@@ansarmiah6049 Learn the first 20 square roots and then approximate anything higher by dividing the input by 100 and multiplying your pre-correction answer by 10 (since sqrt(100) = 10). So for sqrt(1629) ≈ sqrt(16*100) = 4*10 = 40. 1629-1600 = 29, so the approximation from this video would be 40+29/80. It wouldn't surprise me if the trick still works if it's not exactly the closest square, but it's probably a bit less accurate the farther the square you use is from the number you're trying to sqrt.
@@ansarmiah6049 Let's say I didn't know 62^2 = 3844, calculating sqrt(3844) ≈ sqrt(36*100) = 6*10 = 60 and 3844-3600 = 244, so after the correction I get 60 + 244/120 ≈ 62.03. So you see that it even works if you don't use the closest square. You just need one that is relatively close.
It also works for a negative correction: 59^2 = 3481. Let's say we didn't know that, using the technique we get sqrt(3481) ≈ sqrt(36*100) = 60 and 3481 - 3600 = -119, so after the correction, we get 60 + (-119/120) = 60 - 119/120 ≈ 59.01
@@ansarmiah6049 Using that approximation can be somewhat imprecise for larger numbers, but you can apply the correction multiple times to make it more precise: Lets consider 545^2 = 297025. Assuming we didn't know that: Sqrt(297025)≈sqrt(36*10000) = 600 297025 - 360000 = -62975 1st approximation gives: 600 - 62975/1200 = 550 - 2975/1200 = 548 - 575/1200, so let's roughly say 548 548^2 = 300304 297025 - 300304 = -3279 2nd approximation gives: 548 - 3279/1096 = 546 - 1087/1096 ≈ 545.01
@@bompinghead9865 it's differential calculus Method of approximation to be exact. Differential is used to approximate the values of different things and one of them is squares.
Let me try to help. Let's say you have a perfect square x², with x being a natural number. Now you have an imperfect square (x+a)² where a is an irrational number such that 0
Let me try to help. Let's say you have a perfect square x², with x being a natural number. Now you have an imperfect square (x+a)² where a is an irrational number such that 0
It's because it's calculus. You can find the approximate difference. Dx or dy = difference in x or y y' is the derivative of y, or the slope Dy=y' * dx Let y=f(x)=sqrt(x) In this case, y' = 1/(2*sqrt(x)) dy = dx/(2*sqrtx) Let x be 25 and dx be 2 to represent sqrt(27 dy=2/(2*sqrt25) dy=1/5 sqrt27≈sqrt(25)+1/5 ≈5.2 Sqrt27=5.19 Good approximation
cant you just use linear approximation instead. It works for not only square roots but anything
Its very smart but how do we het the closest perfect suare root quickly?
@@ansarmiah6049 Learn the first 20 square roots and then approximate anything higher by dividing the input by 100 and multiplying your pre-correction answer by 10 (since sqrt(100) = 10). So for sqrt(1629) ≈ sqrt(16*100) = 4*10 = 40. 1629-1600 = 29, so the approximation from this video would be 40+29/80. It wouldn't surprise me if the trick still works if it's not exactly the closest square, but it's probably a bit less accurate the farther the square you use is from the number you're trying to sqrt.
@@ansarmiah6049 Let's say I didn't know 62^2 = 3844, calculating sqrt(3844) ≈ sqrt(36*100) = 6*10 = 60 and 3844-3600 = 244, so after the correction I get 60 + 244/120 ≈ 62.03. So you see that it even works if you don't use the closest square. You just need one that is relatively close.
It also works for a negative correction: 59^2 = 3481. Let's say we didn't know that, using the technique we get sqrt(3481) ≈ sqrt(36*100) = 60 and 3481 - 3600 = -119, so after the correction, we get 60 + (-119/120) = 60 - 119/120 ≈ 59.01
@rikschaaf thanks
@@ansarmiah6049 Using that approximation can be somewhat imprecise for larger numbers, but you can apply the correction multiple times to make it more precise:
Lets consider 545^2 = 297025.
Assuming we didn't know that:
Sqrt(297025)≈sqrt(36*10000) = 600
297025 - 360000 = -62975
1st approximation gives: 600 - 62975/1200 = 550 - 2975/1200 = 548 - 575/1200, so let's roughly say 548
548^2 = 300304
297025 - 300304 = -3279
2nd approximation gives: 548 - 3279/1096 = 546 - 1087/1096 ≈ 545.01
Very good
😮💨
Why does this method work though?
@@bompinghead9865 it's differential calculus
Method of approximation to be exact. Differential is used to approximate the values of different things and one of them is squares.
Let me try to help. Let's say you have a perfect square x², with x being a natural number. Now you have an imperfect square (x+a)² where a is an irrational number such that 0
Ok thats smart
🧠
I didn't really learn anything, because you didn't explain WHY it works. All this video is, is teaching a dog some tricks.
Let me try to help. Let's say you have a perfect square x², with x being a natural number. Now you have an imperfect square (x+a)² where a is an irrational number such that 0
It's because it's calculus. You can find the approximate difference.
Dx or dy = difference in x or y
y' is the derivative of y, or the slope
Dy=y' * dx
Let y=f(x)=sqrt(x)
In this case, y' = 1/(2*sqrt(x))
dy = dx/(2*sqrtx)
Let x be 25 and dx be 2 to represent sqrt(27
dy=2/(2*sqrt25)
dy=1/5
sqrt27≈sqrt(25)+1/5
≈5.2
Sqrt27=5.19
Good approximation