in it is proof that students flourish when they're allowed to explore and share insights freely. You are absolutely correct also that your teacher should be supporting your learning. May I ask where you are located? Please keep doing what you are doing! Do not let them stand in your way of enjoying Math!
Not specifically, no, but I am not surprised at what you mentioned. I believe that your teacher should do a lot more to put you at the center of priorities, as opposed to his/her personal preferences.
What does it mean to be a mathematician to you? To me it means that 1) you understand what mathematics is 2) you understand what “doing mathematics” is 3) you “do mathematics” on a regular basis How deeply have you really thought about what mathematics is?
For me, to be a mathematician is to bask in a world where the existence of pure beauty depends on neither purpose nor utility-an indulgence in the abstract where each theorem feels less like a tool and more like a truth unearthed. It is about equating being right with happiness. Mathematics isn’t merely the manipulation of symbols or a choreographed performance of proofs; it’s the art of revelation, a quiet triumph over chaos and uncertainty (though not always, as there are many unsolved problems). To me, 'doing mathematics' is akin to hunting for jewels in an endless landscape, where the act itself is perhaps the finest expression of understanding. So, how deeply have I thought about what mathematics is? As deeply as one can ponder infinity-endlessly, and yet always feeling as if I've barely scratched the surface. That's what keeps me going, in a sense, because it is also about putting reason and right as the most absolutely essential life priority.
@@citytutoring Fine, but the video hardly communicates that IMO. This is a bunch of special case tricks for solving school maths problems, which is fine if that was your intent, but don't confuse that with following the math dream!
Hello Professor Cromwell! once again, your emergency video on the sum and difference trig formulas were very helpful, and also helped me ace the test that I had pertaining to Trig Identities! As of now, my Precalculus class is going over Math Induction along with the Law of Sines in Triangles(with limited usage of the unit circle involved). I hope you will be able to make a video fulfilling these topics, but if not due to time constraints, I understand. Irregardless, I hope to see this channel further grow over time!
Great! So, I'll go ahead and plan a video for those topics you need. I cannot guarantee all of the topics, but at least a good portion of them either Saturday or by Sunday.
@@citytutoringHello once again Professor Cromwell! I forgot to mention that topics further covered as of now include the Law of Cosines along with modeling problems in triangles, solving triangles, and sinusoidal modeling, with the test on all of this being the Monday after next week(Math Induction, Law of Sines, Law of Cosines, Solving Triangles, Modeling problems in triangles, and sinusoidal modeling). The unit after this will be one of toughest ones covered in the course, and will mostly be covering vectors, which tends to get very conceptual.
The common factor is x + 1, so the common factor x + 1 = 0, but when you solve for x, you get x = -1. So in a situation like that where it is a factor that is of an equation type, you have to remember to solve for x. Make sense?
People unqualified to teach math teaching math is better than people qualified to teach math not teaching math. The problem is people qualified to teach math make more money not teaching math.
You make a good point! At the end of the day, students deserve teachers who are truly dedicated to their learning. If qualified math professionals are drawn away by higher salaries in other fields, that is understandable because it is a failed system. We need teachers who also have the incentive and passion to be there. Too many students are left behind when teachers prioritize their own advancement over students' needs.
Ok sir u slove my question problem total 1to 25 numbers than answer come 15 number example 1 4 5 6 7 8 9 12 13 14 15 17 19 20 21 second coming numbers 4 5 6 7 8 9 10 11 12 16 19 20 21 23 25 than what next 15 number come on 1to25 numbers u slove my problem sir I am u new subscriber
Hello! No, the website is just under construction. It should be available by next week. Thanks! Is there anything we can help you with in the meantime?
Hello sir I am in grade 11 i have a humble request can you pls help in trigonometric questions where they involve large manipulations and smart handling as i always try to solve them but i realise that i'm lacking somewhere Thank you.
when doing equations, think of '=' as a binary operator, exactly like '+' etc.
distributivity, commutativity, etc all hold.
(2x = y)
-> -- not equal! we multiply times 1/2 and distribute it to both sides
(1/2)(2x = y)
=
((1/2)2x = (1/2)y)
the main thing is to use "=" as a binary operator, and use "->" when you take an equality and modify it. this makes a lot of sense when you start to try to turn this into code. division is special. when you divide by something, you need a side-condition that it's not 0.
(a = 2b)
->
(a = 2b)/b, b != 0
=
(a/b = 2(b/b)), b != 0
=
(a/b = 2), b != 0
notice that for two things to be equal, the side conditions need to be equal as well. i wish that Lean would allow me to do things like this, but it treats '=' as more than a binary operator. it would make equation solving easier to do things like "step 2, divided by z, and z != 0".
15:26 ive been waiting to hear those words for so long now. i started enjoying math because after failing my adv algebra class, i was more convinced it had to do with the teacher or the lessons it self. So i started doing pure math on my own and learning it at my own pace, im doing complex analysis and stuff because its so cool, words dont really describe just how amazing math really is. I dont really like the way my teachers teach because their WAY of teaching is so formulaic, predictable, and doesn't really give students an intuitive understanding. I feel bad for the students who wont ever get the chance to see how mindblowing the subject really is. Its all a big memory game that we must play until we graduate. They treat it more like a job. It js a job sure, but for instance: about two or three days ago in precalculus we were reviewing logs and exponentiation equations. I shared a log property to the teacher where if you bring the log into the denominator, you can switch the base and the input. I thought it was cool but my teacher immediately went on the offensive because she assumed i was questioning her education. She says she has a masters degree in math and all that. As someone who didnt mean any offense when I said that, I was caught off guard because she made me look bad infront of the class. I feel like since she is a teacher, she should be encouraging my learning and taking advice from us instead of turning it down and dismissing my comment as some rude insult. They just dont seem to share the interest in math which is also ironic because they're teaching the subject. Ive only had two teachers who have an interest in what i was doing. Those are the teachers we need. One of them even gave me his old calculus book which i study from time to time. My other math teacher is very knowledgeable and knows many people so whenever i see him, i learn something new through our insightful exchanges.
Tell me something cool about maths
@@ZDTF I got into complex analysis when wanting to solve the basel problem, the approach with complex analysis is very unique. One of the things you learn in complex analysis ties in with line integrals and vector calculus. It is very useful in solving nonelementary improper integrals and infinite sums. You have to prove the contour goes to zero which at first sounds kind of boring but with vector calculus, it was really easy and not tedious. There is also a very rewarding feeling when using fancy notation such as a circle in between an integral sign. Time flies. Now that I think about it, you learn so much from learning one thing in maths.
@@Epic_Nam0071 ngl
I didn't understand anything after I got into complex analysis
Am I cooked
@@ZDTF I mean it took me like a few months to learn it. Maybe learn how to evaluate infinite sums. The digamma function and basel problem were my starting points. Have like a goal and learn what you need to solve it. Theres a lot of good videos including visual ones and calculators.learn what interests you. Learn it visually and numerically.
@@ZDTF I started with integrals and derivatives
For your first point it is probably clearer to say: a*b=0 implies a (inclusive) or b = 0. The way you wrote the "Fundamental Truth" would imply that for example in the case 6x+6=0 that 6=0 becuase it is a common factor which is a contradiction as 6 is not equal to 0.
what was your PhD paper on and what's your current research work?
I did not pursue a PhD paper, as I did the Masters program in Pure Mathematics at University of London, and then afterwards decided to concentrate on researching on Abstract Algebra as well as Number Theory.
@@citytutoringUh oh. You're in trouble now. All your opinions are now invalid
@@Flaystray how?
@@chadlj i think they were joking
rare find, i've been blessed to find this video
Cross multiplying is multiplying over the equals sign. You meant write the fractions as a single fraction.
Yes, indeed! Sorry about that.
I am just so relieved that i can vent about this in a comment section with someone that shares the same problem because i dont really feel like i can last the rest of the year in that class where learning seems to be discouraged.
Thank you for sharing your experience! It's inspiring to hear about your journey into math on your own terms. Your story shows just how powerful self-directed learning can be-especially when traditional classroom settings don’t always nurture that spark of curiosity. You’re absolutely right: real learning happens when students get to explore concepts, ask questions, and connect with ideas in a way that resonates with them personally. I will let you in on something: unfortunately, most math "teachers" actually hate mathematics. I say this as someone who has taught Math for teachers planning to become teachers at the college level. Many teachers specifically prepare for one area of Math, and learn a very "recipe" like approach, so when you throw them off guard, they get upset.
It’s unfortunate that some teachers can feel defensive rather than supportive when students bring fresh perspectives to the table. Mathematics should be a process where questions and ideas are welcomed-not dismissed. The fact that you’re diving into complex analysis and finding joy in it is proof that students flourish when they're allowed to explore and share insights freely. You are absolutely correct also that your teacher should be supporting your learning. May I ask where you are located?
Please keep doing what you are doing! Do not let them stand in your way of enjoying Math!
@@citytutoring Your response really means a lot. I'm in Texas, have you reviewed schools here?
Not specifically, no, but I am not surprised at what you mentioned. I believe that your teacher should do a lot more to put you at the center of priorities, as opposed to his/her personal preferences.
I will now be a loyal subscriber to you. Hope your channel continues to grow so that more and more people can gain and spread your knowledge.
Thank you! Much appreciated. Where are you located?
What does it mean to be a mathematician to you? To me it means that
1) you understand what mathematics is
2) you understand what “doing mathematics” is
3) you “do mathematics” on a regular basis
How deeply have you really thought about what mathematics is?
For me, to be a mathematician is to bask in a world where the existence of pure beauty depends on neither purpose nor utility-an indulgence in the abstract where each theorem feels less like a tool and more like a truth unearthed. It is about equating being right with happiness. Mathematics isn’t merely the manipulation of symbols or a choreographed performance of proofs; it’s the art of revelation, a quiet triumph over chaos and uncertainty (though not always, as there are many unsolved problems).
To me, 'doing mathematics' is akin to hunting for jewels in an endless landscape, where the act itself is perhaps the finest expression of understanding. So, how deeply have I thought about what mathematics is? As deeply as one can ponder infinity-endlessly, and yet always feeling as if I've barely scratched the surface. That's what keeps me going, in a sense, because it is also about putting reason and right as the most absolutely essential life priority.
@@citytutoring Fine, but the video hardly communicates that IMO. This is a bunch of special case tricks for solving school maths problems, which is fine if that was your intent, but don't confuse that with following the math dream!
Hello Professor Cromwell! once again, your emergency video on the sum and difference trig formulas were very helpful, and also helped me ace the test that I had pertaining to Trig Identities! As of now, my Precalculus class is going over Math Induction along with the Law of Sines in Triangles(with limited usage of the unit circle involved). I hope you will be able to make a video fulfilling these topics, but if not due to time constraints, I understand. Irregardless, I hope to see this channel further grow over time!
Thank you! I am so glad for you that your exam went well! Yes, I think I will be able to do that this weekend. Where are you located, if I may ask?
@@citytutoring You’re very welcome! I am located on the West Coast of the US, in the Bay Area, California specifically.
Great! So, I'll go ahead and plan a video for those topics you need. I cannot guarantee all of the topics, but at least a good portion of them either Saturday or by Sunday.
@@citytutoringHello once again Professor Cromwell! I forgot to mention that topics further covered as of now include the Law of Cosines along with modeling problems in triangles, solving triangles, and sinusoidal modeling, with the test on all of this being the Monday after next week(Math Induction, Law of Sines, Law of Cosines, Solving Triangles, Modeling problems in triangles, and sinusoidal modeling). The unit after this will be one of toughest ones covered in the course, and will mostly be covering vectors, which tends to get very conceptual.
@@BonFlanders Oh that's even better then! Gives us a lot more time to do things for sure!
bruh he looks like harry potter
in the equation where is the product of independent terms (x+1)(x+2)(x+3)=(x+1)^2 (x+6) , the solution is also 0 right?
The common factor is x + 1, so the common factor x + 1 = 0, but when you solve for x, you get x = -1. So in a situation like that where it is a factor that is of an equation type, you have to remember to solve for x. Make sense?
@@citytutoring i understand that x=-1 but isnt 0 also solution
@@safwanebouftass2806 Yes, correct!
These are obvious facts.
Make a maths series for bachelor students plssss sirrrrr plszzzz
People unqualified to teach math teaching math is better than people qualified to teach math not teaching math. The problem is people qualified to teach math make more money not teaching math.
You make a good point! At the end of the day, students deserve teachers who are truly dedicated to their learning. If qualified math professionals are drawn away by higher salaries in other fields, that is understandable because it is a failed system. We need teachers who also have the incentive and passion to be there. Too many students are left behind when teachers prioritize their own advancement over students' needs.
*Host of this site* X and x represent different variables. Write a letter in the same case to represent the same variable.
Yes, sorry -- they should all be lowercase variables, no capitals in this case.
Ok sir u slove my question problem total 1to 25 numbers than answer come 15 number example 1 4 5 6 7 8 9 12 13 14 15 17 19 20 21 second coming numbers 4 5 6 7 8 9 10 11 12 16 19 20 21 23 25 than what next 15 number come on 1to25 numbers u slove my problem sir I am u new subscriber
Hello, your website seems to be down.
Hello! No, the website is just under construction. It should be available by next week. Thanks! Is there anything we can help you with in the meantime?
New Subscriber😊🎉🎉🎉
Thank you! Where are you located?
Hello sir I am in grade 11 i have a humble request can you pls help in trigonometric questions where they involve large manipulations and smart handling
as i always try to solve them but i realise that i'm lacking somewhere
Thank you.
Try AMC or AIME level problems. They will humble you.
@@mathbrah I train students for Math olympiads with such problems. They are fun!
What do you define a common factor?
gg