Calculating Square Root by Hand (Early Grades)

[@bowlineobama]

I learned this method 50 years ago from my Chemistry teacher. I later found out that this method is based on Binomial Expansion (a+b)squared. Not many knew this long division method in school these days. Thanks to KZbin, this method has been revealed. I love this method.

[@Necrozene]

It was not taught generally in class, but my primary school maths teacher taught me!

[@pbworld7858]

@@Necrozene When I was in primary school, nobody even knew what a square root was.

[@Necrozene]

@@pbworld7858 I was very lucky I had a few excellent teachers who fed my curiosity.

[@Necrozene]

@@pbworld7858 I even had a teacher who taught me the formula for the Nth Fibonacci number with the phi in it. A friend was verifying that by hand in Chess club! And it worked!

[@Necrozene]

But he never bought Cantor's diagonalisation!

[@johnbutler4631]

My dad showed me this method many years ago, and I've never met anybody else that knows this method. This is the first video that I've run across that explains it step by step.

[@SpiritofMathSchools]

If you watch any more of our videos, please let us know if your dad would approve!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@BubbaGooch1]

I too learned and later forgot this method years ago. I was amused that the presenter used chalk, which broke, while working the problem. That really brought back the 60s tome.

[@SpiritofMathSchools]

You know the problem is hard when the chalk breaks.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@SpiritofMathSchools]

@@bowlineobama Care to expand upon your point? The more perspectives the better!

[@halesbellss]

@@bowlineobamashe asked you to explain.

[@fre0n1978]

My high school biology teacher (this was 30 years ago and he was near retirement at the time) showed the class this process. I was awestruck, but couldn’t remember the process. I’ve been idly wondering how to do it ever since, and this is exactly it! Thank you so much!

[@alittax]

It's so wonderful to read comments like these! People looking for a long time for something they remember (like a song, recipe, or this, or something else), and it emerges for them from the Internet!

[@SpiritofMathSchools]

Happy to help reel in those (almost) lost memories!

[@SpiritofMathSchools]

@@alittax One of the many reasons we love being able to share these videos online to people around the globe! Well said.

[@HedelTorres]

Back when i was in grade six (in Canada), i went to Ecuador for the summer. I was bored as everyone was in school. So my mom enrolled me in school there for a couple of months. In that short period, my math skills jumped to a Canadian grade 8 level. I learned how to do square root by hand. When i got back to Canada, i went back to learning long division, and in grade eight, we learned to use calculators.

[@SpiritofMathSchools]

The lost art of doing mental math or calculating solutions to challenging problems by hand is one of the reasons our parents say they keep coming back!

[@ianboard544]

We learned this when I was in 6th grade. Years later, I used Newton's method: start with any reasonable guess, then iterate: new guess = 1/2( guess + number/guess). It converges quite rapidly.

[@SusanaSoltner]

Heron's formula

[@ianboard544]

@@SusanaSoltner I didn't know that - thanks.

[@3Cr15w311]

The advantage of the "divide and average method" is if you make a mistake, it will work out if you don't make more mistakes. With the way presented in the video (the way I first learned square root), any mistake will ruin the result from there on.

[@impCaesarAug]

@@SusanaSoltner Heron's method. Heron's formula is the area of a triangle, in terms of its sides.

[@SusanaSoltner]

@@impCaesarAug Thank you for this distinction.

[@boeingpilot7002]

This is how I remember doing it in high school -- many moons, ago -- thanks for the refresher!

[@commoveo1]

Feel same. Been a bit and I feel as you, just a little reminder to do elementary problems! Want a nice square concrete pad and although few concrete workers remember and quiet likely never did by the fun of me when I mentioned hypotenuse they get a big laugh at there 10th grade drop out lol. He who laughs first laughs last right lol. Bless their hearts lol. I always like the 3,4 and 5 or even double the number helps. What I really love is running a say three foot diameter pipe through a floor system lol. Usually take my measurements home lay out on piece of cardboard then bring in to work and always fits so nice nothing even gets mentioned lol but that’s fine huh. I will give anyone who may be interested the Pipe fitters hand book is small and like anything the more you do it you get even noticed less but who wants noticed if it all works nicely. I was a Union Ironworker and modest. Again thank you for the refresher, very nice ❤️. Calculators are very handy lol.. Left 4 men to form up for a metal building and wanted the exterior sheets to run down the side of the concrete pad to eliminate water 💦 running inside the building. Many ways of laying out and having one nice square corner sure simplifies ✌🏼. Sincerely Grateful, HB

[@bowlineobama]

@@commoveo1 This method is based on Binomial Expansion (a+b)squared method.

[@lesnyk255]

I learned this method in 7th grade, back in 1962 or -63. It wasn't part of the curriculum, but I asked our teacher, Mrs Galloway, if there were such a manual method, and she showed me after class. I'd long since forgotten it when I stumbled across this video. The Babylonian method is another way - much simpler to flowchart, but involves ever more lengthy long divisions.

[@johnchristian7788]

What was part of the curriculum? Square root using a log book or square root using a calculator? Did you use a calculator in class in 1962?

[@lesnyk255]

@@johnchristian7788 Consumer-grade electronic calculators wouldn't be invented for another ten years. We were probably shown where to look up tabulated values in a handbook. Use of log tables wasn't introduced until high school (grade 9-10). My dad showed me how to use a slide rule at some point, but I don't remember when. Geez, this was over sixty years ago - I don't remember when they taught what.

[@johnchristian7788]

@@lesnyk255 It's funny to think that even before calculators became popular, they didn't teach square root by pen and paper. They should really include in the curriculum in all countries. I used to love using log tables.

[@lesnyk255]

@@johnchristian7788 Well, personally, I wouldn't go back to using log tables, slide rules, or manual typewriters except maybe at gunpoint. There are easier ways to get rough manual estimates of square roots if you've left your calculator or iPhone at home - polynomial approximation, for example, or the Babylonian method. This video was a bit of a nostalgia rush - 7th grade, Walpole NH JHS... long time ago....

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@spottymaldoon]

I had forgotten this completely how to do this. I tried doing cube roots the same way (except taking three digits instead of pairs and multiplying next stage by three instead of two) and it works. I don't think many kids learn this today. Thank you for a very clean, clear explanation of a rather tedious process!

[@ralphhenderson5276]

I went through five bad videos before I found yours. One guy even helpfully blocked the view of the whiteboard while he explained what was on it. It took about a minute to catch on watching you. Thank you!

[@SpiritofMathSchools]

We're thrilled you found this helpful! If comprehension happens quickly, it means the approach and teaching strategy is the right one.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@bowlineobama]

@@SpiritofMathSchools This method is based on Binomial Expansion (a+b)squared method.

[@thecalculator80]

I got a PhD in physics 50 years ago, and this is the first time I have seen a handcalc. Using easier numbers, this would help a student understand what a square root is. That readies them for using their calculator knowledgeably. There is so much junk math on KZbin that are just PEMDAS trick questions, not real math. Thanks for the real thing.

[@SpiritofMathSchools]

We appreciate such a learned perspective! Our goal is to get kids to use their mental math muscles as much as possible instead of relying primarily on a calculator.

[@douggale5962]

My schools never taught this, and I always wanted to know how to do it by hand.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. It is very easy. I can show it to you in a few minutes. This teach makes longer than it is.

[@DamirKipkaev]

Our math teacher showed us this method in extra classes. Everything was almost the same, except that she said that you can not only multiply by 2, but also add. For example, 48 * 2 = 96. But you can get 96 by adding 8x + x (88+8 = 96), which was usually intuitive, since we put two dots when we were guessing the number for multiplication. Exactly the same in the second case: 487 * 2 = 974, but you can get the same thing if you add 7 to 967. Thus, 967 + 7 = 974. It always works. That is, once again. When you have decided on a digit, multiplied, calculated the difference, and you need to multiply the top number by 2, we don’t have to do this. You can take the number that was the last one on your left and add it with the digit that you put the last (its own last digit).

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@DamirKipkaev]

@@bowlineobama Thanks =)

[@TakeAbackPak]

I learned this almost 45 years ago. Thanks for refreshing my memory! Wonderful.

[@SpiritofMathSchools]

Happy to help provide you with a blast from the past!

[@vanlepthien6768]

I learned this from my 3rd grade (4th year) math teacher. He made math fun. Subsequent math teachers varied in quality, but I didn't have another who was that good until university. Even if you pick a number that is too high for the next step, the algorithm is self-correcting.

[@pietergeerkens6324]

I too leaned this in Grade 3, at age 9, from my Dad. His explanation wasn't quite as tight as one now finds on the internet, but was sufficient for me to have some fun.

[@bowlineobama]

@@pietergeerkens6324 This method is based on Binomial Expansion (a+b)squared method.

[@cipherxen2]

Instead of doubling you can add the left hand side number e.g. instead of calculating 2*48 we can just add 88+8=96 and at next step 967+7=974 etc

[@robertstuart6645]

Glad you presented this exercise. I remember learning the technique in junior high, however I feel I was starting to forget. Thanks much.

[@SpiritofMathSchools]

We all need a little refresher from time to time. Glad we could help Robert!

[@RndomAhhKid]

You're a top G miss Kimberley, this saved me in the grade eight cambridge checkpointnt test

[@ve2zzz]

An alternative i always use to get the leftside multiplier: Use the preceeding number and ADD the last top numeral. In the Miss Kimberlys example, 1 - Add the leftside '4' to the top '4' to obtain 8. 2 - Add the leftside '88' to the rightmost top '8' to obtain 96 3 - Add the leftside '967' to the top rightmost '7' to obtain 974 And so on. Adding a single-digit number is easier to calculte than multiplying by two.

[@Reignspike]

I was never taught how to calculate square roots. When I was in grade school, I tried a few different ways on my own, and they ended up being very much trial and error. This is a more refined approach that improves on what I figured out, but I see that it still involves some. Thanks for showing it!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. It is very easy. i can show you in a few minutes. This teacher makes it look longer than it really is.

[@Muck-qy2oo]

It would be good if you would explain where this method is comming from. The binomial theorem. One can also use other algorithms as herons method.

[@Jimothy-723]

technicaly, the source of this math is Euclid.

[@tomvitale3555]

I agree. I'd love to see the proof behind this method.

[@Muck-qy2oo]

@@tomvitale3555 a²+2ab+b²

[@robertveith6383]

* coming

[@jwm6314]

Title says "early grades." Clearly you are on the wrong video.

[@gdurandeau1143]

I appreciate your great and simple explanation.

[@richardhole8429]

I have learned this method, for amusement, some number of times without ever having to memorize it. Calculators are king now. Thanks

[@SpiritofMathSchools]

An over-reliance on calculators makes your math muscles weak. We always encourage our students to learn the core concepts and do the arithmetic mentally or by hand whenever possible

[@richardhole8429]

​@@SpiritofMathSchools I tend to agree. Teachers can choose values that can be computed in the head or simple multiplication and long division on paper. Real world math rarely has those convenient numbers. Calculators, I would argue do not make one's math weak as doing the calculations is only the lowest skill on the math "tree." Knowing how to set up the problem is where the math skills shine. I suspect that those who do real world math will rarely use hand calculations, and they will quickly notice when their calculator have given faulty inputs. It is important that students learn the basic arithmetical calculation techniques and practice them in the classroom.

[@antoniopango1841]

Superb! I took sometime shifting through many video clips to find out yours with simple explanation how to calculate the square root.

[@SpiritofMathSchools]

We're so glad to hear that! Thanks for sharing 🙌

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@montyhall-vs3ul]

@@bowlineobama Yuh, we know that But you said it about a thousand times anyway

[@midnighttokyo4894]

Watched so many videos but this is the only one that helped me with this, thanks so much!

[@SpiritofMathSchools]

Glad it helped!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@briansykes2806]

I learned this as part of basic maths at primary age, along with fractions, decimals, etc. I think shortly afterwards I learned how to use a slide rule.

[@SpiritofMathSchools]

Ah, yes, the slide rule! The ancient precursor to the modern-day calculator.

[@coaster1235]

it’s neat to see a long division style algorithm for the square root! what makes long division not too bad is that the subcomputations for each digit (guessing the closest multiple below a given number) all involve numbers around the same magnitude, whereas here it seems getting another digit involves a subcomputation with numbers around a magnitude larger than those on the previous step. i wonder if there’s another long division like algorithm where the subcomputations don’t inevitably grow in magnitude? i also wonder if doing this in base 2 would feel simpler?

[@raymondhagerty1769]

The square root of 69 is ATE SOMETHING 😂

[@smaug9833]

My Dad taught me this method when I was in primary school. Remember it to this day, it has been 16 years.

[@Xiuhtec]

Never learned this when I was in school in the 80s and 90s, likely by then they already just assumed everyone had calculators. I appreciate the method, it's very interesting! (Whenever I've wanted to do this without a calculator I've just basically made educated guesses and worked my way to something close, I have squares memorized to about 25 which helps.)

[@hotironaircraftshop]

The first rule of optimization is to identify the operations that take the most total time, and work on making those faster. If this is an infrequently used procedure, i.e. it won't represent a significant portion of a student's life, then why not teach the conceptually simpler approach of progressively refining an initial guess using a binary search?

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@frankyboy1131]

It's interesting to have a procedure that leads straight to the result without corrections, but I ask myself if in some cases, a trial-and-error approximation by nesting with a good estimation wouldn't work faster ... ?!

[@SpiritofMathSchools]

It's an interesting thought. We tend to focus on accuracy and understanding over pure speed.

[@DLV42]

Your way of manually doing square roots is the way my 8th math teacher Mrs Wilker taught us how to do it . I will study this problem and do more problems like it. Lot of WHACK out ways of finding the square roots . They work, but very CONFUSING You is worth your weight in gold raised to 20^20 power . (HUNDRED QUINTILLION) Thank you.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@jabelltulsa]

Very cool video, and you explained it well. It would definitely take practice and would need math-muscle memory.

[@SpiritofMathSchools]

People underestimate muscle memory, especially when it comes to mathematics! That's part of our approach with our students that we notice makes such a difference.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. It is very easy. I can show you in a few minutes.

[@markdstump]

I haven’t started the video yet and I am interested to see it, but I always like to try things before I watch the video. I mean when it comes to math. So in a few seconds, I came up with an estimate that the answer is just shy of 50, since the number is shy of 2500 and then in under three minutes, I came up with a slightly better approximation of 48.77, which I got from interpolation between 48^2 and 49^2 (having already rounded to 2377^1/2, and rounding 103 to 100…and rounding 2401 to 2400.

[@crisismanagement]

Would a slide rule come in handy for the middle calculations? 😊

[@geralynpinto5971]

I really appreciate your clear and methodical procedure, and very pleasant ways.

[@SpiritofMathSchools]

And we really appreciate the positive feedback! Perhaps you could check out some of our other videos and let us know if there's any other topics you'd like to see in the future?

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@JonesFamilyRanch]

SQRT2500=50, 2376.592

[@Vega1447]

Just use Newton's method x=(x+a/x)/2 where a is the number whose sq root is to be found and x is the current approximation to the sq root. And iterate.

[@bowlineobama]

Better use Binomial Expansion Method (BEM). No need for iterations. BEM gives it to you directly in the long run, when you have very large numbers.

[@Vega1447]

@@bowlineobama Easy to find a starting approximation to the sq root. Then Newton's method converges quadratically.

[@jeffw1267]

It's obvious that the square root lies between 48 and 49, because 48^2 is 2304 and 49^2 is 2401. I can use a linear approximation to determine additional digits. 2376.6 - 2304 is 72.6, and the difference between 48^2 and 49^2 is 97, so 72.6/97 is my linear approximation, which gives me the next digit of 7. So far I have 48.7, and I can use linear approximations to double the number of significant digits with each iteration. But everyone knows this.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. This method is much better in the long run.

[@ScottESchmidt]

Thank you for reminding me why I forgot how to do this.

[@korelly]

There exist also the formula of Heron d'Alexandrie. I am trying to find the cubic root algorithm. Many presentators show it in a complicated unclear way.

[@Mikalinium]

Is there one for cube roots? I rlly want to know

[@johnbo964]

I was taught that in 8th grade in 1970. I'm glad to review that.

[@guessundheit6494]

That is flipping cool, I've never seen that before. Usually I would start with 23.76... x 100, because 10^2=100, and the approximate square root of 24 is manageable in your head (~4.8), thus 4.8x10=48. Then I go halfways on each digit (starting 48.5) to minimize the number of brute force cases to test.

[@SpiritofMathSchools]

Love that we could show you a different way of doing things. That's the beautiful thing about mathematics -- so many ways to reach a solution!

[@guessundheit6494]

@@SpiritofMathSchools I'm of the age where calculators were in school bags but slide rules were at home. You think differently when you learn or have to make good estimations, and not depend on technology. Kids should still learn first on slide rules.

[@SpiritofMathSchools]

@@guessundheit6494 100%. There's definitely a difference in their thinking when they have to do the work! They become conditioned not to look for shortcuts in mathematics and really, in life.

[@stevencarr4002]

I got 48.75 in about 20 seconds. Divide 2376.592 by an approximate square root ie 50. That gets you 47.53184. Average 47.5 and 50 and you get 48.75 Trial and error can get you 3 significant figures very quickly by hand.

[@archimedesmaid3602]

The point here was to do this by pen/paper only

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. It is better than guessing.

[@DeckerCreek]

I learned this method sometime in middle school I believe. That would be in the 1960s. Thanks for the refresher

[@SpiritofMathSchools]

Which other videos brought you back to the 60s?

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@kamikaze9471]

How did you get 5 in the blank at the end? The closest square to 20 is 4 squared = 16.

[@davidgrisez]

I am now 73 years old. In my young years I was able to extract a square root using this method.

[@jamesharmon4994]

I was also taught this method 60ish years ago. I had forgotten it and am SO glad for this video!

[@SpiritofMathSchools]

Happy to help you relive the glory days. Now, it's time to pass this knowledge on to the next generation of students.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@martyknight]

You and I are of the same vintage. I learned this method as a 7th grader, long before digital calculators were invented.

[@SpiritofMathSchools]

@@martyknight Great minds age like fine wine

[@johnedwards4394]

I love these forgotten arcane solutions.

[@SpiritofMathSchools]

Glad we could make your day John. Though we wouldn't categorize this type of solution as a secret!

[@finjay21fj]

As always, when teaching, start simple then use a complex

[@alllevelsmath9974]

thank you! well explained!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@VannyPinkLopez]

Amazing. Thank you, teacher!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@Onoelo23gf]

No need to multiply the upper no by 2. Just add the upper no to the divisor, i.e., 4+4=8. Next time, add 8 to 88 and get 96. Either way.

[@bpark10001]

You ought to see what happens if you apply this on binary numbers! You start as usual, grouping the numbers, etc. On the first digit, it is one for the first pair of non-zero digits (there are only 00, 01, 10, 11 cases). To generate the next test number to subtract, you take the answer you have so far, & append to the right of it 0 1. Why? Appending the 0 to the right doubles the number. Appending the 1 is the test digit. Multiplying by 1 is trivial case, just copy the number! If it "fits", write "1" for the next digit of the answer. If not, write "0" & discard the subtract. (You do not cover the case where even "1" is too large. In that case you need to write "0" in the answer & discard the result of the subtract, leaving the partial remainder intact. Then you being the next 2 digits down alongside the existing remainder & proceed from there.)

[@MissPiggyM976]

Very well explained, many thanks!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@robertsullivan2396]

I was taught this in the 50s and still stretch my brain using this method

[@montyhall-vs3ul]

try stretching your brain doing the method for cube roots No one taught this in grades 1 thru 12. But I got interested on my own When the stress closes in, I often find myself evolving the cube root of a number looks like you are about 5 to 10 years older than I

[@someonespadre]

I do this on my antique calculator by subtracting successive odd numbers. That could really lengthy on paper, though.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@SVJIRLI]

Thank you so much Madam ...

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@jethrobo3581]

Wow! I never knew that calculating a square root could be so fun!

[@bowlineobama]

Yes, it is fun. I learned it a long time ago. This method is based on Binomial Expansion (a+b)squared method.

[@JubeiKibagamiFez]

How does this work for a cubed root or root of the 4th or etc? This is what breaks my brain with root calculations.

[@subibrothersbrewing6598]

brings back memories from grade school

[@cipherxen2]

We teach this method in India at 7th grade

[@SpiritofMathSchools]

The method clearly has an international reach

[@holyshit922]

Question for viewers Can you derive such method for cube roots ? If you really understand why this method works you will be able to derive method for cube root yourself I was taught this method in high school once we were solving quadratic equation (to determine if discriminant is perfect square or to approximate roots) and derived method for cube root myself

[@johnchristian7788]

What country did you go to school that they just told you to find the method yourself? I'm suspecting that instead of multiplying by 2 we should multiply by 3 and use cubes instead of squares in the same method. Not sure if I should group by 3 digits 🤔

[@holyshit922]

@@johnchristian7788 In Poland I derived method for cube root for myself and it was not homework As soon as I understood why method for square root works I was able to derive method for cube root Yes you group 3 digits Yes you multiply by three but square of actual approximation not just actual approximation Instead of appending last digit of next approximation you append square of last digit of next approximation To number created in this way you add triple product of current approximation and last digit of next approximation shifted one position to the left (10a+b)^3 = 1000a^3+300a^2b+30ab^2+b^3 (10a+b)^3 - 1000a^3 = 300a^2b+30ab^2+b^3 (10a+b)^3 - 1000a^3 = (300a^2 + 30ab + b^2)b (10a+b)^3 - 1000a^3 = ((300a^2 + b^2) + 30ab)b

[@archimedesmaid3602]

Crystal Clear Maths has a vid on utube where he examines cube roots by the LD method But he concludes that it is not practical beyond a few digits. This is not true. I have demonstrated that with pen/paper I can find the CR of any number to 25 digit accuracy on one side of one sheet. No calculators involved, no separate worksheets, no erasing, no savant ability, just plain old addition subtraction, multiplication.

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method. For Cube Roots, it is (a+b)cubed. It is easy.

[@SkeeterHawkins]

successive approximations might be easier.

[@RSLT]

GREAT VIDEO! Liked and subscribed ❤

[@emorblx_0474]

Step one near from 23 is 7×3=21? Sorry if i'm wrong

[@senatorstevenarmstrong9821]

It was a number times itself that would get closest, so 4x4 would be the closest you could get.

[@JubeiKibagamiFez]

2:10 Also, how would we do this with the (√2)??

[@R4NBOOKA]

Do it with 2.0000 and add as many zeros as you need

[@JubeiKibagamiFez]

@@R4NBOOKA Oh, okay.

[@vpdisco]

This is a video you can take your time.

[@rickyardo2944]

did I missed something? the last digit: 5 shouldn't that be a 4?

[@zgh46840]

I learned this method in school. Going forward I’m using a calculator.

[@jeffdege4786]

I learned this as a kid, without explanation. I later proved to myself why it worked. But truth is, I never use it. Newton's method converges faster.

[@bowlineobama]

Better use Binomial Expansion Method (BEM). No need for iterations. BEM gives it to you directly in the long run, when you have very large numbers.

[@Vega1447]

​@@bowlineobamaWhy do you think that the Binomial Expansion Method is better than NM?

[@peep39]

Thanks Stevie Nicks

[@trien30]

I learned this from a high school classmate but I didn't get what he did. He wrote on paper so quickly. I didn't have time in class. I think if you're in an east or as Southeast Asian country or somewhere from South America they might have taught this. Asian countries taught tough stuff forvyoung kids that's not taught in the USA or Canada.

[@albaiko]

My dad developed a method to manually calculate the cubic root as well.

[@albaiko]

sedqialbaik.blogspot.com/2006/04/blog-post_114434901914567834.html

[@albaiko]

The Cube Root: A Practical Method to Find It from Any Number The Cube Root A Practical Method to Find It from Any Number Sidqi Mohammed Al-Baik In the Abbasid era, Arabs excelled in mathematics, enriching the facts of arithmetic, establishing algebra and logarithms, dealing with exponents (powers) and roots, and organizing tables. It is not unlikely that they devised practical methods to find the square root or cube root, other than the method of prime factorization, but these were not known to modern mathematics scholars or were not published. However, students following the French curriculum recently learned a practical method to find the square root (as in Syria and Lebanon) while those who studied according to the English curriculum did not. I have not come across a practical method to find the cube root, nor have I found any mathematics specialists who know a practical method for the cube root. Therefore, I worked hard and for a long time, spanning several years, fluctuating between despair and hope, until I discovered this practical method to find the cube root of any large number, other than the prime factorization method. Many may now find it unnecessary to use this method and others by using calculators, which also spared them from many calculations. However, people, especially students, still need to learn different methods. This method may be an intellectual effort added to other mathematical information and facts. Here is this method, which requires knowing the cubes of small numbers from one to nine, which are (1, 8, 27, 64, 125, 216, 343, 512, 729). Method and Steps Divide the number into groups of three digits, starting from the right, after writing the number in the correct format. Start the first stage with the leftmost group, approximate its cube root, and place it above the group. Place the cube of this number under the leftmost group and subtract it. Bring down the second group next to the previous subtraction result and start the second stage. Prepare the root factor according to the following steps in the left section: A. Square the root obtained in the first stage and place a zero before it. B. Mentally divide the number obtained in step (4) by three times the squared root (from step A) by underestimating, and assume this result as the second digit of the root and place it above the second group. C. Multiply this assumed number by the previously obtained root with a zero before it. D. Add steps A and C. E. Multiply this sum by three. F. Add the previous multiplication result to the square of the assumed number. G. Multiply the sum in step (F) by the assumed number, place the product under the number obtained from bringing down the group (step 4), and subtract it. Bring down the third group to the right of the previous subtraction result, start the third stage, and repeat the steps in (5) as follows: A. Square the previous root (both digits) with a zero before it. B. Mentally divide the number obtained from bringing down the group (in step 6) by three times the squared root (from step A). C. Multiply the assumed number (from step B) by both digits of the root with zeros before them. D. Add steps (A) and (C). E. Multiply this sum by three. F. Add the previous multiplication result to the square of the assumed number. G. Multiply the previous sum (from step F) by the assumed number, place the product under the number obtained from bringing down the group (step 6), and subtract it. Continue this process. If a remainder remains after subtraction and no groups are left, add a group of three zeros and repeat the previous steps, placing a decimal point in the root as the result will have decimal parts. Practical Example Cube Root of (77854483) Divide the number: 7 2 4 77,854,483 Approximate the cube root: The approximate cube root of 77 is 4, place 4 above the first group. Subtract the cube: The cube of 4 is 64, place it under the first group and subtract it. 77 - 64 = 13 Bring down the second group: Bring down the second group: 13,854 Prepare the factor: Square the root with a zero before it: 40 × 40 = 1600 Mentally divide 13,854 by 1600 × 3 = 2 approximately Multiply 2 by 40: 2 × 40 = 80 Add 1600 and 80: 1680 Multiply 1680 by 3: 1680 × 3 = 5040 Add the square of the assumed number: 5040 + 4 = 5044 Multiply 5044 by 2: 5044 × 2 = 10,088 Subtract 10,088 from 13,854: 13,854 - 10,088 = 3,766 Bring down the third group: Bring down the third group: 3,766,483 Repeat the previous steps: Another Example: Cube Root of (12895213625) Divide the number: 5 4 3 2 12,895,213,625 Approximate the cube root: The approximate cube root of 12 is 2. Subtract the cube: The cube of 2 is 8, place it under the first group and subtract it. 12 - 8 = 4 Bring down the second group: Bring down the second group: 4,895 Prepare the factor: Square the root with a zero before it: 20 × 20 = 400 Mentally divide 4,895 by 400 × 3 = 1 approximately Multiply 1 by 20: 1 × 20 = 20 Add 400 and 20: 420 Multiply 420 by 3: 420 × 3 = 1,260 Add the square of the assumed number: 1,260 + 1 = 1,261 Multiply 1,261 by 1: 1,261 × 1 = 1,261 Subtract 1,261 from 4,895: 4,895 - 1,261 = 3,634 Bring down the third group: Bring down the third group: 3,634,213 Repeat the previous steps.

[@VincentLauria6]

I learned how to calculate square roots nearly 50 years ago. I’m certain they haven’t taught this for probably 30 years

[@bowlineobama]

It is a lost art, but I am glad that it is in the KZbin forever. This is Binomial Expansion Method (BEM).

[@JubeiKibagamiFez]

2:10 Really? I was really hoping this was gonna be the universal equation that solves any square root, or cubed root, or etc. I've never understood roots because there is no reverse calculation for it like division is for multiplication. I also watched a video a few days ago where I was introduced to n⁰=1 and 0⁰=1. Math is suppose to be about logic, but I feel the more advanced maths are just number manipulation to get a desired answer.... Basically arbitrary like language and to me, arbitration is not based on logic.

[@markgraham2312]

Good job!

[@SpiritofMathSchools]

Have you seen our All About Circles video? kzbin.info/www/bejne/aZO4lYOJqLyie5Ysi=yEa2P_KDJrzMDBs9

[@markgraham2312]

@@SpiritofMathSchools Not yet.

[@SpiritofMathSchools]

@@markgraham2312 We've got a bunch of additional curriculum videos that you might be interested in!

[@rchas1023]

Step 1: Convert to binary. This avoids any need to guess. Step 2: Apply the algorithm for binary numbers. Very fast. Step 3: ( Optional ) Convert to base ten.

[@ThePowerfulOne07]

Was never taught this in school. Must have been a “lost art” in my state 😅

[@SpiritofMathSchools]

Good thing we offer this online!

[@bowlineobama]

This method is based on Binomial Expansion (a+b)squared method.

[@robertwatson818]

We did square root problems my senior year but nothing like this!!!

[@SpiritofMathSchools]

It's never too late to learn a new approach!

[@krwada]

This is how I learned it many years ago when I was in 8th grade

[@paulgithens635]

Eventually after many math classes the love of learning was beaten out of me.

[@toma5153]

Same. Never any explanation or real world examples. Just dreary rote practice out of the textbook.

[@SpiritofMathSchools]

We're sorry to hear that! We find the best way to learn is in a collaborative, group setting

[@journeythruwords4619]

Wait. Shouldn't the last digit be a 4 and not a 5?

[@silver6054]

By "as close as possible" I assume it is, as you say in the first case, as close to but less than. And the amazing statement at the end about square roots never repeat. Well, some certainly do, e.g. a square of a rational, such as 2.25, repeats with infinite 0s. So the divisor changing doesn't guarantee non-termination

[@Hydelsius]

I learned this in math class in high school. It was such a waste of time for all of us as for myself I've never had to use it in my life. Our teacher at the time argued that we wouldn't be walking around with calculators in our pocket, not realizing that smartphones would come out the year after. I did however learn how to play chess and that class.

[@SpiritofMathSchools]

We'd counterargue that learning the process behind the math is never a waste of time. Math is in our daily lives, even if it isn't always in the form of square roots. Calculators are nice to have, but relying too heavily on them weakens our math muscles.

[@alittax]

Can you please provide a proof for why this works?

[@SpiritofMathSchools]

The proof is in the video. We suggest trying it out in your own life and seeing how you it works for you

[@terencetwentyman-jones8861]

Ok, it works and you showed it works....but why?​@@SpiritofMathSchools

[@michaelspinks9822]

Looks like a neat method, but frankly you lost me and I have a strong background in mathematics. May I suggest you redo this video? Writing out a script with queue cards may help. Citing a published source for this trick would be great. Other commentators suggest it is a reorganized Binomial expansion....I tend to agree, though more background would be nice .

[@ScarletKevin]

Really I don't understand how you get that 88 please I want to understand

[@SpiritofMathSchools]

The first 8 in 88 comes from finding what squared number goes into but not over the first pair of digits (23). 5 squared would be 25, which is over 23, but 4 squared is 16 which is as close as we can get. When you move down to the next line, you have to double that 4 (from 4 squared), which is 8. The second digit comes from looking at the number from the next row (776). You need to find what 2-digit number that starts with 8 and multiplied by the same single digit number equals close to but not over 776. If we use 9 for example, 9 x 89 = 801. If we try 8, 8 x 88 = 704. This is as close as we can get, meaning an 8 goes above 76 and 88 goes to the left, just under 776. Hope this helped!

[@c.m.p2943]

I learned this method long long time ago when there were no electronic calculators ,am now 70. y/o ,but instead of multipying by 2 we multiply by 20.Now a day they don't do this method any more.

[@archimedesmaid3602]

Yes. I have always simply multiplied the currently completed root by 20, (20a). then estimate how many times that divided into the current remainder . That is your tentative next digit (b). Add the b to the 20a figure and multiply by b. (20a + b)b Subtract from current remainder, bring down the next group of two, for your next current remainder This simple method can be remembered forever, because you know why you are doing what you are doing It is never taught on utube, because it doesn't appear as sexy. But in our father's time, my method was used, because I eventually saw it in a very old encyclopedia

[@shelonnikgrumantov5061]

What I don’t get is that 9х8 is 72, which is less than 76, obviously, why then you use 8?

[@gangleweed]

I do root calculations a different way ........try doing the 6 root od 41........and I'm 85.

[@mjs28s]

I am pretty ticked off that I was never shown this in any year of schooling. Yeah it might have been rough at a young age, but the mental workout it would be if all kids had to learn this stuff. People would be way better thinkers as grown up as well as following rules for things and how to solve problems, in life not just math as the problem solving skills are applicable everywhere.

[@SpiritofMathSchools]

Is there anything else you wish you saw earlier? We can help share another video for you.

[@Tabu11211]

They didn't teach this in school where I was. :(

[@Tabu11211]

Are all square roots of non square numbers irrational?

[@Merione]

No. You can convince yourself by looking at the problem from the opposite direction: if you take a rational number and square it, will you always get a square number? If it's an integer, yes (2*2 = 4; 3*3 = 9; etc), but if it's not an integer, then no: 0.5*0.5 = 0.25, so there exist non square numbers with rational square roots.

[@Tabu11211]

@@Merione thank you for taking my question seriously. I appreciate your response. Just like everything that is explained it seems obvious in hindsight and I probably should have just thought about it harder. That was a very satisfying and simple explanation.

[@robertveith6383]

All square roots of non-square integers are irrational.

[@Tabu11211]

Ah ok, thats probably what I was intuiting.@@robertveith6383

[@archimedesmaid3602]

​@@Merione???????

[@raymondgordon-c2n]

learned this in grade school 1959

[@SpiritofMathSchools]

Unfortunately, children aren't taught this approach in grade school today and they should be!

[@DLV42]

I grew up learning how to do square roots manually . Kids today do not learn how to do sq. rts. manually. They press the magic button on the calculator.

[@AnnaScott-p1l]

wheres the decimal point end up?

[@cbruata5198]

The decimal will never end since the square root of non perfect square is non terminating as well as non repeating. In otherwords they are irrational numbers.

[@Matlockization]

It ends up between 8 on the left & 7 on the right -> 48.75

[@cbruata5198]

@@Matlockization it is simply a round off or we can say approximation

[@Matlockization]

@@cbruata5198 Well, it depends on when you multiply the answer by itself how close you get to the original number. In this case, you can round the answer off to two decimal places, but as it stands the answer is not an approximation.

[@patrickgregory2826]

Sqaure root of 20 is 5?

[@avinashdesai0206]

You know if that guy from that pie movie only new this