This guy is so clear in everything he says. Most teachers would skip most of the stuff he's explaining because they feel it's obvious. Khan never assumes that anything is obvious and that is why his videos are so easy to follow.
teachers who explain like this are the best ones! The world would be a much easier place if only teachers wouldn't assume that we know the "obvious" things in life. What is obvious to a professor, with a Ph.D. might not be so obvious to a college kid on his first day on the topic.
I just spent about an hour looking at proof by induction in an Elementary Linear Algebra book as well as some notes online from Stanford but both of those sources were a million miles away from this level of intuition! Khan Academy to the rescue! Thank you good sir, very clear, understandable and intuitive.
So, to prove by induction that an equation is true for all inputs: 1. Check that it is true for the first input. 2. Write the equation, and incorporate (k+1) to both sides by following the pattern of the left-side part. For example: In the video, since the left-side part was a series of additions, (k+1) (the next number in the series of additions) was incorporated by ADDING it to both sides. 3. Transform the right-side of the above equation into the form of the original right-side. For example: In the video, the original right-side is a fraction whose numerator is the input * (the input + 1), & the denominator is 2; that's its form. And the proof merely consisted in adding (k+1) to both sides and transforming the right side into the same form of the original right side (a fraction whose numerator is the input [which in this case is k+1 instead of just n] * (the input + 1), and whose numerator is 2).
proving something by mathematical induction isnt that difficult, its the question my professor assigns, he has us proving these ridiculously long sums that requires so much algebraic manipulation that it just makes the problem extremely difficult
I remember the story of how this formula was made. The creator of the proof was causing problems in class and his teacher told him that he had to find the sum of every number between 1 and 100. He started to notice a pattern when he added 1 and 100, 2 and 99, 3 and 98... seeing that he is getting 101, 50 times. So he showed it to his teacher saying the answer was 5050. The teacher didn't believe him and wrote out the whole problem and her results came out equaling 5050.
Thanks again Khan, I can watch your video and understand it. You're making the world a better place. You would not believe how much better this is than my class.
omg this is so much better than my lectures, i question why i pay so much money to go to lectures where i get very lost. my math proofs prof sucks balls i don't understand shit when she explains but everything is so clear now that this guy explains it
Oh god ! I was totally intrigued about this topic. Another people were just teaching me how to solve problems based on it. None of them teaching me how it works. Hats off⚡⚡
In my introduction to higher mathematics class, MTH 311, I can pay attention for like the first 30 mins. Then the next 30 mins I'm either staring at him write a proof on the board while thinking about what I want to get from the vending machine when class is over, or how I'm gonna even attempt to write anything on the next assessment, or anime. Or sleeping. The the last twenty mins we take an assessment where we have to write a proof on what we learned that day and two days ago.
This reminded me of the good old days back in high school when I was the straight-A kid in the class. Didn't think I would forget proof by induction someday, and didn't think I'd need it in software engineering (Automata theory course). Thanks a lot
I wanna cry from the moment of understanding and clarity this video gave me after spending hours trying to understand induction from my discrete math textbook
You have to substitute K+1 for all values of 'n' so it will be K+1[(K+1)+1]/2 = K+1(K+2)/2 Reason for doing this is because 'n' represents all positive integers. The reason why you can't substitute 'n' in place of 'k' is because 'n' represents any positive integer while 'k' represents a specific unknown integer. when one is added to 'k' then a new integer is formed 'k+1'. Therefore K+1 = n when P(K+1) is a function for n.
Thank you. This explained proof by induction to me with the same example as my professor but 1000x easier to understand. No steps were skipped. Again, thank you
The binomial formula is just (x+y)^m = SUM k=0 to m; mCk*x^m*y^k. I'm not 100% sure about this but you must get it into the binomial coefficient form by letting x=n and y=0 using the binomial formula. Therefore, n^2 implies (n+0)^2 = 2C0*n^2*0^0 + 2C1*n^1*0^1 + 2C2*n^0*0^2. Hopefully this help some...
ngl this was still kind of confusing, but it really helped a huge bunch, even though this is just the math for it, without the actual proof structure. 🙏thx
Way better than my professor at the university. Honestly questions the value of higher education. Anyways thanks for this. Definitely the hardest concept in my discrete math course.
Thanks! Really helped me. My prof has a heavy accent so I have a hard time trying to make sense of what he said when he went through this. Now I know perfectly how this works! :)
he took the common denominator '2'. Reason: When taking common denominator for (K+1), he has to multiply '2' to the numerator too so the equation stays balanced. If he didn't do that then the previous step would not be equal to the current step. All he did was added fractions with unequal denominators.
we have an exam for tomorrow I know step 1 but the 2nd step induction is to hard for me and I've been diligently listening to my prof all the time. I wish there's a way for me to ace this subject and completely understand it. your video is informative though. it's getting a bit clearer now.
My math course basically tries to regurgitate all of this in like half of one lesson- I really wish they did a full lesson on it. It's only slowing me down. This is a lot (a LOT) of help though.
Congratulations. Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of TH-camTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.
The people who have to lean this, actually used and need it. But it dosent hurt you to learn it either. Not being able to sit thru this is why we never returned to the moon in this generation after Apollo, we are too lazy.
![[UNCUT] เก็บ "สจ.โต้ง" เอี่ยวมาเฟีย รู้ตัวคนบงการ ก่อนสั่งลั่นไก I คนดังนั่งเคลียร์ I 13 ธ.ค.67](http://i.ytimg.com/vi/o-J_p62D40w/mqdefault.jpg)
This guy is so clear in everything he says. Most teachers would skip most of the stuff he's explaining because they feel it's obvious. Khan never assumes that anything is obvious and that is why his videos are so easy to follow.
Facts
Amazing
teachers who explain like this are the best ones! The world would be a much easier place if only teachers wouldn't assume that we know the "obvious" things in life. What is obvious to a professor, with a Ph.D. might not be so obvious to a college kid on his first day on the topic.
Straight facts
Completely disagree. He explains the problem very well, but the concept itself is impossible to grasp using this video
After muddling through my discrete structures textbook, it is so nice to find 9 minutes and 22 seconds of clarity.
yess true!! I finally understood in under ten minutes what the teacher couldn't do in a week
You are so right. My DS text talks like a politician. It explains things and then you're really confused.
This is how all my Math professors are.
agree.
Unbelievable how confused I was vs how confused i am not now
For what it's worth, you've managed to teach in 10 minutes what most teachers cannot do in an hour.
I just spent about an hour looking at proof by induction in an Elementary Linear Algebra book as well as some notes online from Stanford but both of those sources were a million miles away from this level of intuition! Khan Academy to the rescue! Thank you good sir, very clear, understandable and intuitive.
So, to prove by induction that an equation is true for all inputs:
1. Check that it is true for the first input.
2. Write the equation, and incorporate (k+1) to both sides by following the pattern of the left-side part.
For example: In the video, since the left-side part was a series of additions, (k+1) (the next number in the series of additions) was incorporated by ADDING it to both sides.
3. Transform the right-side of the above equation into the form of the original right-side.
For example: In the video, the original right-side is a fraction whose numerator is the input * (the input + 1), & the denominator is 2; that's its form. And the proof merely consisted in adding (k+1) to both sides and transforming the right side into the same form of the original right side (a fraction whose numerator is the input [which in this case is k+1 instead of just n] * (the input + 1), and whose numerator is 2).
THANK YOU FOR: the colors, the explanation, making me feel better, being great at what you do.
That's what teaching should be about: making people feel confident about a subject, from the lowest level up!
As a sixth grader whose already learned some trigonometry and calculus from Khan Academy, I would agree.
@@user-cv8xu2yk7mhahahaha cap
Yeah, talk about great..
buddy you are god's gift
IKR!!! This, just kept me from failing math class.
@@zeyres4029 you're his mother?
proving something by mathematical induction isnt that difficult, its the question my professor assigns, he has us proving these ridiculously long sums that requires so much algebraic manipulation that it just makes the problem extremely difficult
It has been 13 years since the video was posted, but the value it brings to new generations like me is legendary and immortal. Thank you!!
I remember the story of how this formula was made.
The creator of the proof was causing problems in class and his teacher told him that he had to find the sum of every number between 1 and 100. He started to notice a pattern when he added 1 and 100, 2 and 99, 3 and 98... seeing that he is getting 101, 50 times. So he showed it to his teacher saying the answer was 5050. The teacher didn't believe him and wrote out the whole problem and her results came out equaling 5050.
fell off the wagon on my zoom course, muted the presentation, watched this at 1,5x speed and I was up to speed. Great video!
Thank you soo much Khan! Everybody thinks they know how to explain this but they rarely tie it up so that it makes complete sense.
the factoring out k+1 got me weak
think "FOIL"
I think he's saying 'it's got him weak at the knees' as in, he enjoyed it.
Factoring is your friend. 😊
*bruh your comment got me weak haha*
check out factoring my grouping
Thanks again Khan, I can watch your video and understand it. You're making the world a better place. You would not believe how much better this is than my class.
Wow he is way clearer than my teacher.
It's amazing.
Lol
omg this is so much better than my lectures, i question why i pay so much money to go to lectures where i get very lost. my math proofs prof sucks balls i don't understand shit when she explains but everything is so clear now that this guy explains it
you can also visit the official website (www.khanacademy.org/) to get the full list of subjects!!!!!!!!!
4 years later and ur comment is still relevant
@@mumsazpatel9759 6 years later and the comment is still relevant.
That reveal at the end blew my mind. I didnt even realize that he had exactly rewritten the original formula.
After multiple fruitless attempts to understand this concept, I finally get it. Thanks 🙌
You explained this better in 9 and a half minutes than my teacher did in 3 days. Mr. Khan, i love u bro.
11k views and no dislikes, a testament to your greatness khan :) fantastic tutorial ... this looks extremely tricky to learn from a text book o_o
watching this 10 years after you, with 1 million views. the impact that one 9 minute video has had is crazy
WOW!!! This is definitely something else. The examples are always easier than the task. We're having a test today and this is killing me.
Dude. This video is about 7 years old but IT IS GOLD!!!! Thank you so much!!
12 years
I should just pay you instead of paying for college courses. U explain everything perfectly.
As a sophomore in a college algebra class this is a godsend.
how's life now
This video just solved all my doubts. Always grateful for your videos.
So well explained, and i like the fact you say things multiple times! Helps it stick in my head. thank you very much.
Oh god ! I was totally intrigued about this topic. Another people were just teaching me how to solve problems based on it. None of them teaching me how it works. Hats off⚡⚡
In my introduction to higher mathematics class, MTH 311, I can pay attention for like the first 30 mins. Then the next 30 mins I'm either staring at him write a proof on the board while thinking about what I want to get from the vending machine when class is over, or how I'm gonna even attempt to write anything on the next assessment, or anime. Or sleeping. The the last twenty mins we take an assessment where we have to write a proof on what we learned that day and two days ago.
I don't know how they do it.
I go into a video confused as shit,
5 minutes in it clicks
after the video i know it like the back of my hand.
Love it!
This is extremely useful in computer science and electrical engineering
This reminded me of the good old days back in high school when I was the straight-A kid in the class. Didn't think I would forget proof by induction someday, and didn't think I'd need it in software engineering (Automata theory course). Thanks a lot
As far as TH-cam math tutoring videos go, not rewinding once, like watching this video: practically impossible.
I wanna cry from the moment of understanding and clarity this video gave me after spending hours trying to understand induction from my discrete math textbook
That was clearest possible way to teach that.
Thanks a lot!
You're saving the lives of everyone stuck with terrible professors!
Thank you!
hows life now
Thank you for this. I love you very much.
I had an epiphany of understanding watching this, this was really helpful!
You have to substitute K+1 for all values of 'n' so it will be K+1[(K+1)+1]/2 = K+1(K+2)/2
Reason for doing this is because 'n' represents all positive integers.
The reason why you can't substitute 'n' in place of 'k' is because 'n' represents any positive integer while 'k' represents a specific unknown integer. when one is added to 'k' then a new integer is formed 'k+1'.
Therefore K+1 = n when P(K+1) is a function for n.
I am so lost in my discrete math course but I think I’m finally understanding thanks to this video, thank you so much
So simple yet effective. Absolutely loved the video!!
You've made it so easy
OMG Nice explaination !
I just had the "oh my god it's clicking" moment that every student studying mathematics and science strives for. Thank you so much for this!
Thank you. This explained proof by induction to me with the same example as my professor but 1000x easier to understand. No steps were skipped. Again, thank you
Definitely better than my school teacher...
Now i understand it so well
Thanx a lot khan :)
thank you very much! i have been struggling with this for the past 2 weeks now I think I get it!
The binomial formula is just (x+y)^m = SUM k=0 to m; mCk*x^m*y^k. I'm not 100% sure about this but you must get it into the binomial coefficient form by letting x=n and y=0 using the binomial formula. Therefore, n^2 implies (n+0)^2 = 2C0*n^2*0^0 + 2C1*n^1*0^1 + 2C2*n^0*0^2. Hopefully this help some...
really helped thanks
ngl this was still kind of confusing, but it really helped a huge bunch, even though this is just the math for it, without the actual proof structure. 🙏thx
Thanks a lot really helped me with pre calculus!
Way better than my professor at the university. Honestly questions the value of higher education. Anyways thanks for this. Definitely the hardest concept in my discrete math course.
Thank you, I had been looking for a vid on mathematical induction for quite a while :)
Thanks! Really helped me. My prof has a heavy accent so I have a hard time trying to make sense of what he said when he went through this. Now I know perfectly how this works! :)
I FINALLY understand. I should just watch your videos rather than go to my math lectures.
I was first grade when he uploaded this video n now I’m in my first year college watching his videos 🤭
kahn has once again saved my life (and by life I mean my test grade)
Its Khan.. If someone saves your life.. spell their name right :)
Very helpful. Made way more sense than my lectures
he took the common denominator '2'.
Reason: When taking common denominator for (K+1), he has to multiply '2' to the numerator too so the equation stays balanced.
If he didn't do that then the previous step would not be equal to the current step.
All he did was added fractions with unequal denominators.
Khan academy is the best Salman Khan is doing really good thing by providing free education online for everyone
we have an exam for tomorrow I know step 1 but the 2nd step induction is to hard for me and I've been diligently listening to my prof all the time. I wish there's a way for me to ace this subject and completely understand it. your video is informative though. it's getting a bit clearer now.
That very last step, fucked my mind up man! :(
2024🙌 thank you for this video
This genuinely makes me happy.
Thank you so much, everything is so clear now!
OMG YOU SOMEHOW MADE IT CLICK FOR ME YOU ABSOLUTE LEGEND
Excellent video👏🏾
Man it's like your reading right out of my Pre-Calc book.. This is amazing
thank you khan academy for teaching me more than my prof's
brilliant. I love you
man, 2 hours with my math teacher, thanks for the help!
Really fantastic, thank you!!
How did you get the formula of n(n+1) /2 ,
Cuz in my exam formula wasn't given. Is it question?
Thx helped me study for my test
Love your content. amazing job my mam. keep up the good work!
please add correct English subtitle to your videos, it's helpful for non-native english speakers like me, thanks
Khan academy pre-calc lectures + a pre-calc textbook = yet another grade of math that I will skip
"true for two, true for three, true for four". Now THATS a tongue twister.
I'm now a 2nd Year Secondary Education Student Major in Mathematics and it is only by now that I've understood this topic well....
Oh my I finally understand this clearly !!
How can you tell if you can use induction
This is great, really helped me, thanks khanacademy!!
My math course basically tries to regurgitate all of this in like half of one lesson- I really wish they did a full lesson on it. It's only slowing me down.
This is a lot (a LOT) of help though.
THANK YOU SO MUCH YOU SAVED ME THANK YOU
thx , this is how it is supposed to be explained
watching this in my math class right now with ms. Tran OMG
Finally I have understood the concept.Thank you Khan!
It has been years since I have had to work on proofs. This makes it much easier to keep up my college math skills. Thanks!
Good explanation, I totally got it.
Thanks
lovely video, the pretty colours were very helpful
much better than my lecturer
Please do a video on strong induction
THISSSSSSSSSSS
Congratulations.
Notification: I would like to inform you that I plan to include this video among the videos reviewed in my article "Analysis of TH-camTM Videos and Video Comments on Mathematical Proof Methods". Kind regards.
I like the Alternative proof better, it is more intuitive.
I wish I had teachers of this calibre at school.
The people who have to lean this, actually used and need it. But it dosent hurt you to learn it either. Not being able to sit thru this is why we never returned to the moon in this generation after Apollo, we are too lazy.
Im from the UK as well, but I studied it last year, in Year 12
AWESOME! Your handwriting is very neat! :D thanks to this video, I finally understand! Ty! You make it sound so simple(:
Thanks Khan!
You the man, Khan!
You did not mention why we should use 'proof by induction' in the first place? In which situations can we use this?
Statistics. It's how they get a standard number for probability or a constant number for algorithms.