How to Approximate Square Root of a Number

[ricco]

Next time my math teacher says my answer is wrong I'll say, "but it was pretty darn close"

[mrhtutoring]

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.

[tolbryntheix]

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.

[victoriaspade]

Doing math on the chalk board. Miss those days❤

[ryanolsen]

Never seen a math teacher accept “pretty darn close”

[hongminh]

Ok now let me introduce Taylor's series...

[sobhitkumar]

"Let me teach you some Calculus without Scaring you away-"

[mryashoo]

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 😭😭😭

[ubuandeyelbme]

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.

[misterrhombus]

For those that want to learn more. This is called Local Linear Approximation

[karthikk]

This is derived by using derivatives

Let f(x) = √x, x = 100. ∆x =+2
f(x+∆x) = ∆x.(df(x)/dx) + f(x)

[KamiXTwelve]

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

[ryan.y]

Somehow, the chalk hitting the board sounds satisfying

[Michaelmouse]

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.

[MyAmygdala_]

Wow I've been searching for an easy method to find square root but no one said like is very very easy..!
I tried for √109 and I got 10.45 as approximate value and the correct value is
I'm going to send this to all my frnds who r struggling
Thank you soooo much ❤❤

[Aman-swlw]

Really good trick this would really help me in my exams as calculators are not allowed. Thank you again sir

[noa]

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)

[LTV_inc]

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.

[waltervetri]

Thank you for the lesson, Sir .Learning every day .Vetri South Africa 🙏🇿🇦🙏

[Kingboo_]

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