I just started studying math at university and I understand what people mean when they say school math teachers tend to be bad - my math teacher just throws out formulas on the board and tells us to memorize them, that's lame, you don't learn anything just from memorizing technical formulas. That's why Sal is so awesome, because he really stresses the importance in getting intuition for things.
Thank you so much! This was very helpful! This helps me with Binomial "Expansion" with is essentially the same thing...isn't it? At least to expand the binomial but not solve it. Sorry if I sound like I confused myself ^^;
Thanks a lot, that's really seems great! I have never seen that at school, I have just memorized the formulas for (a+b)^2 and ^3. Although I still don't really understand WHAT are we choosing when we write "n choose k" in this formula.
your a LIFE SAVER!!! i have a test tomorrow.. but i missed my first binomial theorem class with my teacher and i have no clue about it!!! THANK YOU SOO MUCH!!! helped alot.. i will tell you how i do!! :D
@Xynsawza since both x's are to the 1st power, they will need to be equal to cancel, so (5X)^10 x (4/X)^10 Which will leave you with 5^10/4^10. It is the 11th term.
haha well I'm taking calculus two and while studying taylor and maclaurin series, I came across a series for (1+x)^k and it made a reference to this theorem. Never in my life had I seen such a thing as "n choose k" where does it come from?
True but the formula for combinations comes from the binomial theorem, then u multiply out the n! to get permutations, then the n! is the factorial if all sets r different or same
dude thanks for uploading this video,you solved my problem with this questions,i would like you to upload more videos about the "TRIGNOMETRIC FUNCTIONS".
It is! Imagine, if you will, (b+a)^4 -- The result would be the terms to the right of the a^2b^2 followed by a^2b^2 and closed by the terms that were originally before a^2b^2.
Actually yes. Look at it this way: say a and b are reals. Then assume that (a+b)^n is not equal to (b+a)^n, in other words: you don't need to get symmetry. Now say a+b = q, then b+a = q, therefore q^n = q^n, therefore (a+b)^n = (b+a)^n which is contrary to the assumption made. Therefore the outcome must be symmetrical. Q.E.D.
awesome... you just cut my discrete mathematics studying time by quite a bit with this video! -.- cause my really prof. sucks soo much in which he can;t even speak proper English... -.-
thank you very much for this but there is a problem at 5:59. instead of using X u should have used a and instead of using y u should have used b. just thought ide let u know
Thanks :) And I'll try to explain it. First of all, the logical explanation. n! is the amount of ways you can arrange n things. How many ways can you arrange nothing? One way. The only problem is that this is kind of ''0 / 0 = 1'' logic. You could also say it is undefined as arranging depends on ''an object''. This also comes back into the definition of n! and the formal explanation I will give in the next comment as I am running out of space.
hm. i think u should do using the "pattern" in binomial theorem instead of formulas so that ppl can remember that much easier. and when doing the expansion 1st do it vertically,then write the expansion horizontally finally. And for nC2 = [n(n-1)]/2!(remember by nC2 having 2 "n" , nC3 = [n(n-1)(n-2)]/3!(remember by nC3 having 3 "n",etc) would be easier. others u can just use calculators for nCr terms. it's only my opinion. thanks
i'm used to a different notation of combinations relying on arangements "combinations of n taken k times = arangements of n taken k times / k factorial " so for example, C of 5 taken 3 times = A of 5 taken 3 times / 3! = (5*4*3)/6 = 10 it's written as big C then a subscript 5 and superscript 3
n! = n * (n-1) * ... * 1 FOR n >= 1. So do not try to insert it in the regular equation. But what we can do is define 0! seperately to ''make equations work''. There does similarly not to be a ''proof'' 1 is not a prime number; mathematicians chose it to make the theorem of arithmetic work. n! = n * (n-1)! for n >= 1 follows from the original definition. So 1! = 1 * 0! 1! must be 1 as you can clearly say there is ONE way to arrange ONE object. Therefore 0! =1. To make equations like this work.
UNRELATED QUESTION - what do i do if its (2x^2)^2 ? i'm confused about the order of operations..do i multiply the two powers first and then do 2^4 =16x^2?
I just started studying math at university and I understand what people mean when they say school math teachers tend to be bad - my math teacher just throws out formulas on the board and tells us to memorize them, that's lame, you don't learn anything just from memorizing technical formulas.
That's why Sal is so awesome, because he really stresses the importance in getting intuition for things.
I am VERY VERY thankful to the one who wrote Turkish subtitles for this. Our education system is shit. Thanks for the video ofcourse!
your welcome..even though i did not write the subtitles. :)
and yeah......everyones education system is shit! its not just you :D
I don't care if this video took another 12 minutes, you were a lot of help! Thanks
One of great teacher which i study that is just you my wizard:))))))))
This video deserves more likes.
I didn't realize you could do (a+b)^3 by just multiplying the result of (a+b)^2 by (a+b). That really helped, thanks a lot.
I've been out of school for almost a year now. Left the beginning of the second semester of algebra 2. It's amazing how much I've forgotten.
Thank you so much! This was very helpful! This helps me with Binomial "Expansion" with is essentially the same thing...isn't it? At least to expand the binomial but not solve it. Sorry if I sound like I confused myself ^^;
You can use the Pascal's Triangle to make it easier rather than multiplying a lot.
Yeah, but I think that is taught later on
Thanks a lot, that's really seems great! I have never seen that at school, I have just memorized the formulas for (a+b)^2 and ^3. Although I still don't really understand WHAT are we choosing when we write "n choose k" in this formula.
Why can't all math teachers be this good?
That's just how society works
Wow, very good video. I like how you took us through the mundane part of it too. It helps appreciate the faster way.
I now understand what my asinine teacher had failed to teach. It also helps that it's being explained in such a sexy voice.
Sweet. I need this in proving the differential of x^n. (I know you did a video on it but I am trying to prove it before watching you do it.)
your a LIFE SAVER!!! i have a test tomorrow.. but i missed my first binomial theorem class with my teacher and i have no clue about it!!! THANK YOU SOO MUCH!!! helped alot.. i will tell you how i do!! :D
How u did? U didn't say about it
You didn't tell us tho
AWESOME VID(: way better than what my textbook was trying to convey!
Feels good to refresh my maths
I've never learned the binomial Theron before. What level of math do u learn this in?
Well Iam grade 11 cbse rn and this is my first time studying this sooo.......mabye in grade 11? Idk
Nice proof by contradiction! By the way, could you also explain why 0! = 1?
This is basically nCr right??
yes but without a calculator
You wouldn't believe how happy I was when I found out about button in PreCalculus! aha!
@Xynsawza since both x's are to the 1st power, they will need to be equal to cancel, so (5X)^10 x (4/X)^10
Which will leave you with 5^10/4^10. It is the 11th term.
Does this work also for negative (x-y)^10? I tried (x-y)^3 but I think I got wrong signs
Yes, it works for negative, with alternating (+) & (-)
e.g.: (a-b)³ = a³-3a²b+3ab²-b³
Please build a school in future so that my future kids can attend
You're the man.
@ACfireandiceDC exactly, is there a video on stuff like that, cause it's over my head...
THANK YOU DUDE its so much simpler than doing it with real numbers for an example so thanks very much for doing it with lettters
keep up your good work khan
@StaticShock1100 if I'm not mistaken, he made this program himself
haha well I'm taking calculus two and while studying taylor and maclaurin series, I came across a series for (1+x)^k and it made a reference to this theorem.
Never in my life had I seen such a thing as "n choose k"
where does it come from?
True but the formula for combinations comes from the binomial theorem, then u multiply out the n! to get permutations, then the n! is the factorial if all sets r different or same
what was the video before this? I need to know why you are using factorials?
is there a trinomial theorem, lol
dude thanks for uploading this video,you solved my problem with this questions,i would like you to upload more videos about the "TRIGNOMETRIC FUNCTIONS".
It is! Imagine, if you will, (b+a)^4 -- The result would be the terms to the right of the a^2b^2 followed by a^2b^2 and closed by the terms that were originally before a^2b^2.
What program do you use to do this?
my mind was blown... thats awesome
Actually yes. Look at it this way: say a and b are reals. Then assume that (a+b)^n is not equal to (b+a)^n, in other words: you don't need to get symmetry. Now say a+b = q, then b+a = q, therefore q^n = q^n, therefore (a+b)^n = (b+a)^n which is contrary to the assumption made. Therefore the outcome must be symmetrical. Q.E.D.
Sal is my math jesus everytime i have an exam the next day
that is very good computer writing
great explaination!
thank you!
awesome...
you just cut my discrete mathematics studying time by quite a bit with this video!
-.- cause my really prof. sucks soo much in which he can;t even speak proper English... -.-
Very good video, thanks so much for making me understand it better.
thank you very much for this but there is a problem at 5:59. instead of using X u should have used a and instead of using y u should have used b. just thought ide let u know
He is singing a plus b. Very good pedagogical technique
great work buddy many thanks
what about "expand (a+b+c)^3 and give the coefficient for the term corresponding to x^2 where x=a=2"
thank you so much I love it, keep going, u r doing well
do you have a video show binomial theroem by the caculator.
Thanks :) And I'll try to explain it. First of all, the logical explanation. n! is the amount of ways you can arrange n things. How many ways can you arrange nothing? One way. The only problem is that this is kind of ''0 / 0 = 1'' logic. You could also say it is undefined as arranging depends on ''an object''. This also comes back into the definition of n! and the formal explanation I will give in the next comment as I am running out of space.
hm. i think u should do using the "pattern" in binomial theorem instead of formulas so that ppl can remember that much easier. and when doing the expansion 1st do it vertically,then write the expansion horizontally finally.
And for nC2 = [n(n-1)]/2!(remember by nC2 having 2 "n" , nC3 = [n(n-1)(n-2)]/3!(remember by nC3 having 3 "n",etc) would be easier. others u can just use calculators for nCr terms. it's only my opinion. thanks
so square is not "to the power of a 2"?
Time to work at mcdonalds.
what does this have to do with sequences?
@TajikThunderstorm even I prefer that... by the way, can I know about you?
thnx healped a lot hope to pass my 11th exam but in simple words it is the process of expantion of (x+a) whole to the power n m i correct
Thanks, if you were my teacher I might actually know how to do math. :D
how do you expand (a-b)^7, what signs do you use??? please help test tommorow
wish this was recorded in 360, 480 or 720 resolution rather than 240.
great teaching
Thank you for the free lesson. Really appreciate it =]
I'm learning this in Grade 11, Standard Level Maths (age 17-18)... but I do the IB. Might be different depending on the education system
why is the quality so bad?
i had no idea it was that simple :) thanks
A number raised to the third power is indeed the same as saying the number is cubed.
i'm used to a different notation of combinations relying on arangements
"combinations of n taken k times = arangements of n taken k times / k factorial "
so for example, C of 5 taken 3 times = A of 5 taken 3 times / 3! = (5*4*3)/6 = 10
it's written as big C then a subscript 5 and superscript 3
Yo I remember this from preschool! Binomial cubes ftw
I tend to use the C function on my calculator rather than the entire Combination formula.
Thank you so much! Very simple
No it doesn't depend. k=0 is standard in the binomial theorem, as otherwise you would blow up the formula and get entirely different results.
n! = n * (n-1) * ... * 1 FOR n >= 1. So do not try to insert it in the regular equation. But what we can do is define 0! seperately to ''make equations work''. There does similarly not to be a ''proof'' 1 is not a prime number; mathematicians chose it to make the theorem of arithmetic work. n! = n * (n-1)! for n >= 1 follows from the original definition. So 1! = 1 * 0!
1! must be 1 as you can clearly say there is ONE way to arrange ONE object. Therefore 0! =1. To make equations like this work.
And I think I just proved it! :) Awesome. On to your d/dx(x^n) proof vid to make sure.
UNRELATED QUESTION - what do i do if its (2x^2)^2 ? i'm confused about the order of operations..do i multiply the two powers first and then do 2^4 =16x^2?
Ans = 4x^4 .U start with (2 Xsquared)squared . Square 2 , square x^2
how can you hate on this video of all videos?
in this case yes in the binomial theorem yes i believe. but if it were trying to solve a summation of something else then no.
great job
Thank you so much for the video.
Nope I mean the way I stated it: combinations r based on the binomial theorem
I understood everything up until the end. what in the world is a factorial? That whole part just... confused me
CopyDrawn factorial is basically the number before the factorial sign multiply arithemically until 1 e.g. 4!=4x3x2x1=24
could you not also just use pascals triangle????
well explained
indians are the god of math! my math teachers indian and hes awesome
I think my algebra exam just got saved
I want to ask can someone tell me that how to calculate the 'E' in calculator?
Love it, just a bit messy ~~
,m
I've always thought binomial theorem was too hard.
but it's easy, like sunday morning.
The quality of this is bad, looks like you're still using a chalkboard
Keep in mind this was made 6 years ago! Khan's video quality has vastly improved.
Factorials r based on permutations which r based on combinations...just google it
Thank You Khan
"slap yourself in the brain"
lol
Thank you very much,sir
the pascals triangle method is way easier
Until you're asked to find a specific term of an expanded binomial or a binomial raised to a power greater than 10
THANK YOU SIR!
11:47 I don't think that's why you get that symmetry!
5:11 best quote ever
5:11
Top geezer :D
life saver
Just saw the date it was uploaded, when the Sichuan Earthquake happened, :(
Meant 'derivative' instead of 'differential' there.
Reduced my burden ty
4:28 challenge accepted!
PLEASE COME TO MY SCHOOL AND TEACH US!! PLEASE!!! Dx