How to Estimate the Square Root of ANY NUMBER!

[@aryansrivastava727]

Using sandwich theorem of limits you can also think that it will lie between 10 and 11

[@marcelgraf5520]

im eating a sandwich rn

[@danny89620]

Using a brain you can see that it will lie between 10 and 11

[@yoyohonewysingaa]

You don't need to use any godamn theory of limits for estimating something so trivial. One glance at 103 will tell you it's square root lies between 10 and 11 since it is >100 and

[@aryansrivastava727]

@@yoyohonewysingaa there is a big difference between tricks and the actual theory so keeping that in mind, I always avoid and advise against using any tricks lest the original concept may come to fade in the minds of the solver. I still use the whole procedure to find the capacitanc when dielectric is inserted instead of formula due to my love for logic.

[@yoyohonewysingaa]

@@aryansrivastava727 But just the fact that the root lies between 10 and 11 is a very trivial finding, which is anyways obvious is what I am saying.

[@MathTidbits]

the method described is usually known as the newton raphson method. here is another way of understanding the same method and will clarify the reason for multiplying 2 in the denominator,without bringing derivatives into the explanation. let N be the given number. in this case N = 103 let PS be the perfect square closes to the value of N and in this case PS = 100 let SQR N be the square root of N let SQR PS be the square root of PS,in this case=10 SQR N = average of N & PS divided by SQR PS SQR N = (N + PS ) / 2 /SQR PS SQR N = ( 103 + 100 ) / 2 / 10 SQR N = 203 / 2 /10 SQR N = 101.5 /10 SQR N = 10.15 bonus explanation : if the given number N is between two perfect squares,the criteria of which perfect square to choose is not base on the closeness of the linear distance to either perfect squares, but whether the given number N is on the left side or right side of the geometric mean of the 2 perfect squares. if your given number N is exactly the geometric mean of the 2 perfect squares , then choosing either perfect square will give you the same approximate answer. example: if the given number N = 20.3, the perfect squares are 16 and 25 . the linear midpoint is 20.5 , since 20.3 is less than 20.5 ,intuitively you will think of using 16 as your closes perfect square. in fact, the answer you get by using 16 will give greater error than using 25. the reason is that the critical point is not the arithmetic mean of 20.5 , but the geometric mean of 16 & 25 which is 20. since 20.3 is in the right side of 20, the correct choice is to use the right perfec t square which is 25. that will give you lesser % of error.

[@ClassicGabriel1]

Am i suppose to learn this in 6th grade?… Everyone here sounds like a highschooler 😅

[@sadiqakiani5868]

Thank uuu! I was looking for a deeper explanation and finally foind one

[@juls.faithh]

@@ClassicGabriel1you learn it in middle school its not thst hard dw

[@ClassicGabriel1]

@@juls.faithhcurrently in 7th, and you’re right. honestly not that hard.

[@-Fluffy-]

I just estimated 10.2 and figured I wouldn't need to go that deep in...

[@Zxter]

Deeper is better

[@rishipatel7953]

🤨

[@Nameuserg]

​@@Zxterkwite profile wtf

[@Adrian.006]

Why did you think 10.2 and not 10.3 or 10.1?

[@homelander21]

It will choke your throat​@@Zxter

[@thundershort997]

Well i have a method for those who memorize log table if u want to calculate any square root let say√2 so we let x=(2)1^2 take log on both sides log x=1/2log2 log x=0.15051499783 x= antilog(0.15051499783) x=1.41421356237 this is the scientific method to calculate the square root now if u put √2 in your calculator u get √2=1.41421356237 which is not even in approx. JazakAllah❤❤❤

[@bluebow9347]

You need a calculator to do antilog(0.15051499783)

[@thundershort997]

@@bluebow9347 well if we really greatly memorize anti log table so we can do it without calculator

[@koks3543]

This is called Heron Algorythm after inventor and constructor Heron from Alexandria. He publicated it in work "Metrica" between 10 and 70 year AD.

[@perlkoblin8177]

This trick does not work for small values as the limited development behind lies on the hypothesis that √(1+x) is close to 1+1/2 x for small values of x (x= the number / closest perfect square).Hence for 2 we get the approximation= 2

[@goated_koja]

If you use the method on 5, you get 2.1, and if you use a calculator, its approximately 2.23606, so it does.

[@Slash4608]

​​​@@goated_koja You get 2.25 using this method how you got 2.1? For example perfect under 5 is 4. Root 4 = 2. So if apply this we get 2 + (5-4)/(2×2) 2+ 1/4 2 + .25 2.25 maybe you did smthing wrong

[@goated_koja]

@@Slash4608 but still my point stands, it is 2.23606, compared to 2.25

[@ApertureScience_Opt]

This is linear approximation by differential. that is equation of tangential line at 100

[@speakingsarcasm9014]

You can use it for 2 that way. 2=1+1 indeed but the parts are equal. It works only if one of the parts is much smaller than the other... for e.g. 103=100+3 and 3 is much smaller than 100... or 227=225+2...

[@kianmath71]

You can also use calculus to do it and actually you can do it in your head because the numbers work out well

[@Jjkfan834]

My brain ain't braining

[@danielschoolofmusic1382]

Best trick bro I am an neet aspirant this trick helps me in physics very much Thanks bro

[@adw1z]

say we want to find sqrt(n), for some non-square n write n = a^2 + b, for positive integer a, and any integer b such that |b| is as small as possible. Then: sqrt(n) = (a^2+b)^1/2 = (a^2(1 + b/a^2))^1/2 = a(1+b/a^2)^1/2 ~= a [ 1 + b/2a^2 - b^2/8a^4 ] by binomial = a + b/2a - b^2/8a^3 Of course, the more terms you expand, the more accurate the approximation will be. I derived up to 2nd order, whereas this video up to first order (truncated at b/2a) Here, n = 103, a = 10, b = 3 ==> sqrt(103) ~= 10 + 3/20 - 9/8000

[@qwertyqwert2818]

very helpful!

[@RX_GL]

If you want to put this in desmos, here's what you need: b=sqrt.(a) a=10e_1+e_2 c=sqrt.(a-e_1) d=sqrt.(10e_1) f=d+e_2/2d e_1 can be from 0 to whatever you want, but e_2 can only be from 0 to 9. If you want to check how accurate it is, put another formula, sqrt(10e_1+e_2).

[@Splatbro24]

It’s summer right now but I’m saving this for when school starts

[@SWLaw14]

Why the calculator can find the exact number and I can't?

[@jasomega2446]

Thank you

[@blurr1903]

Just use f(a)-f’(a)(x-a)

[@JacksonPlayzGames]

Ummm that’s what he did lol. The 10 part of his answer is f(a) where a is 100 in this case and the function is the sqrt(x). f’(a) would be 1/(2*sqrt(a)) which equals 1/20. And x-a is the 103-100 part and that multiplies with the 1/20 we got earlier to become 3/20. Final answer is 10+(3/20) which is what he got.

[@AndrewX981]

Is this Newton method 🙄

[@Bookops]

My brain farted

[@Wutheheooooo]

​@@AndrewX981it's the taylor serie

[@Neilharvi3]

Why am i shocked yet entertained 😭💀

[@user-th5qv2eo2b]

This method improves with accuracy with larger numbers

[@sudhagupta4933]

Smart trick man Cool video Great job Keep going!

[@mofarah1325]

What a kind soul 😂

[@hercules5813]

¿So the sqrt of 120 is 11? A more accurate method would be to use 2n for the denominator if the remainder is

[@ItzApple3214]

Could you explain in a more simple way? I didn’t really understand tbh 😅

[@tixonochek]

Thanks so much!

[@kingbluecr7102]

(-b + root of (b^2 -4ac)) / 2a

[@AndrewGuyGD]

heres a square root i found: 21√9. 21 must be divided by 3. 21 • 3 = 7 then 7 must multiply 9. 7 * 9 = 63 the answer is 63

[@namangarg8637]

We don't say "ten point fifteen" , it's says as "ten point one five" coz after point there is no tens or hundred digits

[@maristodal4119]

It's just for convenience. It doesn't always have to be said like that in informal or learning settings like these.

[@bujintei_kenaith8787]

🤓

[@vaccino3359]

This is a true stfu moment

[@panjak323]

well actually if you look at the decimal as a power of tens, the 15 is valid

[@Doraemon-jm5oo]

🤓🤓🤓

[@sandanth6122]

The aproximate value of these numbers (and this process) can be derived using differentiation

[@Peppa233pog]

Only works well for numbers bigger than 100 inside square root

[@speakingsarcasm9014]

Say, f(x)=sqrt(x), use the first two terms of the Taylor series to approximate f(x+h) = f(x) + h*f'(x)/1! which is reasonable for a small quantity h.

[@DeepanshuGupta-gl2sm]

* THIS TRICK IS BEST WHEN THE NUMBER IS GREATER THAN THE CLOSEST PERFECT SQUARE. *

[@FranklinLee-t3k]

And of course, do not simplify root 103. Since 103 is prime number.

[@-radioactivity]

You can use calculus for approximation.

[@speakingsarcasm9014]

He does that just with a layer of abstraction.

[@f.r.y5857]

Use squeeze theorem

[@TwitterIsReal]

Sprt(103) Between 100 and 121

[@JamesLawrenceSamante-rm7ty]

I cant understand your solution where did you get the 2

[@Aidensk7]

Mention that the closet perfect square has to be lower or else you have to subtract

[@jyotimanhar9752]

I’m in sixth grade and I have a assignment on this and I went through so many videos and spent so much of time until I found this 1 minute short.

[@ChunTeng4Jfivefivefivetwofour]

i think this only works for numbers above 4 because 3 doesnt really work

[@sockfreak2003]

Linearization

[@owenslamansa4378]

Why did you double 10 for the denominator?

[@sscjessica]

Why do we miltiply by 2 here, is this a rule used everytime?

[@Pangethecat]

I tried it but i think im doing it wrong because what i got using this is wrong from the calculator

[@masumraja6300]

Thanks❤🎉🎉🎉😅

[@What_A_Great_Dude]

The estimation doesn't work on other numbers such as 80.

[@zahranf.a.9864]

If the number is closer to the higher root (for ex 80, 15, or 24) you can just estimate it like 8,9 for √80; 3,9 for √15; and 4,9 for √24. Or if you can sense it, it's easier.

[@What_A_Great_Dude]

@@zahranf.a.9864 That's what I'm trying to say

[@zahranf.a.9864]

@@What_A_Great_Dude You can use instead of putting 2n on the denominator, you put 2n+1. For example √80 = 8,(16/(8×2+1)) = 8,(16/17) which is 8,941 and we know that √80 exact answer is 8,944. I read this on the other comment and i forgot the explanation.

[@What_A_Great_Dude]

@@zahranf.a.9864 Oh yeah I totally forgot about that approximation

[@joanntunandao3631]

I did not understand when its in the newmeratir and denominator

[@graceyamba9502]

How did get the two

[@prithvisinghpanwar007]

very great trick for exams

[@potato_1523]

If the whole number is different than 10 like 20, is it 20 × 2?

[@Justkidding09]

√103=10 R=3 10+3/30=1/10=0.1 so 10.1

[@kaushik9539]

i'am in class 11th our teacher cleared this concept within 39secs #lord physicswallah

[@mofarah1325]

😂😂

[@AnthonyDoesYouTube]

And where did we get the 10 * 2 from???

[@e_pi_i_is_-1]

It came from differentiating the √x function. In short terms it's a linear approximation of x.

[@BOMBON187]

Its kind of works and is very limited. Doesnt really work high 3 or 4 digit numbers.

[@SivakamiMurugesh-v1u]

But if we take √24 then √24 = 4 + 8/8 which make the ans 4 which is wrong.

[@hael03]

broooo you saveddd meeeeee😭

[@novayarussia]

Just use the Taylor expansion, geez

[@Unknown_1096_]

(1+x)^n=1+nx+...... acurate upteo 2 to 4 decimals

[@abnormal4471]

Teacher boutta give you -1 score on the essay for that

[@eggi4443]

why is the 10 doubled?

[@Bjowolf2]

It comes from the derivative of f(x) = x^2, which is f'(x) = 2x (!) This function then gives you the slope of the tangent at every point of the graph of f(x) ( a parabola ), and this line is then used as a local linear approximation around the x-value, where you already know the square ( it doesn't have to be an integer btw. ) From there it's easy to calculate backwards to get a good estimate of what the square root of the number in question is - and this involves dividing the relative distance with 2 instead.

[@mega9178]

I don’t know wtf he said but double the closest square root of the original nimber

[@WerewolfLord]

It should be double plus 1, so for 10 it should be 21, so the approximation for 103 should be 10 3/21 rather than 10 3/20. The difference between two perfect squares is the square root of the smaller number +1. If you add up consecutive odd numbers (starting from 1), you get all the perfect squares. n² is the sum of the first n odd numbers so the difference between it and the next square is the "n+1"-th odd number. It's similar to taking the derivative graphically. We're taking the y=sqrt(x) curve and approximating it with straight lines connecting the perfect squares.

[@vageeshalalkarna1206]

Simple double means multiply by 2..

[@jerrymemes2207]

Finding the square root of a number involves setting the area of a square to (in this case 102) and finding a side length. You can subtract the area of the nearest whole number squared (in this case 10) leaving you with two rectangles with an area of x * 10 and a square with an area of x^2. These two rectangles and the square have an area of 2 and x is a decimal. Because a decimal times a decimal is a smaller decimal we ignore it for ease and this gives us the equation 2*10*x = 2 which we add on to the 10 to get the estimated value of sqrt 102 Here is the estimated equation beginning to end: 2ny + n^2 =102 2×10×y +10^2 =102 20×y = 2 y = 1/10 And n+y = sqrt 102 Here is the actual equation without estimating following the same steps: 2ny + n^2 + y^2 = 102 n + y = sqrt 102

[@Unknown_1096_]

Instead of this you can use bionomial theorem

[@VeggieRice]

this is kind of a primer for folks who don't have a method, or at least that's how it came off to me. people think in diff ways. binomial is efficient, this works too

[@Unknown_1096_]

@Protocol 7 yep u are right but for harder numbers long division method takes forever

[@dinokiller9186]

Isn't this the local linear approximation

[@penguin22penguin22]

PLEASE I NEED HELP WORH DECIMALS NOT WHOLE NUMBERS

[@Mh-gq7rs]

Ok but why did we need to double the 10

[@sasmitagrawal]

Root 120??

[@adityamishra0706]

Do √35

[@BAD_LS]

Following this instruction it would be 25, root of 5 5+10/(2x5) = 6 5.9161 would be the almost exact result

[@andrew___3233]

​@@BAD_LS can you do it for 99, please

[@BAD_LS]

@@andrew___3233 ok, but you should try it by your own :D as we can't use 10 squared for 100, we use 9, and know that the square of 9 is 81. What means we have something with 9.x 99-81=18 ; x=9*(18/(2*9))=9 , what means we have 9.9 and 9.95 is aprox. the result Hope you can do it too

[@andrew___3233]

@@BAD_LS thank you

[@ZeonplayzYt]

@@BAD_LS please root 24 it’s coming 4.1 for me

[@Heywjebbs]

It's just binomial approximation 😮‍💨😮‍💨

[@pukeface7371]

Dude, schools out don’t make me think

[@codygolden7074]

Was on board until 10.15 I'm confused how'd he get that sorry I'm a bit slow

[@yeahgamer6190]

"cockulator"

[@cindorranimates]

Bro t bro turned into roplex gaming

[@EYX6437]

I can smell the Sharpie from here

[@SanskarYadav-yw2cu]

Not working with number root 15

[@LB0206]

I feel like either I'm getting something wrong or it doesn‘t work for _/99

[@WerewolfLord]

Yeah, if your number is (n+1)²-1 like (9+1)²-1=99, the estimate is n+((n+1)²-1-n²)/2n so it is n+2n/2n or just n+1. This gives a result for 99 as 9 18/18 which is just 10. The number to divide by isn't 2n but should be 2n+1 so the estimate for 99 should be 9 18/19, or 9.9473... while the real value is 9.9498... .

[@mohammedkerrou]

Bro i dont get it i feel dizzy 🥴😵

[@toroddlnning6806]

pick a more difficult number and you will see that this is not so easy

[@Dio-qr6mu]

5.2

[@thilaks5666]

Used this method for √61 Calculator=7.81 His method=7.85 Do i made any mistake?

[@silasfilms]

bro this doesn’t really work for numbers less than 8 or so

[@kimmson6356]

Doesn't always work. Try 190

[@sosm917]

help with square root of 5

[@sompallisunitha3533]

Do it for 93

[@Kali-bs7oj]

That’ll get you the estimate 9.66666 The actual root is 9.64 so not bad

[@perialis2970]

me in the back of the class choosing which side i should go to My friends (the jedi): we shall remember what each root is approximately up to 200. My smart friends (the dark side): we will calculate every step of the way.

[@mothserp]

Where did he get 10•2 from?

[@WaterDrinker5000]

For this technique, you multiply the number by 2. It is always 2.

[@muz7958]

The dot means multiply, and he said we have to double 10 so he just multiplyed by 2.

[@probropalzlive6961]

Here's a better explanation: Newton's Method is a method to approximate zeroes of a function. For a guess 'a' such that a function f(x) evaluated at x=a is close to zero, a better guess 'b' can be approximated by taking the formula [a - (f(a)-f'(a)], where f'(a) is the value of the derivative of f(x) at a. When we take x=sqrt(103), what we are really saying is find x such that x²-103=0, through algebraic manipulation. Taking f(x)=x²-103, we then get f'(x)=2x through basic calculus. Then, it is simply a matter of recognizing that the square root of an irrational is closely approximated by the square root of the largest perfect square and a little bit. Thus, by taking our initial guess as 10, we can easily see that b = a - (f(x)/f'(x)) = 10 - (100-103)/(100×2). Because the f(x)/f'(x) part is always negative because the first guess is lesser than the actual number, he automatically converts the minus sign to a plus. This method is applicable to nearly all functions. Hope it helped!

[@Bjowolf2]

sqrt(b) ~ a + ( b - a^2 ) / 2a, where b ~ a^2 It's the first linear part of the Taylor approximation of the squareroot function around the value a^2 - simply using its tangent at this point as a local linear approximation of the squareroot function. This works really well, when the second part is "small"- typically numerically less than 0.3 or so, but it can still give you a fairly good idea for even larger steps. sqrt(110) ~ 10 + 10/20 = 10.5 Real value: 10.488... So it's only off by c. 1 pct.

[@Bjowolf2]

​@@probropalzlive6961 Correct, and well explained, but you should have "10 x 2" in your denominator instead of "100 x 2" 😉

[@jimjenke3661]

Lost me at the fractional denominator determination.

[@BryceStLouis-et5gn]

Same. Why doubled, why not trippld?

[@probropalzlive6961]

Here's a better explanation: Newton's Method is a method to approximate zeroes of a function. For a guess 'a' such that a function f(x) evaluated at x=a is close to zero, a better guess 'b' can be approximated by taking the formula [a - (f(a)-f'(a)], where f'(a) is the value of the derivative of f(x) at a. When we take x=sqrt(103), what we are really saying is find x such that x²-103=0, through algebraic manipulation. Taking f(x)=x²-103, we then get f'(x)=2x through basic calculus. Then, it is simply a matter of recognizing that the square root of an irrational is closely approximated by the square root of the largest perfect square and a little bit. Thus, by taking our initial guess as 10, we can easily see that b = a - (f(x)/f'(x)) = 10 - (100-103)/(100×2). Because the f(x)/f'(x) part is always negative because the first guess is lesser than the actual number, he automatically converts the minus sign to a plus. This method is applicable to nearly all functions. Hope it helped!

[@B-Yself]

You don’t need to estimate there is already a method

[@zeze9899]

√103=10.15

[@angeloalvarez5520]

Why are we doubling ten to make twenty??

[@Guppy_Flower]

bruh,,,how the fuck would i know the previous perfect square of the number 7414??

[@WazzupWazzup-gg7jd]

It’s actually 10.148

[@seraphym8683]

Nice

[@elevatedemotions4548]

Much easier way than this

[@quertiqr]

I got really good at this in high school and said 10.4

[@amielmagday2646]

how the hell he got 10.15 in 3,20, and 10

[@Ligmaballls987]

Bro be teaching class 6 techniques

[@oliverserta]

as a 10 year old I kno algebra times divided a plus I just need how to do this and I'm finished put my brain can't find how to do this

[@satyanarayanaprashanthi3403]

Why you write 20 their

[@Hi_Brien]

10.15-ish is my guess before watching.

[@Sushmas888]

I didn't understand

[@adamgwinnowicz]

Why does this work

[@googelman]

does the same apply if there are numbers outside the radical symbol like 3√56 ?

[@gondoo_plays7627]

Try it

[@kruGrockz]

You just need to bring the 3 back inside the root by squaring it and multiply with 56, which makes √504. Now you can do it.

[@blist3826]

So would this answer be correct or incorrect?

[@maksschmidt]

Depends on what you’re doing. If what you’re doing allows for some error, then probably. Also the estimate gets more precise with larger numbers. I think the worst case with positive integers is a little under 6.1% error. So if what you’re doing can tolerate up to 6.1% error then this method works great.

[@Slash4608]

It is correct you can check calculator for confirmation

[@genepirate7114]

just taylor series expand it...

[@nackro3476]

Let’s introduce Taylor series

[@_jintojoseph_8106]

What about√18 it won't go like this

[@Standard____]

This method gives you 4.25 which is very close to the actual 4.2426