use calculus, not calculators
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- čas přidán 28. 08. 2024
- Use calculus, not calculators! Tangent line approximation (aka local linear approximation) and differential to approximate the irrational number sqrt(100.5)
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10 ... Didn't use any calculus either :) I think that's the engineer's answer :)
As a software engineer, I say the answer is 10.0000000023
@@IamBATMAN13 mfw computers are calculators
10.0 is a great approximation to 3 significant figures.
e * e = 2e = 4
@@sinpi6332 e + e = 2e, e * e = e^2,
either way everyone knows e = 3.
For people who remember their Taylor series, it is very fast to factor out the 10 and use (1+x)^n ~ 1 + nx where n=1/2 and 1+x = 1.005. You can get 10 * 1.0025 = 10.025 without writing anything down.
This also works for other powers, like cube roots, etc., which are of this form, and if you’ve forgotten it, the linear term in the Taylor series (or Maclaurin series if you prefer) is easy to calculate as well.
(10 + h)^2 = 100.5
100 + 20h + h^2 = 100.5
20h + h^2 = 0.5
h^2 is very small
20h ~ 0.5
h ~ 0.025
=> sqrt (100.5) ~ 10.025
I remember doing this
as a sort of Newtonian calculus
early on in learning the calculus.
Matt Parker's method!
@@General12th
Like all naming conventions - we tend to associate it with the person we saw it from first or the person our teacher associated it with.
In my case Newton in yours Matt Parker.
Very nice. Approximation by linearization makes more intuitive sense to me, but clearly the differential method is way more awesome.
They’re essentially the same
10.0249 to be more exact
Using long division
Next digits are 688 (ie 10.0249882). So 10.0250 is closer to the given approximation than 10.0249.
Still cool you used long division though!
Thank you sir! i knew this method from somewhere
4m48s to do something you can do in 4s with calculator.
That intro with "Square root of 100 is NICE number " got me,..
My guess before watching: approximating via tangent lines.
Let f(x)=√x. We know f(100)=10 because 10^2 = 100.
f'(x) = 1/(2√x), f'(100)=1/20=0,05 which is pretty small compared to 100, so:
f(100,5)≈f(100)+0,5•0,05 = 10 + 0,025 = 10,025.
It should be close enough considering the f(100)/f'(100) ratio is big
Use binomial theorem.. It will be much easier and faster.
If it’s so fast, could you explain how?
Ten and a tiny bit. Done
It's aproximately 10.
Do you Americans not get thought the square root extraction algorithm in school? Do they just throw you a calculator for elementary algebra?
American get taught nothing
🤓
Hmm
Yeah, not standard at all here in the states. Really weird too because it's very similar to long division, which we do learn. It's the first thing I think of when I see this though. "sure, how many digits you want?"
What happened over there (also where?), did they give you an algorithm and a quiz in a week, or did they explain why what you're doing works?
@@jkid1134 If you have a perfect square, then you’ll get a result part of the natural number set.
If you have a square roots that yields you a irrational number, then you can just approximate with 2,3, whatever number of decimals.
root 100.5 is very easy to do, I could send you a video of the algorithm we use if you would want.
Using calculus gives you better understanding than using calculator...
I got 8030/801 ~10.02496878 using the continued fraction sqrt(1+x) ~ 1+x/(2+x/2). Checked and it's accurate upto 6 decimals
I guessed 10.02 which is 100.4004. So I was within 0.1% using only my brain cells.
Use binomial expansion 😨
The world’s most cumbersome lapel mike.
1st comment
I did, from the thumbnail, √(100∙5)… ah well, learned something regardless 😅