My thoughts exactly. Mathematics as it is taught in school is about precision not approximations, regardless of how cool they are. One of the reasons i hated it.
It is 102^(1/2) =(100+2)^(1/2) =[100*(1+2/100)]^(1/2) =100^(1/2)*(1+2/100)^(1/2) =10*[1+(1/2)*2/100±...] (Maclaurin series, ignore higher order terms as they tend to 0) ≈10*(1±0.01) ≈10.1
It seems needlessly complicated. Why say it's approximately equal to 10 + 2/(2*10)? The remainder "2" will always cancel out if you put it on the numerator and the denominator. Easier to say it's about 10 + 1/10.
For those who want to know, this is an application of the newton method, which allows you to find the root of a nonlinear function very quickly (Each iteration roughly doubles the amount of digits that are accurate). In this case, the function with root sqrt(102) is f(x)=(x^2) - 102. In the Newton Method, you pick a starting point x_{0} and get a better approximation of the function's root by inductively defining x_{n+1} = x_{n} - (f(x_{n})/f'(x_{n})) In this case, we apply 1 iteration with x_{0} equal to the smallest whole-numbered root smaller than sqrt(102), which gets us close enough to the real root for the newton method to substantially improve the guess. We can check that the equation used in the video is the same by plugging in the values: x_{1} = x_{0} - (f(x_{0})/f'(x_{0})) = 10 - (-2/2*10) = 10.1 The newton method works pretty generally with differentiable functions (careful: its bad if the root is also a stationary point, it can be adjusted to rectify this flaw though) because the geometric intuition is that it puts a tangent line on the starting point and computes the intersection with the zero line, which is really close to the functions intersection with the zero line (its root) if the starting point is already close to the root of the function. See Newton Method on wikipedia for a nice graphic showing this concept.
@@telemans107 yes, you can look at this using just first digit Taylor approximation: the important thing is that the error of the Taylor series gets small very quickly in almost all cases when the current value is close to the root of the function, which allows this method to converge.
You should try an advanced mathematic course, theres ton of numerical estimations for unsolveable equations. Its called numerical estimation. But usually you can increase number of iterations and can get an accuracy as accurate as possible, down to 10**(-(15+)) if you want.
taylor series is a generalisation. If you take the constant and linear term of taylor series, it is linear approximating. All this video uses is f(x+h) = f(x) + h* f '(x) where f'(x) = 1/(2sqrt(x)) , x = 100 and h = 2
@@thailandler1110 Here's how it works: Given y = sqrt(x) Approximate this function near the nearest perfect square, at x=100, with a tangent line that touches it and locally matches its direction. The point slope form of a line is: y = m*(x - x0) + y0 x0 = 100, and y0 = 10. All that's left to find is m, the slope of the tangent line, dy/dx. y = sqrt(x) = x^(1/2) If y=x^n, then dy/dx = n*x^(n - 1), which means: dy/dx = 1/2*x^(1/2 - 1), which can be rewritten as: dy/dx = sqrt(x)/(2*x) Evaluate this at x=100: m = dy/dx = sqrt(100)/(2*100) = 10/200 = 1/20 Thus, the tangent line is: y = 1/20*(x - 100) + 10 Plug in x=102: y = 1/20*(102 - 100) + 10 Result: y = 10.1
It should be noted that the 2 in the denominator is NOT the same 2 as in the numerator. The denominator 2 is a constant and is always used regardless of the remainder from the number to be square rooted and the closest perfect square.
Or approximation using first principle of derivative. f(x+h) = f(x) + f'(x)/h (h is very small) Also (sqrt(x))' = 1/(2*sqrt(x)) That's why there's "2".
It is also horribly unprecise and there is no need to actually use it. You can actually do it from both sides: sqrt(102)=sqrt(100+02)=sqrt(10*10+02) ~10+02/(2*10)=10+2/20=10,1 sqrt(102)=sqrt(121- 19)=sqrt(11*11- 19) ~11 -19/(2*11)=11-19/22~10.136 The further you are away from the the actual squared number (100 or 121) the wronger it gets. 102 is really close to 100, which is why the result is somewhat good. You could technically also do something stupid like picking a square that is not adjacent and get something horribly wrong (: sqrt(102)=sqrt(49+53)=sqrt(7*7+53) ~7+53/(2*7)=10+53/14~10,79 I approcimated 102 with 49, which is 7*7 (:
Wow I've been searching for an easy method to find square root but no one said like this.......this is very very easy..! I tried for √109 and I got 10.45 as approximate value and the correct value is 10.4403...... I'm going to send this to all my frnds who r struggling Thank you soooo much ❤❤
for anyone wondering this is actually the beginning of the Taylor series of the square root function (around the closest perfect square, 100 here). The next term in the sum (for more accuracy) would be -(x-a)²/(8a sqrt(a)) with "a" being the perfect square. Here (x=102, a=100) it would give 10 + 2/(2*10) - 2²/(8*100*10)=10.0995 (exact value)
what is a and x in your formula?If a=2 then it would be in your formula: 2²/(8x2x2²).So if you say a=10 then in your formula:10²/(8x10x10²). This is what I understood from your formula.Can you make it clear please?
@@copculerkral1157 The number of which you take the square root is x and the closest approximation you use is a. So in this case x = 102 and a = 100. The sum of the first two terms in the Taylor Series is f(a) + f'(a)/1! * (x-a). Since f(x) is sqrt(x) in our case, you get the result.
But this does none of that. It's a set of instructions to get an approximate answer. To show different ways of thinking about numbers, he would need to show why it works - or better yet, walk the viewer through the thought process for figuring out the trick.
@@adityakamat9856 You just dropped jargon. Doing that does not help anyone "to learn a different way of thinking about numbers". Also, the fact that you thought you were telling me something I didn't already know tells me that you did not understand my point.
I like how passionate and smart teachers like you say "Pretty darn close" or "close enough" while teachers who just reads and gives an example are often the one's that says everything has to be precise. Kind of crazy lmao
I’m fifty years old and through out my years I’ve had trouble with math. I’ve had several teachers over the few years I was in school and they all seemed to rush through the process. I really enjoyed this video and will be watching more. Thanks for your sharing and time.
For anyone who are interested in this algorithm, this an application of newtons method iterated once. Netwons method is a way to approximate a value x such that f(x) = 0. In order to make such a function we make it be 0 when x is correct: f(x) = n-x^2 x ≈ guess - f(guess)/f'(guess) Repeat with the new guess being the current x. This works for many other cases as well, i.e. third root: f(x) = n - x^3 x ≈ guess - (n-guess^3)/(-3*guess^2) A little more complicated, but still highly doable.
They used this trick in the first quake game I believe. But they also used other tricky bit wise logic as well. This was used to normalize a vector I think for light calculation.
@@jo54763 I don't get why we'd use this method? I'd just find the largest root numbers of each then simplify it until I get the smallest surd. Kinda useless in my opinion.
Is this another way of saying: Without my calculator I feel helpless? In retrospect I wonder how we people born up to the early 1960s (so joining the school system up to 1970 - as electronic calculators dropped into an affordable range during the first half of the 1970 decade only) could cope with calculating anything beyond simple addition, subtraction and multiplication (say up to 20×20).
I worked at an inner-city school where the math teacher didn't believe the kids could learn (of course they could). He gave them calculators for the simplest things, thus making sure they DIDN'T learn. That's what you're doing to yourself.
@@arcanine_enjoyer I don't even like math that much, but seeing how he wrote root(102) = root(100) was so painful to watch, it really made my brain hurt for a bit.
You might as well say it's 0 because it's only approximation. The point is that without saying what's the upper bound on error of your estimate, it's meaningless.
@@Ennar But its not completely meaningless because there are no applications of estimating a square root where an upper bound on the error actually matters. Estimating a square root is for casual conversation and super quick rough working. For any application where you need some level of confidence in the answer you can just use a calculator.
@@callanc3925 I'm sorry, but every single thing you wrote is *completely wrong*. What OP wrote is that they are confident that the correct answer is somewhere between 10 and 11 which is estimating bound on the error. What I wanted to point out that they seem to completely ignore the importance of this by saying "it's only an approximation".
you can tell by the way the chalk glides with ease across the board that this man has put many many sticks of chalk through the ringer. we're talking *holding with your nails* levels of point precision. very talented.
Can’t teach a “why” into a 1minutes video. If u want to know a “why”. Search up “taylor series” and see how is it going. The reason most low level math class in middle school etc doesn’t teach into the detail is because the detail is so complicate and unnecessary when trying to solve the real world problem. No one is using definition of derivatives all the time to solve basic polynomial. They just memorize the rule. But once get used to it, it may be a great idea to come back and see how it work under the hood. This is good because not everyone is interest in math, it would be overkill to go into detail for everything about math. It should be left for passionate students to learn it by themselve. People that are nớt interested in math can just use the short cut method for their daily life
The act of writing does more than waste paper and materials. Writing reinforces learning by pairing abstract concepts and thoughts with physical motions that are unique to each written sentence.
I have a similar story. To make it short: I didn’t really understand the fundamentals of Calculus until my daughter’s AP math teacher taught limits some 30 years after I learned it in college. He took 2 weeks to explain limits where my professor made us memorize (and recite on a test) the formal Epsilon/Delta definition of a limit and then moved on. It is about the teacher.
here is why this is a good approximation: imagine the closest whole square root as X and the remainder as Y. we are interested in finding: √(X²+Y) he is suggesting this approximation: X + Y/2X let's see what this approximation will give us if we raise it to the power 2: (X + Y/2X)² = X² + Y + Y²/4X² the only difference between this expansion and what we were taking the square root of is the third term (Y²/4X²). and if X is larger than Y this value will be very small. that is why this approximation is good for X>>Y
One of the flaws of this method is that the approximation is only accurate near a square number. If you add a +1 in the denominator of the first derivative term, you get a much better approximation across the entire domain. This corresponds to the piecewise linear approximation of sqrt(x)
We do it differently. For example, to find sq.root of 56. We take the integers whose square is immediate less and immediate more i.e. 49 = 7², and 64 = 8². So our required number is 7 + (56-49) / (64 - 49). Which is quite close
You can also use the next larger square if it's more convenient. E.g., if we want the square root of 97, we can still start from 10, and then add -3/(2×10) and get a similar approximation. In fact, if the square of an integer you picked is closer to the target number (as 100 is to 97, compared to 81), the approximation will be more accurate.
Thanks sir... Today, I found a new trick that's applicable for almost every root I hv tried different values and the answers are exactly same, differ in 0.1 or less.. Thanks.. 🙏❤❤❤
I was just now 5 min ago having an existential crisis because I apparently completely forgot how to do the long multiplication method on paper until I realized not only was I doing it backwards, I was also applying long addition method rules to it (also doing that backwards as well ffs) so I felt stupid. Thank God I could actually understand and follow through easily with this example. I'm not completely hopeless after all 😂
Nice, that’s the linear approximation method . Take the nearest number (100 in that case) and write the equation of the tangent line to the graph of sqrt of at x= 100 . The equation will be as follows: y= (x-100)/20 +10 Finally substitute 102 and you’ll get 10.1 Also you can use this method to approximate the values of unpopular angles of trigonometric functions .
Or, if you can remember three holidays, then you can easily remember three approximate square roots. Valentines, St Patricks, and Halloween. SR of 2=1.4, SR 3 = 1.7, and 10=3.1
@@looming_ I work for an ad tech company. What are people (besides math teachers) using it for? I don't even know anyone who has ever used it in real life.
@@drewmcmahon2629 I use it vaguely as a GC. And I mean very sparingly but I do use it. Say I have a wall that’s 65’ long and need to put decorative trim up and want to do even spaces…. Yea you can split the difference then split it again then again as most contractors would but it’s easier for me to go ok I need 8 boards at approximately 8’ apart plus an additional one as my start piece…. Again the situation rarely comes up but I would be lying if I said I never used it. Curious if anyone else does? 🤔 I also use the Pythagorean Theorem ALL the time. I thought it was useless learning that in school but as a GC I use it constantly. 😅
that's because you don't really need to worry about being precise with engineering and design... things like bridges, tunnels and skyscrapers are typically just eyeballed due to the size 🌉🌁🙌🏽👀. They don't make tape measures long enough... 📏 👷🏽♂️
If the goal is to approximate the square root, saying "approximate the number and take the square root" is not useful. Also this is essentially what he is doing.
There is a much more intuitive way to work this out, although it is slightly harder. It is known as the concept of small change (an application of differentiation). Only read on if you know how differentiation works :) . Basically, say we want to estimate the value of 3 root 1004 . To do that, let x = 1004, so we have an equation for y = x^(1/3). Differentiating that, we get (1/3)(x^(-2/3)). Using the increments formula, that the change in y = the derivative multiplied by the change in x, we can find the value of root 1000, which is 10. Note that the change in x is 4 (from 1000 to 1004). Then, by substituting x for 1000 and the change in x for 4 in the increments formula, we can estimate the value to be 10+1/75. Please reply if this doesn’t make sense and I will try to explain it to you :) .
Sure I'll show you :). Basically, the increments formula is that the change in y divided by the change in x equals dy/dx. Multiplying by the change in x, we now get that the change in y equals dy/dx multiplied by the change in x. By subbing in dy/dx= (1/3)(x^(-2/3)) and multiplying both sides by the change in x, we get the change in y equals (1/3)(x^(-2/3)) multiplied by the change in x. Subbing in the change in x=4 and x=1000, we get an estimated value of 10+1/75.
this comment deserves millions of likes. If ANY if my math instructors/teachers had shown us this trick in such a simple way, say anytime between kindergarten to 7 years of grad school for physics ... ? And I finally see it in a youtube video? I might have liked math instead of endured it.
Next time my math teacher says my answer is wrong I'll say, "but it was pretty darn close"
HAHAHAHAHAHAHAHA
Is approximate answer
Technically he's right if it's to 1 d.p
Haha
My thoughts exactly.
Mathematics as it is taught in school is about precision not approximations, regardless of how cool they are.
One of the reasons i hated it.
After making the video, I realized I used the = sign by mistake. I meant to say the approximate square root of 102 to square root of 100.
Did you invent this method?
@@mrhtutoring UCLA invented it?
It is 102^(1/2)
=(100+2)^(1/2)
=[100*(1+2/100)]^(1/2)
=100^(1/2)*(1+2/100)^(1/2)
=10*[1+(1/2)*2/100±...] (Maclaurin series, ignore higher order terms as they tend to 0)
≈10*(1±0.01)
≈10.1
@@ishanshah230889 I appreciate it. Thank you.
It seems needlessly complicated. Why say it's approximately equal to 10 + 2/(2*10)? The remainder "2" will always cancel out if you put it on the numerator and the denominator. Easier to say it's about 10 + 1/10.
For those who want to know, this is an application of the newton method, which allows you to find the root of a nonlinear function very quickly (Each iteration roughly doubles the amount of digits that are accurate). In this case, the function with root sqrt(102) is f(x)=(x^2) - 102. In the Newton Method, you pick a starting point x_{0} and get a better approximation of the function's root by inductively defining
x_{n+1} = x_{n} - (f(x_{n})/f'(x_{n}))
In this case, we apply 1 iteration with x_{0} equal to the smallest whole-numbered root smaller than sqrt(102), which gets us close enough to the real root for the newton method to substantially improve the guess. We can check that the equation used in the video is the same by plugging in the values:
x_{1}
= x_{0} - (f(x_{0})/f'(x_{0}))
= 10 - (-2/2*10)
= 10.1
The newton method works pretty generally with differentiable functions (careful: its bad if the root is also a stationary point, it can be adjusted to rectify this flaw though) because the geometric intuition is that it puts a tangent line on the starting point and computes the intersection with the zero line, which is really close to the functions intersection with the zero line (its root) if the starting point is already close to the root of the function. See Newton Method on wikipedia for a nice graphic showing this concept.
I understand but didn't understand 😂
Me too😅
Isn t this a Taylor expansion approximation ?? Using only the first derivative ??
@@telemans107 yes, you can look at this using just first digit Taylor approximation: the important thing is that the error of the Taylor series gets small very quickly in almost all cases when the current value is close to the root of the function, which allows this method to converge.
@@tolbryntheix4135
Thank you .I have stadied that 40 years ago in Rabat Morocco but still in touch.
Doing math on the chalk board. Miss those days❤
R u Indian?
we still use it
You talking as if we do maths on telebooks 😂
that is racist@@invincibleghost__23
@@blueshot333well most schools use whiteboards and now BenQ screens
Never seen a math teacher accept “pretty darn close”
You should try an advanced mathematic course, theres ton of numerical estimations for unsolveable equations. Its called numerical estimation. But usually you can increase number of iterations and can get an accuracy as accurate as possible, down to 10**(-(15+)) if you want.
@@herman7880Which will be pretty darn close
Pretty darn close is the entire idea behind limits in calculus
Mine did
In 4th grade she didn’t accept me putting 0.33 for 1/3 but accepted 0.35 for 1/3
@@h4m1d39 to be fair, if you took the taylor series to finity, you would be 0 of. That's pretty darn close if you ask me!
Ok now let me introduce Taylor's series...
Don’t need a series, just do linear approximation
@@joj0ee But Taylor series is more accurate
Newtons method is far superior than taylor series for approximating numerical roots
taylor series is a generalisation.
If you take the constant and linear term of taylor series, it is linear approximating.
All this video uses is f(x+h) = f(x) + h* f '(x)
where f'(x) = 1/(2sqrt(x)) , x = 100 and h = 2
😂
"Let me teach you some Calculus without Scaring you away-"
Exactly
I want to learn then😁😊
@@thailandler1110 Here's how it works:
Given y = sqrt(x)
Approximate this function near the nearest perfect square, at x=100, with a tangent line that touches it and locally matches its direction.
The point slope form of a line is:
y = m*(x - x0) + y0
x0 = 100, and y0 = 10. All that's left to find is m, the slope of the tangent line, dy/dx.
y = sqrt(x) = x^(1/2)
If y=x^n, then dy/dx = n*x^(n - 1), which means:
dy/dx = 1/2*x^(1/2 - 1), which can be rewritten as:
dy/dx = sqrt(x)/(2*x)
Evaluate this at x=100:
m = dy/dx = sqrt(100)/(2*100) = 10/200 = 1/20
Thus, the tangent line is:
y = 1/20*(x - 100) + 10
Plug in x=102:
y = 1/20*(102 - 100) + 10
Result:
y = 10.1
Thats how to teach students about math. Make it simple and clear, not terrifying and complex.
Not in my lifetime! Been there, tried that.
I solved All Kinds of Equations in School And College.
Now I am Security Guard and my Salary gets "Square Root" on a very first day 😭😭😭
It should be noted that the 2 in the denominator is NOT the same 2 as in the numerator. The denominator 2 is a constant and is always used regardless of the remainder from the number to be square rooted and the closest perfect square.
Thx. I was wondering about that.
It was the least clear part of the demo, in my opinion too.
That was REALLY confusing and not clear in the video !
Thank you !
@@Gottenhimfella I was thoroughly confused at first lol
Thank you. I was running some other numbers and didn't get even close, but your comments helped to clear it up.
For those that want to learn more. This is called Local Linear Approximation
Thank you
Or approximation using first principle of derivative.
f(x+h) = f(x) + f'(x)/h
(h is very small)
Also (sqrt(x))' = 1/(2*sqrt(x))
That's why there's "2".
Thank you
It is also horribly unprecise and there is no need to actually use it.
You can actually do it from both sides:
sqrt(102)=sqrt(100+02)=sqrt(10*10+02)
~10+02/(2*10)=10+2/20=10,1
sqrt(102)=sqrt(121- 19)=sqrt(11*11- 19)
~11 -19/(2*11)=11-19/22~10.136
The further you are away from the the actual squared number (100 or 121) the wronger it gets. 102 is really close to 100, which is why the result is somewhat good.
You could technically also do something stupid like picking a square that is not adjacent and get something horribly wrong (:
sqrt(102)=sqrt(49+53)=sqrt(7*7+53)
~7+53/(2*7)=10+53/14~10,79
I approcimated 102 with 49, which is 7*7 (:
It is called Binomial Approximation.
(1+x)^n is approximately 1+nx if x is very small as compared to 1.
I Wasnt Even Paying Attention To What He Was Doing Because I Was Just Imagining How SMOOTH It Sounds When He Writes With The Chalk
Oh, you're one of *those* people. I always paid attention. Shrugs
@@mariatorres9789 What’s wrong with listening to chalk sounds? its a yt short anyway, it’ll replay
Wow I've been searching for an easy method to find square root but no one said like this.......this is very very easy..!
I tried for √109 and I got 10.45 as approximate value and the correct value is 10.4403......
I'm going to send this to all my frnds who r struggling
Thank you soooo much ❤❤
Happy to help
Somehow, the chalk hitting the board sounds satisfying
Ik
asmr
It helps with concentration as well.
I hated it. Felt like someone is banging my front door during a hangover.
Yeah brings back some good memories of learning in school. 😊
He's like Bob Ross of maths. I thought he was going to say we're going to put a happy little number justttt here at one stage.
Square roots do make some happy little trees.
😂😂
Gfc
what is Bob Ross?
@@mrjodoea really cool drawer
It's just the zeroth iteration of Newton ralphson method. 😅
Yup! They both function via a linear approximation.
I get the feeling this guy is shredded.
for anyone wondering this is actually the beginning of the Taylor series of the square root function (around the closest perfect square, 100 here). The next term in the sum (for more accuracy) would be -(x-a)²/(8a sqrt(a)) with "a" being the perfect square.
Here (x=102, a=100) it would give 10 + 2/(2*10) - 2²/(8*100*10)=10.0995 (exact value)
what is a and x in your formula?If a=2 then it would be in your formula: 2²/(8x2x2²).So if you say a=10 then in your formula:10²/(8x10x10²).
This is what I understood from your formula.Can you make it clear please?
@@copculerkral1157 The number of which you take the square root is x and the closest approximation you use is a. So in this case x = 102 and a = 100. The sum of the first two terms in the Taylor Series is f(a) + f'(a)/1! * (x-a). Since f(x) is sqrt(x) in our case, you get the result.
That is not the exact answer, the exact answer is √102 or the infinite Taylor expansion
@@awvz_1194 Yeah :)
@@awvz_1194 Yep, what i was trying to say is that this value is the exact result of the sum i wrote (the first 3 terms)
Nice! As a retired engineer i think more people need to learn algebra, not to master it but to learn a different way of thinking about numbers.
But this does none of that. It's a set of instructions to get an approximate answer. To show different ways of thinking about numbers, he would need to show why it works - or better yet, walk the viewer through the thought process for figuring out the trick.
Nah, I'll just remember Arbitrary rules and tricks.
This isnt algebra
@@ApesAmongUs This trick is based on binomial approximation.
@@adityakamat9856 You just dropped jargon. Doing that does not help anyone "to learn a different way of thinking about numbers".
Also, the fact that you thought you were telling me something I didn't already know tells me that you did not understand my point.
I like how passionate and smart teachers like you say "Pretty darn close" or "close enough" while teachers who just reads and gives an example are often the one's that says everything has to be precise. Kind of crazy lmao
I've never been as interested in math as I have been the past 2 days watching random math shorts
I’m fifty years old and through out my years I’ve had trouble with math. I’ve had several teachers over the few years I was in school and they all seemed to rush through the process. I really enjoyed this video and will be watching more. Thanks for your sharing and time.
What is good learning on YT is that you can replay until you get it!😂
For anyone who are interested in this algorithm, this an application of newtons method iterated once.
Netwons method is a way to approximate a value x such that f(x) = 0. In order to make such a function we make it be 0 when x is correct: f(x) = n-x^2
x ≈ guess - f(guess)/f'(guess)
Repeat with the new guess being the current x.
This works for many other cases as well, i.e. third root:
f(x) = n - x^3
x ≈ guess - (n-guess^3)/(-3*guess^2)
A little more complicated, but still highly doable.
Thank you for that explanation❤
They used this trick in the first quake game I believe. But they also used other tricky bit wise logic as well. This was used to normalize a vector I think for light calculation.
@@IhsanMujdeci that makes sense, as a lot of calculators also use this algorithm with more iterations and a look up table.
I was having flashbacks of newton's method but wasn't sure glad i was remembering in the right direction
What is the real world application for square roots?
This is derived by using derivatives
Let f(x) = √x, x = 100. ∆x =+2
f(x+∆x) = ∆x.(df(x)/dx) + f(x)
Thank you professor for accepting the answer when it is "pretty darn close".
the larger the number is the more precise this gets
accurate*. The larger the number, the more accurate this method gets.
@@Inferno.522 The better you are at preventibg mistakes, the more precise this gets 🤣
EDIT: Case in point, just gonna leave it there
@@jo54763 I don't get why we'd use this method? I'd just find the largest root numbers of each then simplify it until I get the smallest surd.
Kinda useless in my opinion.
@CherryZzz I wasn't commenting on the method, just adding to the accuracy vs precision statement.
@@cherryzzz6229 how do you think calculators work? Pulling digits out of a magic hat?
And this RIGHT HERE is why I ALWAYS carry my calculator at all times with me.
Dude smart devices 😅
What?
It's called a phone
Is this another way of saying:
Without my calculator I feel helpless?
In retrospect I wonder how we people born up to the early 1960s (so joining the school system up to 1970 - as electronic calculators dropped into an affordable range during the first half of the 1970 decade only) could cope with calculating anything beyond simple addition, subtraction and multiplication (say up to 20×20).
I worked at an inner-city school where the math teacher didn't believe the kids could learn (of course they could). He gave them calculators for the simplest things, thus making sure they DIDN'T learn. That's what you're doing to yourself.
I'm still trying to draw the square root sign, then we can get to the numbers. 😂
This guy is so great for doing this on yt. Say what you want about the normal stuff, but this kind of universal teaching is invaluable.
Like others have said, it's just the first two terms of the Taylor expansion, but for the love of God, please don't abuse the equals sign like that
Guys I found the mathematician.
Abuse the equals sign? How the hell do you do that?
@@arcanine_enjoyer he wrote an untrue statement using equal sign
@@arcanine_enjoyer he wrote √102=√100
@@arcanine_enjoyer I don't even like math that much, but seeing how he wrote root(102) = root(100) was so painful to watch, it really made my brain hurt for a bit.
My 2 second approximation would have been “10.something low” and I would stand by it because it’s only an approximation. 😂
You might as well say it's 0 because it's only approximation. The point is that without saying what's the upper bound on error of your estimate, it's meaningless.
@@Ennar But its not completely meaningless because there are no applications of estimating a square root where an upper bound on the error actually matters. Estimating a square root is for casual conversation and super quick rough working. For any application where you need some level of confidence in the answer you can just use a calculator.
@@callanc3925 I'm sorry, but every single thing you wrote is *completely wrong*. What OP wrote is that they are confident that the correct answer is somewhere between 10 and 11 which is estimating bound on the error. What I wanted to point out that they seem to completely ignore the importance of this by saying "it's only an approximation".
Thank you for the lesson ,Sir .Learning every day .Vetri South Africa 🙏🇿🇦🙏
Really good trick this would really help me in my exams as calculators are not allowed. Thank you again sir
Always to happy to hear that it gets used.
you can tell by the way the chalk glides with ease across the board that this man has put many many sticks of chalk through the ringer. we're talking *holding with your nails* levels of point precision. very talented.
Hogoromo chalk, that’s the special Japanese invention that’s much coveted around the world.
wringer
This was my typical math teacher, growing up in the 70s, never teaching anything about why, so I just memorized.
You are absolutely correct!!!
Can’t teach a “why” into a 1minutes video. If u want to know a “why”. Search up “taylor series” and see how is it going. The reason most low level math class in middle school etc doesn’t teach into the detail is because the detail is so complicate and unnecessary when trying to solve the real world problem.
No one is using definition of derivatives all the time to solve basic polynomial. They just memorize the rule. But once get used to it, it may be a great idea to come back and see how it work under the hood. This is good because not everyone is interest in math, it would be overkill to go into detail for everything about math. It should be left for passionate students to learn it by themselve. People that are nớt interested in math can just use the short cut method for their daily life
It's a goddamn RUclips short.
No way I got this right after my big test.
When math is "pretty darn close," I'm all on board.
Jeez, it's been so long since I've heard writing on a chalkboard, that really takes me back to elementary school.
The act of writing does more than waste paper and materials. Writing reinforces learning by pairing abstract concepts and thoughts with physical motions that are unique to each written sentence.
I always said its about the teacher.
You just taught me something in 60secs that I could never learn in High School. Cheers.
I have a similar story. To make it short: I didn’t really understand the fundamentals of Calculus until my daughter’s AP math teacher taught limits some 30 years after I learned it in college. He took 2 weeks to explain limits where my professor made us memorize (and recite on a test) the formal Epsilon/Delta definition of a limit and then moved on. It is about the teacher.
Good instructor
I’m an engineer and even 10 would have been pretty darn close 😂
Finally a good short on my feed. Thank you sir.
Nah, this method doesn't work
here is why this is a good approximation: imagine the closest whole square root as X and the remainder as Y. we are interested in finding: √(X²+Y)
he is suggesting this approximation:
X + Y/2X
let's see what this approximation will give us if we raise it to the power 2:
(X + Y/2X)² = X² + Y + Y²/4X²
the only difference between this expansion and what we were taking the square root of is the third term (Y²/4X²). and if X is larger than Y this value will be very small. that is why this approximation is good for X>>Y
We need people like him ❌ we need to become smart like him✔️
Thanks sir, now I'm gonna impress my classmates and math teacher being the class square root human calculator😅😅😅
Great 👍
I wish my teachers were like “ya that’s pretty darn close, good job”
This is actually pretty terrible. Try other values above 102.
@@macfrankist No lmao
@@thanosnoctem4473 Yeah try with 114.
Sorry I stand ridiculed.
One of the flaws of this method is that the approximation is only accurate near a square number. If you add a +1 in the denominator of the first derivative term, you get a much better approximation across the entire domain. This corresponds to the piecewise linear approximation of sqrt(x)
“approximation cannot be accurate” but “close”
No I myself don't do it either way.
@@ocayaroHe is accurate in his approximation.
I was thinking.. there is your answer LMAO!
The old chalk board is more fun than the estimated answer.
"Google, what is the square root of 102?" Also works.
Not on a test
@@tanishianandit 100% works on a test, just gotta be quiet
its can be proofed by eror of √x by taking derivative
Newton-Raptson iteration method left the chat.
Last many people commented on his triceps, he bought a New full sleeves shirt
We do it differently. For example, to find sq.root of 56. We take the integers whose square is immediate less and immediate more i.e. 49 = 7², and 64 = 8². So our required number is 7 + (56-49) / (64 - 49). Which is quite close
Average of taylor expansion from both sides eh
Wow, something on youtube where I can learn something. Incredible.
This is so satisfying. It’s exactly how to do math.
bro fork everything eles i love how smoothly you write mate
he is like my asian uncle who ask about my math marks
I have literally never needed to know the square root of a number irl. Thanks school.
I did. Thanks, school!
Is it just me that the noise that the chalk made is extremely satisfying 🤤🤓
You can also use the next larger square if it's more convenient. E.g., if we want the square root of 97, we can still start from 10, and then add -3/(2×10) and get a similar approximation. In fact, if the square of an integer you picked is closer to the target number (as 100 is to 97, compared to 81), the approximation will be more accurate.
Thanks sir... Today, I found a new trick that's applicable for almost every root I hv tried different values and the answers are exactly same, differ in 0.1 or less..
Thanks.. 🙏❤❤❤
In the beginning, I figured it’s around 10 and figured I was pretty darn close.
I was just now 5 min ago having an existential crisis because I apparently completely forgot how to do the long multiplication method on paper until I realized not only was I doing it backwards, I was also applying long addition method rules to it (also doing that backwards as well ffs) so I felt stupid. Thank God I could actually understand and follow through easily with this example. I'm not completely hopeless after all 😂
That's called approximation, done through differential calculus, when i calculate imperfect roots, my friends think im Einstein or sumthin
Damn it, now i kinda miss math classes
My eyes glazed over almost immediately.
If you're approximating, you should really be using the ≈ symbol instead of the = symbol.
Nice, that’s the linear approximation method . Take the nearest number (100 in that case) and write the equation of the tangent line to the graph of sqrt of at x= 100 . The equation will be as follows:
y= (x-100)/20 +10
Finally substitute 102 and you’ll get 10.1
Also you can use this method to approximate the values of unpopular angles of trigonometric functions .
Using binomial expansion
(x+y)ⁿ = 𝚺ⁿₖ₌₀ (n k) xᵏyⁿ⁻ᵏ
or 𝚺ⁿₖ₌₀ (n k)xⁿ⁻ᵏyᵏ
where (n k) = ₙCₖ = n!/(k!(n-k)!)
√(ϕ+ε) = (ϕ+ε)¹ᐟ²
= 𝚺⁰⁰ₙ₌₀ (½ n) ϕ¹ᐟ²⁻ⁿ εⁿ
≈ 𝚺¹ₙ₌₀ (½ n) ϕ¹ᐟ²⁻ⁿ εⁿ
= (½ 0)√ϕ + (½ 1)ϕ⁻¹ᐟ² ε
= √ϕ + ε/(2√ϕ)
√(100+2) ≈ √100 + 2/(2•√100) = 10.1
The first sqrt(102)=sqrt(100) is triggering me so hard
Waaaaaah waaaaaah waaaaaaaah
@@Megadumbyog I only see you crying
@@KebabTM I ate your family
Or, if you can remember three holidays, then you can easily remember three approximate square roots. Valentines, St Patricks, and Halloween. SR of 2=1.4, SR 3 = 1.7, and 10=3.1
Also a good way to remember Valentines day
Cute trick.
It's binomial expansion. First two terms by transformation. Sqrt 102 = 10 sqrt 1.02 = 10 ×(1+0.02)^0.5 = 10 × ( 1 + 0.02/2 + ...) = 10 × ( 1 + 2/(100 ×2))
can’t wait to tell my math teacher next time that i was pretty darn close lol
It's impossible to get the exact value of a square root of a number
I've literally never needed the
Sq root of anything even once in my life.
Cause ur cleaining toilets
Whaaat how not?!
what do you do for a living?
@@looming_ I work for an ad tech company. What are people (besides math teachers) using it for? I don't even know anyone who has ever used it in real life.
@@drewmcmahon2629 I use it vaguely as a GC. And I mean very sparingly but I do use it. Say I have a wall that’s 65’ long and need to put decorative trim up and want to do even spaces…. Yea you can split the difference then split it again then again as most contractors would but it’s easier for me to go ok I need 8 boards at approximately 8’ apart plus an additional one as my start piece…. Again the situation rarely comes up but I would be lying if I said I never used it. Curious if anyone else does? 🤔
I also use the Pythagorean Theorem ALL the time. I thought it was useless learning that in school but as a GC I use it constantly. 😅
Nah yo why did I understand this process in less time than I ever could in high school
POV: Teacher asks you to round to two decimal places
10.10
Brilliant! I love approximation methods.
Where I live, young children are taught to approximate before they calculate. It's such a healthy approach.
I'd just say roughly 10 and be done with it
Sounds good to me too
It's the first 2 terms of the binomial expansion.
Not the binomial theorem, but the taylor expansion.
@@Daniel31216it's the first 2 terms of both. Seriously I got bored one day and checked for it myself on my whiteboard lol.
@@somedude1666 Great to know! Now I have the urge to check it out myself.
It is pretty close. Thanks for the tip! Subscribed. All.
i actually never studied math but i understood everything .only if all teachers could explain it this way
in engineering we draw a squiggle and call it 10
that's because you don't really need to worry about being precise with engineering and design... things like bridges, tunnels and skyscrapers are typically just eyeballed due to the size 🌉🌁🙌🏽👀. They don't make tape measures long enough... 📏 👷🏽♂️
Use newton raphson method for x^2-n and n^(1/2) is your number
If the goal is to approximate the square root, saying "approximate the number and take the square root" is not useful. Also this is essentially what he is doing.
Thats how we using applications of derivatives 😅😊
It just pretty darn close lol best tutor
There is a much more intuitive way to work this out, although it is slightly harder. It is known as the concept of small change (an application of differentiation). Only read on if you know how differentiation works :) . Basically, say we want to estimate the value of 3 root 1004 . To do that, let x = 1004, so we have an equation for y = x^(1/3). Differentiating that, we get (1/3)(x^(-2/3)). Using the increments formula, that the change in y = the derivative multiplied by the change in x, we can find the value of root 1000, which is 10. Note that the change in x is 4 (from 1000 to 1004). Then, by substituting x for 1000 and the change in x for 4 in the increments formula, we can estimate the value to be 10+1/75. Please reply if this doesn’t make sense and I will try to explain it to you :) .
i get the differentiation but dont get the increment formula part \\
Sure I'll show you :). Basically, the increments formula is that the change in y divided by the change in x equals dy/dx. Multiplying by the change in x, we now get that the change in y equals dy/dx multiplied by the change in x. By subbing in dy/dx= (1/3)(x^(-2/3)) and multiplying both sides by the change in x, we get the change in y equals (1/3)(x^(-2/3)) multiplied by the change in x. Subbing in the change in x=4 and x=1000, we get an estimated value of 10+1/75.
@@jimmy-j6465 thankyou for taking the time out and explaining. I got it.
🍁
@@SujalShan-ot8gc You welcome. I'm glad this was helpful :) .
Why does math make more sense in a RUclips video then 4 years in school
this comment deserves millions of likes. If ANY if my math instructors/teachers had shown us this trick in such a simple way, say anytime between kindergarten to 7 years of grad school for physics ... ? And I finally see it in a youtube video? I might have liked math instead of endured it.
@@NavigatEric Surely you have binomially expanded? (1 + x)^0.5 is approx 1 + 0.5*x, that's school level.
You pay attention to RUclips. In class your checking out the babes and your mind is elsewhere
I always stop at the first step 😂
This channel double as an Asmr channel😂
That was awesome
Binomial approximation
Thanks. I'm good at math but this was never taught to me
For more precision, you can add more terms of the taylor series of square root function
Ty I like that
Thanks
Derivatives gives you approx results.
He should tell the derivatives instead of this magic.😂
He is using derivatives, in the form of the taylor series. He's just using the first two terms of it.
Great video 👍 sir
He is actually doing a binomial expansion shortcut.
Love how smart people can break things down to simplify things like this. Great explanation.
Never met a man so smooth with his chalk.
This guy is NOT a math teacher, he's actually a very famous actor from Korea!
Max holloway after retirement 😂😂😂