You cannot imagine what it felt to come across this channel.. As an international student learning in a different language.. You are t God sent.. Just finished my calc 1 and passed.. I really appreciate..
Just wanted to leave a comment saying how amazing this guy is and he is the only reason I passed Calc 3. I had a teacher who wasn’t very good at teaching but professor Leonard helped me be one of the few that passed with a good grade
Hey, please comeback professor. We seriously need you. Lectures on Trigonometry, Complex variables and Partial differential equations are needed. You are our only hope.
Professor Leonard, thank you for a basic Introduction to Mathematical Induction. Learning Mathematical Induction is very helpful in Discrete Mathematics, Advanced Calculus, Number Theory, Linear Algebra and future course in Mathematics. Mathematical Induction helps students to write proofs in Advanced Mathematics. Mathematical Induction helps me to fully understand Advanced Calculus.
you are everything i needed for my discrete mathematics course. immaculate explanation, humanly most explicit explanation and breakdown of each and every activity... best mathematician youtuber in history
I love your Calculus series ( I've watched all of it). It is the reason I passed Calc I and II. I can't wait until you are back in a classroom in front of students again. I miss your sense of humor.
I can't said anything. I will meet this guy in person . I downloaded his videos from Cal 1-3 and statistics. I will watch this video for life. Imagine studying in college without knowing the language of lectures(arabic). All I have to do is watching P. Leonard videos and come for quiz or exams laughing at my colleague. 1 million subscribers is not enough to give. I don't know how to thank you.🤝🙏
Thank you Prof.....you are a blessing to students ❤❤❤❤❤❤❤❤💃💃💃💃💃💃💃💃💃I thank you a lot Prof, I managed to get my Bachelor of Education degree via your videos
Professor Leonard I am a big fan of the work you are doing! Please consider making a series on content related to proofs. Such material is really scarce to come across on the web even to this day unfortunately. It's a shame too because proofs are the bread and butter of understanding mathematics at a deeper level. :(
Hey Professor!! As of writing this comment I have taken my final and passed Pre-Calculus!! I will be taking Calculus 1 for Winter Semester 2021! Thank you so much for these videos they truly did help alot especially with Distance Learning being a thing. I'll see you in your Calc 1 Playlist. Thanks again!!
Yet again Professor Leonard makes something so approachable that zybooks made feel so convoluted. Watching these videos always makes me wish I had time/money to go after a math degree with Professor Leonard teaching the classes!
wow... Still coming through for me way after Calculus... You're really the MVP. Thank you for all your videos, please keep going on this discreet mathematics/number theory lessons!
Goodness Leonard, this felt like a watershed moment for me (and the previous Series/Sequence and Asymptote videos) I feel like I have a tangible grip harnessing infinity God bless you
Love your videos, and have used them to help prepare me for teaching my own calculus class! I do want to mention that you want to be careful about assuming what you are trying to prove. For example, at 19:25 when you start setting 1+5+9+... + (4k-3) + [4(k+1)-3] equal to what you want it to be. Instead, best practice would be simply manipulating the first part of your equality until you get it to the point that it looks like what you want to prove. For me, I did it like the following: 1+5+9+... + (4k-3) + [4(k+1)-3] = 2k^2 - k + 4(k+1) - 3 = 2k^2 - k + 4k + 4 - 3 = 2k^2 +4k + 2 - k + 2 - 3 (splitting up the +4 here into two instances of +2) = (2k^2 +4k + 2) - k - 1 (association property and also simplified 2 - 3 to be -1) = 2(k^2 + 2k +1) - k - 1 (factor out the 2 in the grouped terms) = 2(k+1)^2 - k - 1 (factored the grouped polynomial) = 2(k+1)^2 - (k+1) (factored out a negative in the non-grouped terms) and at this point we can conclude that 1+5+9+... + (4k-3) + [4(k+1)-3] = 2(k+1)^2 - (k+1) by the transitive property, and have proved the theorem by induction. thanks again for all your videos, prof!
Good day Professor, It sometimes goes unnoticed when amazing people do amazing things. While this pandemic has caused all our lives to change, your videos became a source of inspiration to us all students. Not many can be as detailed in content and explain with such enthusiasm. Thank you for being the essential worker we needed by delivering excellence on absolutely every video uploaded. May your 2021 be im'PECC'able("see what i did there-HA HA HA"-Preofessor Leonard).
Thank you so much Proffesor Leonard, finally now I understand the basic concept of this mathematical induction. I will retake exam discrete math and hope i can answer the questions. Thank you sir
Trigonometry with you, prof Leonard, would be such a blessing. Sadly I must proceed and finish it alone. Guess I'll have to go back to it when you do it!
Another way to explain it would be subtract the series of n=k terms from the series of n=k+1 terms and the only thing standing is the k+1th term on the left hand side and the difference between the right hand side terms...simplify the right hand side and show the right hand side (differences) equals the k+1th term.
Thanks so much! I have an exam in Discrete Mathematics to take today, and I know this will be very important for it! Really glad you had a video on it, especially from just a few days ago!! Hope you are well as you can be!
Your videos are having good effect on my colleagues and I, We thank you so much 👏. We consider if we can have a playlist on Abstract Algebra with subtitles: Mapping, Binary Operation, Relations, Mathematical Induction, Divisors, Irrational Numbers, Rational Numbers, Complex Numbers, Congruence Modulo N, Groups, Rings and Fields.🙏
lectures about linear algebra would be great! many thanks for these lectures, appreciate much what you are doing and for free. everything is explained in a great manner. if the donation is possible I would be glad to support as much as I can. I am sure there are many who would support such a great job. there are many lectures and lecturers but this is far above them.
Great idea, giving a subsequent example after the base case. It drives home the point that the SUM OF the left side at the nth term = formula on the right side with n plugged in. For example at n=3, the correct workup is 1+3+5 = 3^2, its NOT 2(3)-1 = 3^2. That's obvious. But at n=1, students might not get the distinction since 2(1)-1 does equal 1^2..... wrong process leads them to a true statement and confusion.
I studied this topic on my own about 3 years ago. This is such an important topic in mathematics. Is it needed for calculus 1? Not really but it will expand your understanding of proofs at a very basic level.
This topic can sometimes be brought up in some calculus courses. My engineering physics program in Sweden started out with a course called "Differential Calculus", which basically serves the same purpose as "Calculus 1", and almost all old exams in that course have asked the students to do a proof by induction.
@@Peter_1986 : Mathematical induction is interestingly difficult to grasp. It opens the door to a world of proofs. It is important to know the basics of proofs, especially for students majoring in mathematics.
Good Afternoon Professor Can you please assist me with these questions Question 1 An education official is interested in the percentage of the adult population over 20 years, in his city, who has read at least one book during the last year. How can he investigate this question using descriptive statistics tools? Question 2 A social researcher is interested in the number of children in the households of City A that receive welfare. He does not have funding for a census nor access to official city records. How can the researcher accomplish his task using descriptive statistics tools? Question 3 In an organization, most of the employees earn at the lower end of the salary scale.State a chart you would use to illustrate the salary data. Describe this salary chart using the appropriate statistical terminology. If Maya, a potential employee, is seeking general information about the company salary scale, what statistical information would you give her and why? Question 4 A quality engineer is curious about the thickness of the paint on the Model A cars produced by her company. Explain how she can obtain a reliable measure of the thickness of the paint, stating clearly the characteristics of the variable she should use. Question 5 Devise a research question. Use this question to explain what is (a) representative sample (b) random sample (c) variable (d) measurement/data type
Another great lecture Professor , thank you. As a patreon may I ask if you have, in your playlist anywhere, an explanation for the 'general binomial expansion?' Regards.
What if the central of domino not falling? All textbooks don’t give examples of what if the induction failed? Can somebody do 1 example showing induction fail?
Yes! This is Amazing, incredibly useful for CS students, Please do consider a Discrete Mathematics series, it would help us a lot.
yeah discrete was quite the class. a good series on it would be super helpful for lots of students.
Hmmm
Define "discrete". Because when you take (or anyone takes) Theory of Computation, it's all graph theory application to automata.
Yeah that would help a lot. I like how this guy dumbs things tf down
Agreed!
You cannot imagine what it felt to come across this channel.. As an international student learning in a different language.. You are t God sent.. Just finished my calc 1 and passed.. I really appreciate..
I fear we have lost you sir. What a devastating loss. I hope you are well and I will always continue to look for your return. A long long time fan.
This man is truly getting me through my degree. Thank you for all you do.
Just wanted to leave a comment saying how amazing this guy is and he is the only reason I passed Calc 3. I had a teacher who wasn’t very good at teaching but professor Leonard helped me be one of the few that passed with a good grade
Hey, please comeback professor. We seriously need you. Lectures on Trigonometry, Complex variables and Partial differential equations are needed. You are our only hope.
Professor Leonard, thank you for a basic Introduction to Mathematical Induction. Learning Mathematical Induction is very helpful in Discrete Mathematics, Advanced Calculus, Number Theory, Linear Algebra and future course in Mathematics. Mathematical Induction helps students to write proofs in Advanced Mathematics. Mathematical Induction helps me to fully understand Advanced Calculus.
you are everything i needed for my discrete mathematics course. immaculate explanation, humanly most explicit explanation and breakdown of each and every activity... best mathematician youtuber in history
Best explanation of mathematical induction EVER!
I love your Calculus series ( I've watched all of it). It is the reason I passed Calc I and II. I can't wait until you are back in a classroom in front of students again. I miss your sense of humor.
If you have taken notes from these lectures can please share them with me my email id is rawatvanshika26@gmail.com
Superman never disappoints. Thank you sir.
Professor Leonard is the TRUTH! Thank you as always.
Its 2 am at night
And i.am.watching your video and doing maths..your are.just amazing professor... Thank you.so much❤️
I can't said anything. I will meet this guy in person . I downloaded his videos from Cal 1-3 and statistics. I will watch this video for life. Imagine studying in college without knowing the language of lectures(arabic). All I have to do is watching P. Leonard videos and come for quiz or exams laughing at my colleague. 1 million subscribers is not enough to give. I don't know how to thank you.🤝🙏
Crystal clear lecture. You are a gift to the math community.
Thank you Prof.....you are a blessing to students ❤❤❤❤❤❤❤❤💃💃💃💃💃💃💃💃💃I thank you a lot Prof, I managed to get my Bachelor of Education degree via your videos
Professor Leonard I am a big fan of the work you are doing! Please consider making a series on content related to proofs. Such material is really scarce to come across on the web even to this day unfortunately. It's a shame too because proofs are the bread and butter of understanding mathematics at a deeper level. :(
the way you set everything up made it click in my brain.. thanks so much
Hey Professor!! As of writing this comment I have taken my final and passed Pre-Calculus!! I will be taking Calculus 1 for Winter Semester 2021! Thank you so much for these videos they truly did help alot especially with Distance Learning being a thing. I'll see you in your Calc 1 Playlist. Thanks again!!
This man gives the best explanations on RUclips
Wooow I've been looking for Mathematical Induction... Please do lots and lots of videos on it
Saving my life once again. Using induction in my Bridge to advanced math class and linear algebra class. After this video I'm golden.
I love your lecture. I’m from Saudi. I studied statistics 101 so it was wonderful time I spent during watching your lectures that time.
honestly, you are doing a great job.
One of the best and most in-depth explanations on this short topic. Prof. Leonard should do a video course on Mathematical Thinking.
Yet again Professor Leonard makes something so approachable that zybooks made feel so convoluted. Watching these videos always makes me wish I had time/money to go after a math degree with Professor Leonard teaching the classes!
U've made me understand the concept in fullest
Best explanation I’ve found from dozens of other videos
Subscribed ,thank you for your support.
Professor L,
I really hope you will go over Trigonometry after College Algebra.
Thanks for all your hard work!!
He did
best explanation i have found on this. thank you
THIS IS EXACTLY WHAT I NEEDED. GREAT EXPLANATION :)
wow... Still coming through for me way after Calculus... You're really the MVP. Thank you for all your videos, please keep going on this discreet mathematics/number theory lessons!
Goodness Leonard, this felt like a watershed moment for me
(and the previous Series/Sequence and Asymptote videos)
I feel like I have a tangible grip harnessing infinity
God bless you
so good to see you uploading continuously, professor.
best wishes
Professor, can you please put a Linear Algebra series on your todo list?
this would be something I would watch
He has already confirmed in his video "Professor Leonard's Important Announcement 2019" that he will make Linear Algebra videos in the future.
@@Peter_1986 oh sweet
@@Peter_1986 amazing
@@adamharoon6021 Try James Hamblin for linear algebra
I just wanted to say thank you so much! you'er angel sent from Haven... I've never understood math faster than this.
God bless you prof Leonard
Gaining alot lately
Love your videos, and have used them to help prepare me for teaching my own calculus class!
I do want to mention that you want to be careful about assuming what you are trying to prove. For example, at 19:25 when you start setting 1+5+9+... + (4k-3) + [4(k+1)-3] equal to what you want it to be.
Instead, best practice would be simply manipulating the first part of your equality until you get it to the point that it looks like what you want to prove. For me, I did it like the following:
1+5+9+... + (4k-3) + [4(k+1)-3]
= 2k^2 - k + 4(k+1) - 3
= 2k^2 - k + 4k + 4 - 3
= 2k^2 +4k + 2 - k + 2 - 3 (splitting up the +4 here into two instances of +2)
= (2k^2 +4k + 2) - k - 1 (association property and also simplified 2 - 3 to be -1)
= 2(k^2 + 2k +1) - k - 1 (factor out the 2 in the grouped terms)
= 2(k+1)^2 - k - 1 (factored the grouped polynomial)
= 2(k+1)^2 - (k+1) (factored out a negative in the non-grouped terms)
and at this point we can conclude that
1+5+9+... + (4k-3) + [4(k+1)-3] = 2(k+1)^2 - (k+1) by the transitive property, and have proved the theorem by induction.
thanks again for all your videos, prof!
You are just out of this world thank you so much for this great video
Good day Professor, It sometimes goes unnoticed when amazing people do amazing things. While this pandemic has caused all our lives to change, your videos became a source of inspiration to us all students. Not many can be as detailed in content and explain with such enthusiasm. Thank you for being the essential worker we needed by delivering excellence on absolutely every video uploaded. May your 2021 be im'PECC'able("see what i did there-HA HA HA"-Preofessor Leonard).
Thank you so much Proffesor Leonard, finally now I understand the basic concept of this mathematical induction. I will retake exam discrete math and hope i can answer the questions. Thank you sir
I would start worshipping this dude if made a linear algebra and discrete math playlist
Trigonometry with you, prof Leonard, would be such a blessing. Sadly I must proceed and finish it alone. Guess I'll have to go back to it when you do it!
youre the best math professor ever
Well understood with good explanation
its unbelievable how quickly my stress dissipates when I see Prof. Leonard has a video covering what I'm stuck on.
Super concise and optimally explained, thank you!
professor Leonard, you are an amazing man
Another way to explain it would be subtract the series of n=k terms from the series of n=k+1 terms and the only thing standing is the k+1th term on the left hand side and the difference between the right hand side terms...simplify the right hand side and show the right hand side (differences) equals the k+1th term.
Thanks so much! I have an exam in Discrete Mathematics to take today, and I know this will be very important for it! Really glad you had a video on it, especially from just a few days ago!!
Hope you are well as you can be!
Your videos are having good effect on my colleagues and I, We thank you so much 👏. We consider if we can have a playlist on Abstract Algebra with subtitles: Mapping, Binary Operation, Relations, Mathematical Induction, Divisors, Irrational Numbers, Rational Numbers, Complex Numbers, Congruence Modulo N, Groups, Rings and Fields.🙏
Happy New and much blessings upon you Sir and family, and Much much appreciation for your efforts
So great
lectures about linear algebra would be great! many thanks for these lectures, appreciate much what you are doing and for free. everything is explained in a great manner. if the donation is possible I would be glad to support as much as I can. I am sure there are many who would support such a great job. there are many lectures and lecturers but this is far above them.
Awesome and clear lecture
Yes I have a better understanding now.
All support for that professor
You give great lectures! I hope to one day make as good videos as yours. You are a great inspiration to young people such as myself.
I love this man
Thank you so much sir
Great idea, giving a subsequent example after the base case. It drives home the point that the SUM OF the left side at the nth term = formula on the right side with n plugged in.
For example at n=3, the correct workup is 1+3+5 = 3^2, its NOT 2(3)-1 = 3^2. That's obvious. But at n=1, students might not get the distinction since 2(1)-1 does equal 1^2..... wrong process leads them to a true statement and confusion.
Thank you
I wonder when he'll finish his Differential Equations series and start his Linear Algebra vids 🧐
One of the best.
Holy hellllllll!!!!!!!! He finalllyyy did!!!!!!!
Yes 💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃💃
Hahahh love your reaction.
This makes me happy!
this helped a ton
Please start a complete course of modern algebra.
YOU TEACH BETTER THAN MY PROFESSOR!!!!!
Wow you explain this so well compared to my teacher. Thank you
I really hope you'll someday do a series on tensor calculus and differential geometry.
I doubt that it's his domain.
@@Czmlol I bet it is!,,
5:33
"4 does equal 4 - that is really kinda cool!"
Haha!
Oh please, get to Trig soon!
Sir you're the BEST!
Thank you bro!
Thanks, L!
Professor after pre calculus i think you do algebra or partial differential equation or real analysis
Outstanding sir 😍😍😍
This is amazing vedio
thank you so very much ❤
Thank you for the amazing video, it helps me a lot ,Mr Kent ,oh...Professor Leonard :D
Thank you so much! ♥️
How many proofs do I need to do for arms like that? Please provide a proof lol
lol @ "I like to treat my assumptions as fact" said every economist ever 3:10
Please do a course on discrete math!
Love you!
I studied this topic on my own about 3 years ago. This is such an important topic in mathematics. Is it needed for calculus 1? Not really but it will expand your understanding of proofs at a very basic level.
This topic can sometimes be brought up in some calculus courses.
My engineering physics program in Sweden started out with a course called "Differential Calculus", which basically serves the same purpose as "Calculus 1", and almost all old exams in that course have asked the students to do a proof by induction.
@@Peter_1986 :
Mathematical induction is interestingly difficult to grasp. It opens the door to a world of proofs. It is important to know the basics of proofs, especially for students majoring in mathematics.
can you explain the strong mathematical induction, please 🙏?
I didn't understand it 😭😭, and I have final exam after week in discreet mathematics😭😭
Thanks
Good Afternoon Professor
Can you please assist me with these questions
Question 1
An education official is interested in the percentage of the adult population over 20 years, in his city, who has read at least one book during the last year. How can he investigate this question using descriptive statistics tools?
Question 2
A social researcher is interested in the number of children in the households of City A that receive welfare. He does not have funding for a census nor access to official city records. How can the researcher accomplish his task using descriptive statistics tools?
Question 3
In an organization, most of the employees earn at the lower end of the salary scale.State a chart you would use to illustrate the salary data. Describe this salary chart using the appropriate statistical terminology. If Maya, a potential employee, is seeking general information about the company salary scale, what statistical information would you give her and why?
Question 4
A quality engineer is curious about the thickness of the paint on the Model A cars produced by her company. Explain how she can obtain a reliable measure of the thickness of the paint, stating clearly the characteristics of the variable she should use.
Question 5
Devise a research question. Use this question to explain what is (a) representative sample (b) random sample (c) variable (d) measurement/data type
Another great lecture Professor , thank you. As a patreon may I ask if you have, in your playlist anywhere, an explanation for the 'general binomial expansion?' Regards.
Not a single trigonometry function video in precalculus 73 video series... Why sir ??
Because you will know trigonometry is a few videos away if you actually watch his videos. He already said that a few videos back
@@eobardrush2112 thanks for replying & for the info. Brother....👍
Wow absolutly prof😯
What if the central of domino not falling? All textbooks don’t give examples of what if the induction failed? Can somebody do 1 example showing induction fail?
Hi !! Love your videos. Could you please to a Linear Algebra series? Thanks !!
You are the best!!!! Ty
When are you covering Laplace transform in differential equations?
What if there's some sequence in which the pattern remains until n reaches a very large number and we aren't aware of it?
This guy saved my ass times on end!!
Are you gonna be posting any lectures about Advanced calculus (Laplace trans. & fourier trans.) please?