How to Approximate Square Root of a Number

[@ricardogarcia9847]

The guy is amazing. These lessons brighten my day and let me recall all of my college years of math.

[@420sakura1]

Yeah. All of them are basic maths but he's teaching them with no bells & whistle and straight to the point

[@JennyWoo-vg1mu]

Thanks for teaching me. That"s is interesting process. Enjoy your every lesson again and again. This is NOT the answer I learn, but the PROCESS I truly enjoy it. So amazing !

[@paultvshow]

What? Did you learn this in college? I learned this simple trick in middle school.

[@dhwang101]

Yup you learn this in school and Fun story, this is how Richard Feynman beat the abbacus 😂

[@alishozi1975]

8

[@ComposedBySam]

For anyone wondering why this works, these are the first two terms of the binomial expansion.

[@yousciencelab5307]

Of course. Ralph-Newton also works

[@KYosco]

​@@yousciencelab5307Newton-Raphson method

[@victorpaesplinio2865]

Or the famous Taylor series of sqrt(x+h).

[@Ennar]

This works because derivative is defined so it gives the best linear approximation: f(x) = f(a) + f'(a)(x-a) + o(x-a). In this particular case, sqrt(n^2+x) = n + x/(2n) + o(x).

[@davicanto2899]

I have a better method: n=number s=closest perfect square number srqt(n) ≈ sqrt(s) * (3n+s)/(3s+n) Exemple: sqrt(138) ≈ sqrt(144) * (3*138 + 144)/(3*144+138) = 11.74736

[@marytredinnick3366]

Im 60 and have BS in Elementary and Special Education.Now that Im retired I'm enjoying learning algebra. I soooo wish i could have had you as a teacher. You're so good at explaining each step. Thanks so much❤

[@bt2gr8k72]

Bigger the number,it's relatively more difficult to find next number which is a perfect square. Also, error tolerance varies accordingly.

[@IrrelevantGuy]

Yeah, that's a given. But this works great for figuring out the approximate value of smaller numbers, which will of course help you become faster at hand calculation (this is helpful especially for us Asians haha)

[@MohamedBenamer940]

Use Halley method instead

[@brodymiller9299]

At large numbers the second derivative of sqrt(x) is lower, so this becomes more accurate even if you don’t get the exact closest perfect square

[@dhess34]

Thanks, Captain Obvious.

[@b.a.dieudonne4501]

Wow I started looking at your videos to keep my mind sharp. I am now forwarding them to my children. Amazing - you are a fantastic teacher thank you 🙏🏽

[@mrhtutoring]

You are so kind

[@gerrysecure5874]

Nope, he just says how to do it, not why it works. Draw 2 overlapping squares fixed at one corner, the correction are the 2 borders.

[@asingh9540]

Me too.... hahaha 😅

[@aaronaaron5013]

This is magnificent... the elegance of mathematics never stops marveling me.

[@EEEEEEEE]

E‎‎‎‎‎‎‎‎‎‎‎‎‎‎

[@aaronaaron5013]

@@EEEEEEEE EEE ?

[@anneashley5110]

Buy the Trachtenberg book on mathematics. Devised by Polish mathematician whilst POW in Germany. Wrote theories on cigarette papers to stay sane. Escaped and eventually married rich Countess then dedicated his life teaching under privileged children easy ways of mathematics in academies in Switzerland.

[@reigen-]

I like your teaching ❤🎉

[@mrhtutoring]

Thank you! 😃

[@thalesnemo2841]

So straight forward and simple ! My recollections are that the textbook method was too complex !

[@MikolaZak]

Я не розумію мову викладача, але він так добре пояснює, що інтуїтивно зрозуміло все. Дуже добрий викладач!

[@brettkowalski]

This guy is bringing back my migraines I used to have when I was doing math competitions in middle school and high school.

[@YourAveragePlay3r]

Really appreciate this guy popping up on my feed once a day, extremely helpful.

[@Martini16REAL]

This is the coolest thing ever. I have tried to figure out how to find an approximation of a square root and now I can.

[@ChrisM541]

Very well explained. If only we all remember that we can use this 'way of thinking' in many, many more cases in our lives.

[@kevintarrant5854]

I was wondering the same thing, but can you actually name one. ?

[@honeyartstudios]

These daily clips might help me overcome my math trauma

[@Kelsey-qh7rh]

You taught this WAYYY better than my online math lessons

[@JSSTyger]

I would like to offer a similar method. Let G = guess and E = error and we want to find the square root of C. C = (G+E)² = G²+2GE+E². With a small enough E value, E² will be close to zero. Our equation now becomes an approximation. C~G²+2GE and E~(C-G²)/(2G). Now that you have your approximate error, simply add it to G to get your final estimate. If you choose to get more accuracy, you can revise your guess. This method can also be extended to cube roots, etc by knowing the binomial expansion formula and eliminating the terms that have powers of E greater than 1.

[@sussybaka6921]

Lol 🤣

[@navamgarg]

Thankyou so much, you r in school, college or above?

[@JSSTyger]

@@navamgarg I graduated with a minor in math in 2004. I took over 3 years worth of university math. But the funny thing is, I learned this on my own AFTER my schooling because I kept my old books. There was an interesting section on approximating square roots in my Numerical Methods book that never got covered and this was it.

[@navamgarg]

@@JSSTyger Great, where are you from? And i guess you are 39 years old, is it right?

[@JSSTyger]

@@navamgarg USA

[@anuragswain5910]

Just wow 😍.....Quite amazed by your simplified teaching style sir..... Please keep us guiding through the gigantic universe of Maths

[@mrhtutoring]

Thank you for the nice comment!

[@krishnagarg1313]

Thank you sir for such helpful tricks😊😊

[@rinkudamanrd]

ah. this is basically using the derivative to make a tangent line approximation simplifed! nice video!

[@stephenhousman6975]

To those of you taking calculus this approximation is a first order Taylor series for the square root of x.

[@عبدالرحمنالحربي-ل4غ]

شكراً لك. شرح جميل أعاد لنا الذكريات القديمة في المدرسة

[@kingminato5219]

We can use the tailor's series , f(x) = ✓x f'(x)= 1/✓x For each number we must find the perfect square a which is closer ( x > a) f(x) ~ f'(a) + 1/2 × 1/✓a × (x-a) f(138) ~ f(121) + 1/2 × 1/11 × 17 f(138) ~11,7 We get the first decimal of ✓x , for more precision check the tailor's series .

[@luisclementeortegasegovia8603]

Excelent 👍 Beautiful algorithm.

[@mrhtutoring]

Many thanks!

[@jmolvera8337]

Super, I will memorize this procedure! Thank you professor!

[@mineapple1165]

This is actually really useful because sometimes i forget my calculator at home and everyone in my class doesn't want me to use the one they own Thank you sir

[@mrhtutoring]

Very happy to hear that it's useful 😀

[@jpo3811]

Multiple succinct high quality demonstrations.

[@stonetee1845]

you are indeed a maths genius. pls where do you get these shortcuts from. it's just incredible. wow

[@NicholasOfAutrecourt]

It's a derivation from differential calculus.

[@stonetee1845]

Okay, l see

[@stonetee1845]

Thanks

[@Daniel31216]

He used the first two terms of the Taylor series. It's not normally used for stuff like this, but the series can be used to approximate any function you want.

[@BBalasa]

Wow , great universal solution.. Thank you..

[@kenbihler7175]

finally ! thank you ! I've been doing bad approximations all my life !!!

[@vibushithirthankar]

Thank you sir❤.. this is very helpful 🙏🏻

[@villageidiot2372]

Love these lessons, great explanations.

[@grantgautney7900]

The tapping sound of chalk on a black board takes me back.

[@evilsensei8262]

Bold of you to assume I know how to do fractions 😅. Awesome video!

[@mrhtutoring]

Thanks! 😃

[@olerask2457]

In general f(x) ~ f(a) + f'(a) * (x-a). That is 1. sqrt(x) ~ sqrt(a) + (x-a)/(2*sqrt(a)) 2. cuberoot(x) ~ cuberoot(a) + (x-a)/(3*(cuberoot(a))^2), etc. Fx. cuberoot(75) ~ cuberoot(64) + (75-64)/(3*(cuberoot(64))^2) = 4 + 11/(3*4^2) ~ 4.2292, where cuberoot(75) ~ 4.2172.

[@MyAmygdala_]

Thank you 😊

[@mrydobon]

I think you have to show this Taylor series expansion if you want to teach this approximate solution to a square root. Otherwise it is better to just teach the linear interpolation solution, which is intuitive just by looking at the number line. It is not as accurate, but it doesn't require any calculus to understand. And we're talkin about approximate solutions either way. In this example, 12 + (138-144)/(144-121) = 11.74. That's still a pretty good approximation.

[@niggydiggy3992]

I dont speak mincraft enchanting table

[@humblehombre9904]

WHY couldn’t they have taught us this in high school? Nice one, thank you Teacher!

[@harrys2331]

if you took calculus in high school, this should have been covered.

[@kaichousan8626]

This is just the best method, instead of doing the hard stepdown like approach

[@opufy]

It’s so satisfying the chalk-work

[@mathandcflmao]

That was super helpful!!!! Thanks ❤

[@velmurugank4871]

If I had a math teacher like you back in school, I would have scored better in my Maths exams.

[@sarabana3093]

Nice. My Math's teacher from Iran called this method "BANNA" in Persian which is very use full at Entrance Exam

[@DeepanshuGupta-gl2sm]

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

[@marc-bs8bj]

Excellent estimation method!!

[@vimax3858]

I remember when I was in 4th grade and questioned a teacher how to solve roots that are irrational numbers and they had no clue, now KZbin gives it to me.

[@peterirvin7121]

It's very sad that math literacy is not required to teach elementary school.

[@nowonwoo9528]

There's also one easier method to do this that almost yields the same results, albeit the margin of error is higher than this method. For the sqrt(138), find the closest smaller square root, which is 11. And then find the difference between the perfect square and the number; hence, giving us 17. Make a fraction and put the difference on the numerator while twice the square root is the denominator, giving us 22. This would yield to a mixed fraction of 11 and 17/22, or 11.77

[@WildayMATH]

thanks Prof for the bright explained

[@stevenmiller5999]

That was absolutely awesome thank you

[@mrhtutoring]

Glad you enjoyed it

[@jwlee654]

Awesome. I am not even in a math intensive job or course of study, but I will definitely use this!

[@ART7N23]

Sensai ur awesome 😫😫🙌🏻🙌🏻🙌🏻

[@aimranehd]

this is really handy! especially because i have a national math exam at the end of this year, and one of the subjects that will be included in the exam are roots, on top of that they dont allow calculators. thank you so much, mrhtutoring!

[@mrhtutoring]

Good luck on your exam!

[@RivaldoDaviero-dc2vx]

This is really helpful wow😮

[@carolinehammes]

Excellent!

[@ionescunicolae473]

Amazing ! Thank you very much.

[@murdock5537]

Simply great, many thanks, Sir! 138 = (12 - x)^2 ≈ 144 - 24x → x = 12 - 1/4 = 11,75 (don't use x^2, it's very small btw)

[@AL.BUNDY.]

0:56 I like it

[@yvesdelombaerde5909]

Then iterate. Newton-Raphson algoritm. You can generalize to nth root. If a is an approximation and e stands for error then Sqrt(x)=a+e squaring both sides x=(a+e)²=a²+2ae+e² Assuming e is small wrt a: x~a²+2ae e~(x-a²)/2a Replace e, get a better approximation and iterate.

[@victorpaesplinio2865]

This is the first order Taylor expansion of sqrt(x+h). It gives better results smaller the h, unless you go a bit further and add a second order correction √(x+h) = √x + h/(2√x) - h²/(8h^(3/2)) + O(h³)

[@pattyguy]

are you casting spells on us

[@predatorff8274]

f(a+h)= f(a)+h. f'(a+h) use derivatives and consider f(a) as perfect square which is closest to 138 and it's done ; a=144, h=-6

[@n.gineer8102]

If I recall this technique was used by Hans Bethe as described by Richard Feynman the part of his book about working on the Manhattan Project.

[@billjohnson3858]

Linear interpolation is easier to remember and just about as close. Since 144-121=23 and 138-121=17 so 17/23 gives us 0.739 which is the fractional part of 11.739

[@ionbintf]

"Pretty DARN close"...

[@TechnocratSohail]

Very nice..It is quite useful..

[@mrhtutoring]

Thanks a lot

[@Sg190th]

I like this. Much simpler than using the Newton method

[@markyujoco1912]

That is also Newton's method, he was just clarrifying it hahhaahah ....... 😅😅😅😅😅

[@michaelsibson7941]

Careful bout how you talk bout newton.

[@MathTidbits]

@@michaelsibson7941 newton gonna sue you ?

[@tommymyers3183]

You make it look easy.

[@ronaldbell7429]

I'd be interested to see how that equation was derived, to be honest

[@yousciencelab5307]

Ralph-Newton iteration gives you this. Let √N = x N = x² N - x² = 0 f¹(x) = -2x x₁ = x₀ + f(x)/f¹(x) For this case, N = 138, x₀ = 11 or 12, which ever you want to start from. x₁ = 11 - [138 - 11²/(-2 × 11)] x₁ = 11 - [138 - 121/(-22)] x₁ = 11 - [17/(-22)] x₁ = 11 + 0.77 ≈ 11.77 Binomial expansion can also be used

[@aashutoshgoswami344]

It is derived using the derivative of √x Which is 1/(2√x)

[@blurr1903]

It’s newtons approximation formula of f(a)+f’(a)(x-a)

[@floris2042]

its the first order taylor expansion of the square root

[@quandarkumtanglehairs4743]

Love it! Very useful, thank you. Another tool in my box of...tools. I have other boxes, not just of tools, I guess is what I'm saying.

[@mrhtutoring]

Glad it was helpful!

[@erikblaas5826]

I have learned it another way... The part of writing it down is more complicated, but there is hardly making square root calculating in it.. mostly devisions and adding or substracting...

[@kanye_ne]

This is definitely an interesting way to do it. I was always taught to use L(x) to approximate the square root of a number.

[@jacksonsmith2955]

isn't that... exactly what he's doing? assuming that L(x) here means linearizing sqrt with a first order taylor poly

[@Michael.storen]

This video cured my depression

[@mrhtutoring]

Whoa!

[@maxwiebe357]

for a rougher but quicker one, take the lower of the 2 close roots (121), take that as your whole number, multiply it by 2 and that’s your denominator, find the difference of your original and the perfect square and that’s your numerator 11 and 17/22 = around 11.71

[@s_d_0_07]

These Asian teachers are crazy bruh

[@OmahcronOmni]

🤯 wow thank you for sharing this.

[@mrhtutoring]

My pleasure!

[@KyleAlexJohn]

This is beyond useful...

[@javierjones6682]

Great video sir

[@troyjudge8466]

Wow! A actual chalk board, I though that was gone forever.

[@MaltaMcMurchy]

The next time I have a math problem I will go to this guy for help, because after watching his videos, I have learned that he makes a math problem, no problem and no problems help me to go to sleep at night.

[@taciodasilva8291]

There is a 12 years old girl in Bradil that is famous because she developed a method to calculate the square roots where the roots are integer numbers. I belive, now, by coincidence or not, this is the same method, just simplified in her case.

[@vaishnavisardar]

Just do thia :- if you want to find the square of 95 :-do 5 square write one no and carry second then do 9 square and double up the value and add the carried no.

[@christianrazvan]

Maybe next video will be a general case and with a proof? Thank you!

[@AmarDas-wg2sq]

You are phenomenal Is there any couses you provide

[@mrhtutoring]

Not at this time~

[@AYA-GAMING_SHORTS]

There's a 2nd way too *"LONG DIVISION METHOD"*

[@troycongdon]

When I was in middle school and they introduced square roots of perfect squares to explain the concept then told us to use a calculator I asked how people did the work before calculators were a thing. The math teacher wouldn’t (couldn’t) answer the question. As an adult I sorted the answer out but I wish the teacher could have told a younger me this method.

[@420sakura1]

Boomers had slide rulers. Mechanical Calculators for professional work.

[@Mehediprodhan1613]

Those who are wondering how does this method work. This is newton raphson method, it converges easily and a part of numerical calculation.

[@Burntheadversary]

Amazing!

[@mahmodi5timetolearn]

You are brilliant

[@joseroyal877]

Will test this out with another number. Interesting. I wish I had you as my maths teacher.

[@yvesdelombaerde5909]

For 3th root, do the same with the 1st term of (a+e)³ dropping the e² and e³ terms etc.

[@waffle8364]

he's got that good chalk

[@rekkoha-dk1nh]

Thanks, Mr. H.

[@al2o342]

Όμορφος τρόπος σκέψης. Πρακτικός

[@deanmoncaster]

Wouldn't it just be easier to take 138 from 144 then take that result from the root rather than adding a negative which most people can't do.

[@mrhtutoring]

Yes, certainly. I did post another shorts video on the method you commented.

[@99mistarip]

Wow easier now than in middle school

[@asseroy]

L(x)=f(a)+f'(a)(x-a) where a is the closest perfect square to the number we're trying to approximate (a=144) L(x)=√(144)+(1/2√144)(x-144) L(138)=√(144)+(1/2√144)(138-144) =12+(1/2×12)(-6)=12-1/4=11.75 √(138)≈L(138)=11.75

[@felixasanteantwi]

Nice please is there a method to get the perfect square...

[@flashdash5761]

The best part is, this formula is in fact derived using calculus.

[@AftabAlam-yw4eq]

The method we learned was If you want to find square root of X. The it will be (X+Y)/(2.Z) Y is the nearest perfect square and Z is the square root of Y. Which is a more simplifide form of this equation.

[@cwalenta656]

Somewhere between 11/12, I think, but armed with phone I don't estimate anymore, you can actually just straight up google that or even ask siri or alexa.

[@mayankkhi]

Wow how didn't I got to learn this in school😮😮

[@yuridxh]

You can do like this too: 11

[@Test-gs4ni]

This is not science it is majic.

[@Alalpog]

Nicee. Thanks for this.