Not only did I learn a more efficient method of expanding, it also made me wonder what Pascal's triangle is and search for it. excellent channel and video
Liked and subscribed! Thanks - an adult relearning math as a hobby who found your videos thanks to a recommendation from the YT algorithm on my home page.
As soon as we started calculus in Engineering maths i had no idea what was going on because i was never taught pascal's triangle (which was the method the teacher was using). This really helped. Cheers mate.
oh my gosh, this s very easy when you started i didn't understand, but later i understood. thanks very much. God bless you for showing us your skills.we will subscribe and watch more.
I accidentally found your website recently. Your method of teaching math is phenomenal. I wished I had you as a math teacher when in High School. Thank you so much.
Fantastic! I didn’t do this on my maths curriculum back in the day but was a 1st maths student and still enjoy it! This was perfect to take my panicking 16 year old through this topic. It was also a great opportunity to teach her the absolute joys of a scientific calculator. Thank you!
You're just superb. It's the nonchalant manner in which you present your explanation that puts the icing on the cake Being a mathematical genius is an obvious quality of yours. And the simplicity is just superb. Cheers ..
The coefficients found from Pascal's Triangle perfectly match the coefficients found from a binomial-theorem expansion (whenever the exponent is a positive integer)
is there a faster way to do this by any method like factoring or something else? This was good to know but idk if its good to do it this way in an exam where we only have like an hour and a half to finish 100+ questions Thank you for the answer in advance sir!
Yes. You just let beta = -b. Expand (a + beta)^5, exactly as you would with this method. Assuming a is positive: Every odd exponent on beta, will materialize as a negative term when translated back to b. Every even exponent on beta, will materialize as a positive term when translated back to b.
This guy has been helping me so much but i just got a question where theres only one letter thingy in the parentheses and the other is just a number (d+2)to the power of 8
You can put a letter in that number's place. For your example, let t = 2. Thus: (d + t)^8 = d^8 + 8*d^7*t + 28*d^6*t^2 + 56*d^5*t^3 + 70*d^4*t^4 + 56*d^3*t^5 + 28*d^2*t^6 + 8*d*t^7 + t^8 8th row of Pascal's triangle for reference: 1, 8, 28, 56, 70, 56, 28, 8, 1 Now evaluate each t-power at t=2, and have it join the coefficient. Put the d-power aside for later. 8*t = 16 28*t^2 = 28*4 = 112 56*t^3 = 56*8 = 448 70*t^4 = 70*16 = 1120 56*t^5 = 56*32 = 1792 28*t^6 = 28*64 = 1792 8*t^7 = 8*128 = 1024 t^8 = 256 Thus, the result is: d^8 + 16*d^7 + 112*d^6 + 448*d^5 + 1120*d^4 + 1792*d^3 + 1792*d^2 + 1024*d + 256
Request on Convergence vs. Divergence please ? I have to go over it but it will be a bit before I get to it as I'm doing all of Calc I again. Though I 4.0 the class still. Currently 1/2 way through online. I never like being rushed in College. Math is like a fine wine and should be enjoyed ! Though not many would agree with me on that. 🤣
I take it you are talking about convergence and divergence, of an infinite series, as there are other meanings of those words. A sequence (or progression) is a list of numbers, a series is a summation of a sequence. Usually k represents the index of the individual term, and n represents the total number of terms. There is usually a function of k that determines the sequence value, although a sequence can also be recursively defined like the Fibonacci sequence. Some series add up to a finite value, when you have an infinite number of terms, which is called a series that converges, or convergent. Other series either add up to infinity, or continuously flip/flop between the same two numbers and never get closer to zero or to infinity, which is called a divergent series. Some series include an alternator term, of (-1)^k. If an alternating series only converges when it is alternating, it's called conditionally convergent, for instance the harmonic series of 1/k. If a series converges, regardless of whether it alternates, it's called absolutely convergent. The substance of Calculus's work with series, is to analytically determine whether or not it converges, and if so, what value it converges to. There are various tests for convergence, based on properties of the series definition, that may use limits, powers, roots, and integrals, to test which kind of convergence a series has.
Here's an example with a negative: Suppose it starts as (u - v)^3, where u and v are both positive numbers. Let w = -v. Now we've reduced it to (u + w)^3. Expand it out accordingly, with u's power descending from 3 to 0, and w's power ascending from 0 to 3. Coefficients follow 3rd row of Pascal's triangle, which is 1, 3, 3, 1. u^3 + 3*u^2*w + 3*u*w^2 + w^3 Since squared negatives are positive, this means all terms with w to an even power, will equal the same term with v replacing w. The opposite is true for cubed negatives. All terms with w to an odd power, will have a negative coefficient, and otherwise equal the same term. This gives us the result: u^3 - 3*u^2*v + 3*u*v^2 - v^3
![A-Level Maths: D1-01 [Binomial Expansion: Introducing and Linking Pascal’s Triangle and nCr]](http://i.ytimg.com/vi/Bw-9_Rq2bp4/mqdefault.jpg)
If you are watching this video, then you are probably making the best use of RUclips.
Yes!😅
Fully agree.
the hardest thing is to remember this when you don't use it regularly
Hehehe 🎉
"best use" academically speaking. This teacher is Awesome. But, There are other Great things here on RUclips.
I recently found Mr H by chance and I've found his videos clear and easy to understand. Thank you, Mr H.
Thank you! 😊
if u found it easy then ur prolly dumb
Same Thank you very much I would fail math without your videos
Sir , iam 51 years old engineer and mathematician.You are one of the best teacher i saw in my life. Thank you very much
Wow, thanks. I appreciate your comment!
It's true
When your students who does not enjoy maths is inspired by your teaching you know you're a great teacher! Thank you so much!
the fact that an 8 minute video teaches me more than the 2hrs of lesson from my prof lol
i never actually learned pascal's triangle in school. thanks for teaching me!
Happy to help!
@@mrhtutoringsir I follow you but some things i don't understood
I didn’t learn it until college
I was hoping to see a cat...
Never knew this. Thank you for the topic
I didn't know Pascal's triangle in my school. Now only, I learnt it, because of your help. Thank you very much.
Thanks for reminding me of Pascals triangle I leant 35yrs ago. You are awesome.
Literally saved my life. So grateful for teachers that go out of their way to make these videos, thank you!
Not only did I learn a more efficient method of expanding, it also made me wonder what Pascal's triangle is and search for it. excellent channel and video
Glad you are back after 2 years sir!!
Please continue!! We love your work!!
I will try! Thank you
Liked and subscribed! Thanks - an adult relearning math as a hobby who found your videos thanks to a recommendation from the YT algorithm on my home page.
Thanks for subscribing!
As soon as we started calculus in Engineering maths i had no idea what was going on because i was never taught pascal's triangle (which was the method the teacher was using). This really helped. Cheers mate.
Very nice reminder of Pascal's Triangle. Let me add: The Exponent of term to be expanded (call it 'n' ) results in 'n+1' Terms.
Thank you for the additional information
oh my gosh, this s very easy when you started i didn't understand, but later i understood. thanks very much. God bless you for showing us your skills.we will subscribe and watch more.
Way better than my teacher explained it. THANKYOU!
You're most welcome
Thisnis best and most concise explantiom inhave seen of this. Learned something today.
I accidentally found your website recently. Your method of teaching math is phenomenal. I wished I had you as a math teacher when in High School. Thank you so much.
Wow, thank you!
He's very straightforward in his explanations. Nice video.
Fantastic! I didn’t do this on my maths curriculum back in the day but was a 1st maths student and still enjoy it! This was perfect to take my panicking 16 year old through this topic. It was also a great opportunity to teach her the absolute joys of a scientific calculator. Thank you!
Oh and if it helps
I did up to N12 for her to see the pattern. We laminated it along with theorem and applications!
last-minute revision for my GCSE, thanks Mr H, you're a lifesaver
Glad it helped.
Brilliant Method! I'm going to use this technique as much as I can.
Fantastic!
They say you can learn stuff from youtube, but would it be possible to do so, if we didn’t a teacher like Mr H ? My deepest gratitude to you sir.
excellent sir
i love your teaching
you are such a great teacher.. bless me sir
This is now my favourite video from this channel.
Excellent! Mathematics is beautiful!😊
You're just superb. It's the nonchalant manner in which you present your explanation that puts the icing on the cake
Being a mathematical genius is an obvious quality of yours. And the simplicity is just superb. Cheers ..
Thank you so much 😀
I'm going to subscribed this channel because I actually learned something today. Thank you sir!
thank you for the breakdown, didn't think i was going to get this.
No problem!
This clears up so much that I didn’t know. Thanks you!
Awesomess. At school, they just made us memorize the formula. This is beautiful !
Excellent video, Sir!! All my respect
Wow, I did'nt notice that the sum of the exponents led me to the exponent of the binomial, in this case five. Thanks professor!
You're very welcome!
I am very excited about this teacher
I'm a maths teacher and you're the best!
Thank for the awesome comment.
I appreciate it
This is just wow. Thank you very much sir for making it look so simple.
Wow !easy way! thank you for your time sir🙏
I was never good enough at math to do math involving PT in high school, but I still enjoyed this video and understand the concept better now..
Absolutely beautiful. Thank you so much
Great overlay! And explanation :D
Never learnt this before. This is so coooool.
Helpful for me, now i can easily binomial theorem class 11
Absolutely amazing property, thank you sir!!
Happy to see someone who appreciates math.
I can only understand by your action and writing. Thank you sir 🙏🙏🙏
Thank you sir this is amazing! May Allah bless you and grant you a healthy and a prosperous life.
I'm italian and we call it "Triangle of Tartaglia" because Tartaglia was an important mathematician who found this before Pascal.
thank you u saved me my final exam is tomorrow thhhhaaaanksssss
Good luck on your final!
I just thank you very much
Didn't learn this school. Thanks a lot for this learning👍
Happy to help
Your a master sir thank you for teaching sir ❤🎉
It's my pleasure
BRO I THESE 5 MINUTES YOU MADE ME UNDERSTAND THE WHOLE THING WHICH I THOUGHT I WILL FAIL TOMORROW IN TEST
I have an exam today and part of it uses this, did not understand till now so thank you
The coefficients found from Pascal's Triangle perfectly match the coefficients found from a binomial-theorem expansion (whenever the exponent is a positive integer)
You made me understand pascal's triangle thank you so much sir
Tnx , ur video helped me a lot , keep making this videos so that maths can be a child's play
thanks to you, Mr. H, i just might not fail this course.
Brilliant and to-the-point.
Thank you so much I needed this! 🙏
You saved my life ,finally.
You are excellent sri
you're awesome! thank you very much! it would be nice if you did an expansion if both terms were subtracting instead of adding... helped a lot!
simply brilliant
Sir was finding it for a lot of time thanks for this upload
Most welcome
Though I will admit my dream of a good set of notes came true. Having the online stuff now. Great reference.
WOW why no body teach us like this ?????????? great man
Excellent explanation. Thank you.
Thanks King, you made me a little less cooked.
is there a faster way to do this by any method like factoring or something else? This was good to know but idk if its good to do it this way in an exam where we only have like an hour and a half to finish 100+ questions
Thank you for the answer in advance sir!
I wish you the best in life, as well as your endeavors Mr. H
Thank you~
math equivalent of indian guy tech tips, life saver right here :D
wonderful! THANK YOU!
Welcome!
Wow! I enjoyed this 😘
Excellent teaching
Thanks for the comments! I appreciate them.
Are you releasing a video on the binomial theorem?
You are a great teacher.
Agree
If it is (a - b)^n do we use alternating addition and subtraction? E.g. a^3 - 3a^2b + 3ab^2 - b^3 for (a - b)^3
Yes~
thanks man i am looking for this video
Can you do subtraction with powers on both values
Why are the coefficients separated from the variable?
Keep it up
Thank you Sir🙏
Amazing! Thanks a lot!
Thank you too!
Does with work with (a-b)^5 - or is there something similar?
Yes. You just let beta = -b.
Expand (a + beta)^5, exactly as you would with this method.
Assuming a is positive:
Every odd exponent on beta, will materialize as a negative term when translated back to b.
Every even exponent on beta, will materialize as a positive term when translated back to b.
If you want to learn more about this search binomial theorem
This is way more interesting and easy to remember.
U just inspired me to open my own RUclips channel
Music to my ears!
This is amazing!!
Great Sirji
Do video for derivative(also functions) please, i need it as soon as possible, thank you
Thank you for this video
excelente, well explained, than you
This guy has been helping me so much but i just got a question where theres only one letter thingy in the parentheses and the other is just a number (d+2)to the power of 8
You can put a letter in that number's place. For your example, let t = 2. Thus:
(d + t)^8 =
d^8 + 8*d^7*t + 28*d^6*t^2 + 56*d^5*t^3 + 70*d^4*t^4 + 56*d^3*t^5 + 28*d^2*t^6 + 8*d*t^7 + t^8
8th row of Pascal's triangle for reference:
1, 8, 28, 56, 70, 56, 28, 8, 1
Now evaluate each t-power at t=2, and have it join the coefficient. Put the d-power aside for later.
8*t = 16
28*t^2 = 28*4 = 112
56*t^3 = 56*8 = 448
70*t^4 = 70*16 = 1120
56*t^5 = 56*32 = 1792
28*t^6 = 28*64 = 1792
8*t^7 = 8*128 = 1024
t^8 = 256
Thus, the result is:
d^8 + 16*d^7 + 112*d^6 + 448*d^5 + 1120*d^4 + 1792*d^3 + 1792*d^2 + 1024*d + 256
well explained. Thanks
Incredible!!!!
How do you make these videos?
Thank you.
Request on Convergence vs. Divergence please ? I have to go over it but it will be a bit before I get to it as I'm doing all of Calc I again. Though I 4.0 the class still. Currently 1/2 way through online. I never like being rushed in College. Math is like a fine wine and should be enjoyed ! Though not many would agree with me on that. 🤣
I take it you are talking about convergence and divergence, of an infinite series, as there are other meanings of those words.
A sequence (or progression) is a list of numbers, a series is a summation of a sequence. Usually k represents the index of the individual term, and n represents the total number of terms. There is usually a function of k that determines the sequence value, although a sequence can also be recursively defined like the Fibonacci sequence.
Some series add up to a finite value, when you have an infinite number of terms, which is called a series that converges, or convergent. Other series either add up to infinity, or continuously flip/flop between the same two numbers and never get closer to zero or to infinity, which is called a divergent series. Some series include an alternator term, of (-1)^k. If an alternating series only converges when it is alternating, it's called conditionally convergent, for instance the harmonic series of 1/k. If a series converges, regardless of whether it alternates, it's called absolutely convergent.
The substance of Calculus's work with series, is to analytically determine whether or not it converges, and if so, what value it converges to. There are various tests for convergence, based on properties of the series definition, that may use limits, powers, roots, and integrals, to test which kind of convergence a series has.
Where is the binomial theorem video??
Nose bleeding again sir😂thank u again❤❤❤
🤣🤣ofc its this guy teaching math.
No hate this it helped me a lot😈💞💞
Brilliant
What if negative
Here's an example with a negative:
Suppose it starts as (u - v)^3, where u and v are both positive numbers.
Let w = -v. Now we've reduced it to (u + w)^3. Expand it out accordingly, with u's power descending from 3 to 0, and w's power ascending from 0 to 3. Coefficients follow 3rd row of Pascal's triangle, which is 1, 3, 3, 1.
u^3 + 3*u^2*w + 3*u*w^2 + w^3
Since squared negatives are positive, this means all terms with w to an even power, will equal the same term with v replacing w.
The opposite is true for cubed negatives. All terms with w to an odd power, will have a negative coefficient, and otherwise equal the same term.
This gives us the result:
u^3 - 3*u^2*v + 3*u*v^2 - v^3
Now i can aso be one of the toppers😊