@@want-diversecontent3887 actually it was subtracting mixed numbers... stack them up like a normal subtraction problem and "borrow" a whole fraction from the ones column in order to help subtract a larger fraction on the bottom. i was used/trained to find a common denom and turn them both into improper fractions, subtract the numerators and convert the answer back into a mixed number... this was kinda mind blowing to me after a couple decades of practice one way!
This is actually impressively precise. For my own enjoyment I wrote a script that used this method for the root of every number up to a million. The highest margin of error was a little over a fifth for the root of three. With precision trending to increase proportional to the size of the number. By the time you reach thirty the number is accurate to +/-0.1
It gets better if you adjust it a but, if the number is closer to the next highest square rather than the previous lower square. You use that higher square and subtract much like he adds. So 34 would be 6 - 2/12 = 5.83. But who wants to do subtraction, amirite
@@WhatsAGoodName42069 I noticed that as well and came to the same conclusion. Considering I was happy with the root of 87 is nine and a bit before this video. I'll take accurate to a tenth and not do subtraction.
@@Geweldenaar that's memorization my friend. Personally I only know up to 20squared off the dome. And that's fine by me I give up on mental arithmetic in the mid 3 digits anyway.
The trick is actually based in calculus. It’s the Taylor series approximation using two terms for the square root function. Without reading through the hundreds of comments already made I’m sure several other people have probably mentioned this. Just shows you that calculus can be really useful! What’s really neat is that if you use a Taylor series approximation with three terms, its even closer to the exact value of the square root.
Taylor series of 3 terms: f(a)+f'(a)/1!(x-a)+f''(a)/2!(x-a)^2 f(a)=a^(1/2) f'(a)=1/2a^(-1/2)=1/(2√a) (the 1st derivative, which is the slope at point a) f''(a)=-1/4a^(-3/2)=-1/(4√a^3) (the 2nd derivative, which is slope of the slope at point a) x is the point you want to find, and a is the closest number (or any number, but the accuracy won't be the same) Then you have to expand it and find the cube then bla bla bla congratulations you're accurate for maybe the 3rd digit but we will round the first digit anyways
@@ncpolley Basically the trick in this video is sqrt(x^2 + y) is approximately equal to x + y/(2*x). The original square root can be rewritten to be x * sqrt(1 + y/x^2). The Taylor series for something in the form sqrt(1 + z) is 1 + z/2 - z^2/8 + z^3/16 +... Using that Taylor series, the 3 term approximation would be x + y/(2*x) - y^2/(8*x^3) which is a little rough for mental math but doable. The easier way to improve the accuracy would be to use the closest square (rather than the square below) since y in the original square root can be negative or positive, and the accuracy of the Taylor series approximation is better the smaller the "z" term is.
@@duk8227 bro, its from Binomial Theorem .... If at higher level you need more digits... Them Expand it more and more..... Its infinite series....... You can get whatever amt u need...
@@greatmnn3868 u can just type in a calculator and get the correct result. The trick might be interesting in some cases to get the first decimal number but if you are able to estimate the root by taking the root of the next smaller quadratic number you have a reasonable estimation already. In other words, this is useless on any level.
Here's how this works algebraically: Let √c = a + b, where a represents the whole number component of the answer, a ∈ Z, and b represents the decimal component, 0 < b < 1 c = (a + b)² c = a² + 2ab + b² Now solve for b: 2ab = c - a² - b² b = (c - a² - b²)/2a b² (the decimal component squared) will be very small, so we can ignore it therefore, b ≈ (c - a²)/2a in other words, b ≈ the original number, c, minus the nearest perfect square below it (the whole number component squared), all over 2a: which is the formula given in the video for working out the non-whole number part of the answer e.g. √27 = 5 +b 27 = (5 + b)² 27 = 25 + 10b + b² 10b = 27 - 25 - b² b = (27 - 25 - b²)/10 b ≈ (27 - 25)/10 b ≈ 0.2 √27 ≈ 5 + 0.2 = 5.2 As further proof, the approximate answer will always be out from the actual answer by a margin of b²/2a: 5.2 - 5.196... = (0.196...)²/10
If anyone wants to know how this works, it essentially uses the fact (x+a)² = x²+2ax+a² For example, √(37) is a little more than 6 so call that little extra bit a Then √(37) = 6+a Squaring both sides gets 37 = 36 + 12a + a² 1 = 12a + a² Since a is small, a² is small and can be ignored thus a ≈ 1/12 so √(37) ≈ 6+1/12 Approximations are better when a is smallest (the number being rooted is large and close to a perfect square)
This can be improved slightly by choosing the CLOSEST perfect square as opposed to just picking the one below. For the sqrt 138 example, choose 12^2 = 144, then SUBTRACT 6/24 (which is the same as 1/4) from 12, leaving 11.75 as the answer, which is closer than the 11.77 obtained using the original method.
Yeah it can be done for any power if you subtract your real number from the nearest perfect and take the derivative of the constant where the number is x and adding that to the perfect power
It also gets improved just by adding 1 (or subtracting 1 if calculating from the perfect square above) to the denominators, such that sqrt(138) = about 11 & 17/23, which is about 11.739 (compared to the real answer of about 11.747). It works better this way, since (n+1)^2 - n^2 = 2n + 1 (or in this case, 12^2 - 11^2 = 11*2+1 or 144 - 121 = 23) rather than 2n.
Because you were probably talking a lot about your test near your phone/ Siri / Alexa or texting or messaging about it. Your phone and apps listen to you and analyse your messages and then adverts and marketing appear to suit what your talking about.
This is great! I always thought I was 'bad at math' until I took a VERY basic course in my 40s and discovered that, not only can I do math well enough to get by, but it's actually fun. Your video presentation is super clear and friendly. I was able to do all the problems the first time through; going to check out your other videos now. Thank you!
@@Derpalerpa Hi, unfortunately I don't have a link, but it was an in-person class called "Math Modules" at Berkshire Community College in Massachusetts. The textbook is "Introductory Algebra" by Marvin Bittinger - publisher is "Pearson." Last year I picked up the textbook again and started studying on my own. That's going really well, mostly because the book is so well written. Hope this helps!
Math is fun when your “life” doesn’t depend on it. Aka grades. Edit: 13k likes!? Holy... Thank you, everyone. Also, there's a fight in the comment section lmfao. I'm very sorry I caused this chaos.
@@SapphireeCos As I’ve been taught and what I have researched on my own in this very topic, it firmly depends on discipline and willpower. I’ve you’re being judged by your grades, it is partially up to you, to change that very situation and your surroundings.
idk man it's still not fun for me even after school and not needing to worry about grades. After all I really only use the stuff from grade school in my day to day.
For anyone who's curious: This is based on a linear Taylor Series approximation of the square root function. For example- Sqrt(x + dx) ~ sqrt (x) + (1/2)(dx)(1/sqrt(x)) And this is exactly what he's doing
@@JonathanBartlesSWBGaming the formula in the video comes out from the taylor series with an error of the first degree. So they are using the same way to calculate it.
I made a quick excel spreadsheet and graphed the percent errors of this method. The %error curve is a sawtooth pattern, but the peaks look to be decaying exponentially. This is very interesting, thanks for sharing.
Yeah its because the taylor series is just the sum of the derivatives devided by the faculty of the index (nth derivative/n!). As we are only using the first summand of that series, the rest of the series is our error. Now this sum for the sqrt of x looks like an exponential of 1/x, if you look at the series formulation of exp. Thus you get the exponential decrease in the peaks. P.S. thought about it again. It is just wrong.
@@TheMrk790 you know what's insane, anyone that understood what you said knows you're just bloviating and anyone who doesn't, doesn't care about what you're saying.
I guess it works for people who can convert fractions into decimals in an instant, but since I am not one of those people, this method doesn't help me too much.
He did not tell you that in the video, but in case the sqrt is closest to a full number higher than itself, you should use that instead, and your second part of the answer (that comes after the point) will be negative. You'll do the division normally, and then subtract from the big number (i.e. closest full number that stays in front) and you'll get your answer. Example: sqrt of 24, you'll go for 25, and the subtraction will lead to 5 -1/10, which is 4.90. The actual answer is 4.89
@@mike_slav0477 the point of this method is that it's a simpler version of an approximation equation. You'll never get the exact same number unless you round it up. I simply wanted to show that you should use the closer number even if it's a higher one. If you were to use ✓16 it wouldn't work. That's all there is to it
I stumbled upon something similar to this a few years ago and presented my technique to the math class. Teacher was convinced enough to let us use the technique over the slower "multiply 'til you find it" method in the curriculum.
Reminds me of Richard Feynman when an abacus salesman challenged him to a math contest. Ended in a square root contest. Feynman won because the number was close to a perfect square and he could quickly refine the number. That and doing square roots on an abacus is very labor intensive.
It is also possible to start with the closest square root above and subtract the fraction as shown. Example root138 = 12 - (6/24). Root 138 = 11.75. Good luck!
you can also apply this same step to your won approximation. It will be a littler harder to calculate, but it will improve your results :) e.g. use 138-11.75², and then 11.75 + difference / (2*11.75) Because it's Newtons Method, and you can use as many iterations, as you like :)
@@GahloWake Yes, it would be more accurate to do it to the closest square. How the trick works is it uses the common school formula (a+b)^2 = a^2+2ab+b^2, sets a as the lower perfect square so that 0 < b < 1, and then assumes that b^2 is small enough to ignore. If you go to the closest square (sometimes above), what you are doing is reducing the size of b^2, in other words reducing the error of the trick
This actually makes the trick more accurate, trying for root 15 by the addition method will give 4, but the subtraction method you suggested (essentially follow the closest perfect square and do accordingly) gave 3.875, which is very very close.
@@feeldog1019 depends on the instructor. I had a chemistry teacher that would allow you to approximate your answer if you could prove your answer was reasonable. Usually this was only on tests where calculators weren’t allowed, but it’s worth noting. Plus with this method you could trivialize multiple choice questions by going with whatever’s closest to your result.
@@redsgxd Esitmating the sqrt of something large is still still relativly easy using this method, your error just gets bigger. Just Divide your number by 100 until it is smaller than 100. ex 32767=100*100*3.2767 sqrt using the above method (That I will modefy slightly for increased accuracy) 4 (closest square) -3.2767= .7233 so you get 2-(.7233/((2-1)*2))=1.819 then cuz we divided by 100 (or 10^2) twice multiply back by 10*10 and your final answer is 181.9 sqrt(32767) is exactly 181.1..... so still reasonably accurate
It's called aproximation. My astrophysics teacher always says: "ah well it's the same power of 10 right?". It's just a matter of how accurate you require your result to be whether you want to use this or not :P
You actually don't understand how much this is gonna help me. I'm in Algebra 2, and every other concept is pretty easy for me to understand, but I never did pick up radicals and stuff like that, so they keep moving on expecting that everyone already knows how to do this. This is genuinely gonna make my life so much easier.
@@Nectrik I got ~7 by going down from √48 to √36, which is 6, and then the difference between 48 & 36 is 12, so that's the numerator, and twice 6 is 12, so it's the denominator. 12/12 is one, which would make the 6 into an approximate 7. I checked my answer with the calculator and the answer was something like 6.93, which is close enough in my opinion.
@@spoopyscaryskelebones3846 Smidgy Widgy is an infamous art. You think schools are going to teach that? You have to travel out to the himalayas and speak to master Wu to learn Smidgy Widgy.
For smaller numbers, you can get a much better approximation if you calculate a square root of a number that's 100 bigger and then divide the result by 10, so sqrt(2) = sqrt(200)/10. This is because the higher the number is, the better the approximation will be.
Yeah ! but finding the required intuitive square root of 8700 (93) is far more complicated than 87 (9). The final fraction becomes also more challenging (51 / 186). You get of course your loved 93.27 / 10 = 9.327.
In the end, it would be far better to just take a few minutes to learn to do the real method for finding square roots to as many digits as you desire. You are not gaining anything at all by trying to learn these "tricks" What I am suggesting is that you have no business learning "tricks" to do something, until after you have actually learned a functional method to do that something.
Honestly I stumbled across this while I was half drunk.😂😂 And I was somehow able to follow along and get them all right! I even kept the sheet to prove I get it! This taught me better than school did! Props to you mate! This is awesome!! I hated math but getting these answers right boosted the hell out of my confidence!!
lol, me? graduated HS in 1979; never went any further than gen math. FFWD to 2006 I found myself in college earning a BCS/IT which required algebra 1&2 and Geometry; needless to say I did not do so well with sqrt's ;) FFWD-again, to tonight and I find this video while having drank 1/2 bottle of 10 year old single malt Irish whiskey in celebration of my 61st birthday.... WHAM! 'I now can do sqrt's in my head! I'm not fast (yet). Thanks, just doesn't cut it.
After I saw you do the fist one, I did one of my own which is 149, I got 12 5/24. Came back, did all the ones you have before you and got all correct. Thank you sooooo much this was extremely helpful and effective. What good would it do to my test scores if I didn’t subscribe? None. Glad to be apart of the team.
it essentially uses the fact (x+a)² = x²+2ax+a² For example, √(37) is a little more than 6 so call that little extra bit a Then √(37) = 6+a Squaring both sides gets 37 = 36 + 12a + a² 1 = 12a + a² Since a is small, a² is small and can be ignored thus a ≈ 1/12 so √(37) ≈ 6+1/12 Approximations are better when a is smallest (the number being rooted is large and close to a perfect square) (copied off another comment)
@@RajSinghTanwar_ Clearly, the guy mentioned that he copied the comment. I would recommend you to stop being so ignorant and a recalcitrant person as he was only helping the other person. If you're a person who can only say to other people to "go get a life" - then I say you're the one who needs to get a life. Thank you :) Note: pls spread happiness and harmony. It's not hard
I’ve never seen this before, but it’s actually pretty easy to see what that trick is really doing mathematically. All it’s doing is converting the continuous function root(x) and converting it to a piece wise function by connecting the points at perfect squares with strait lines. It works because the inverse function of x^2 has a derivative of 2x, so the slope of each new line is 2x. It’s a pretty neat trick.
One improvement: Add 1 to the doubled number in the denominator. So, instead of calculating the square root of 27 as 5 + 2/10, do it as 5 + 2/11. On average, this cuts the amount that any given solution is off from the actual amount by half.
@@txtce In this case, definitely, but if we were doing, say, the square root of 94, which would be 9 + 13/18 with their method, and 9 + 13/19 with mine, I wouldn't say that 13/19 is that much harder than 13/18.
How about difference in squares? For some reason I cant wrap my head around doubling the square root, i dont understand where that part comes from. But what about this method. I just thought of this but im positive someone else has invented it.I 87 is between 81 and 100 (9 and 10 squared). The differece is 19 and you have 6 remainder. 6/19 is around .315. I know its not as accurate but it makes more sense to me. Also 650. 625=25 676=26 676-625=51 650-625=25 remainder 25/51= .49ish So 650squard root is 25.49
@@tacobender1643 Yep, that is actually the basis behind what I mentioned. The 'doubling of the square root' comes from the fact that the difference between the squares will always be halfway between twice the lower square root, and twice the higher square root. So, in your first example, 2 x 9 = 18, and 2 x 10 = 20. The actual difference is 19, as you noted. So, if you add 1 to double the square root of the lower number, you get the difference between the squares.
this is absolutely awesome, went form completely guessing and taking a good 30 seconds before watching this to being able to calculate roots in like 2 seconds. in 8 minutes of my time. absolutely worth it
It’s nice when teachers can be teachers and not forced into some regimented way of explaining things handed down by an out of touch and overly academic administration.
This is amazing. I believe it’s called “Approximating Irrational Numbers” and this has been the easiest way to find the square root of “imperfect squares” that I have ever seen. I’m super grateful ❤
They should literally teach this in schools. I'd say 7th grade would be a good time to learn this. (Or maybe don't teach it, just so your viewers have an edge over the average person :))
@@eitankalfa7836 calculators should never be allowed in schools. It's good at making you bad at math and nothing more. Everything should be done manually. There are more ways to calculate square root manually than this. They could teach those if this is too difficult.
I got curious and tested out this aproximation method in excel against excel's internal square root number. I calculated the percentual difference between the aproximation and the actual sqrt given by excel. i worked it out from 1 up to 1600 So it happens that the diference (as expected) resets to zero whenever you get to a perfect square (like 36 or 100...), and then increases as the number climb to the next perfect square. When you plot it the graph looks like the teeth of a saw. What's cool is the fact that as the numbers go up the teeth go wider and shallower, so the aproximation get's better and better. The absolute worst case is number 3 that as a square root 1.73205 and the aproximation yelds 2, thus giving a percentual difference of 13.397%. But, for instance, 1598 has a square root 39.97499 an the aproximation yelds 39.98718. That's 0.03048% difference. (at 1600 it resets the differences as it is the square of 40)
+tecmath I should be the one doing the thanking for the brain candy, so... Thank you. ...Keep them coming. BTW, if you want, i can send you the excel file. Regards
+Danana J: Thanks. It's always good to know. Allow me to excuse myself, English is not my native language and i'm not using a spelling checker. Most words that exist in my native language have an English counterpart, so i tend to apply them. Sometimes, such as this one, that is not the case and the word doesn't exist in English (rare). Needless to say we have both "Percent" and "Percentual" (the later meaning that we are talking about something with percent-like caracteristics). Regards.
+Nuno Gonçalves Yes i undertood your use of "percentual". Btw this "amazing" trick done by tecmath is just a Taylor series approximation, which is most mathematicians use to do these "party" tricks.
The material in this video is application of the binomial approximation: sqrt(1 + x) = (1 + x)^(1 / 2) ~ 1 + x / 2. Let the value under the square root be written as a + b, where a is the closest perfect square and b is the remainder (difference from the perfect square). You then have sqrt(a + b) = sqrt(a) * sqrt(1 + b / a) ~ sqrt(a) * (1 + b / (2 * a)) = sqrt(a) + b / (2 * sqrt(a)).
And just like that... 32 years into my life hating math... I now know what a square root is and was able to pause the video and complete each of these problems. Thank you! That was surprisingly fun!
I have had this video in my watch later for approximately 5 years, I always knew I would need it. now im watching after it got recommended to me for searching up a similar topic
It would be nice if you could show why this is true? Differential calculus. So we should thank Newton for this :P Say you need to calculate root(x+ dx) where dx is much smaller than x Here x is the nearest perfect square, and dx is the extra number We say let y = root(x) and y+dy = root(x+dx) Our goal is to find dy, we can differentiate y dy/dx = 1/2root(x) Or dy = dx/2root(x) Hence our final answer is root(x+dx) = root(x) + dx/2 root(x) You can now understand why he is taking that extra number, and dividing by the double of root of the nearest perfect square. This also explains, why you cannot use this method for squaring numbers, where the extra number is comparable to the nearest perfect square, the differential approximation works best when dx is much smaller than x (atleast 1 order of magnitude, else you get horrible answers :P)
you can also try looking up "binomial theorem" if you are not that familiar with calculus. this approximation is the first two terms in the expansion of (a^2 + b)^(1/2). you can also use (a^2 - b)^(1/2) as others have pointed out, if it gives you a smaller "b". the smaller the value of "b" gets compared to "a", the better the approximation gets. this is because it makes the terms you are ignoring in the binomial expansion after the first and second become smaller faster. this is why he says to use the perfect square closest to the number you are taking the root of.
I've noticed that the closer you are to a perfect square, the more accurate the answer gets. When I say closer I don't mean sqrt 2 is very accurate bc it's one away for sqrt 1, I mean fraction wise of the number, 2 to 1 is a big jump of 1/2, so it's not that accurate. But 27 to 25 is a much smaller fractional jump, which is why it's so close to exact.
Actually, you get good results when you are just above a perfect square, but very bad if you are just below. Sqrt(24) -> 4 + 8/8 =5. A better formula, that indeed works well whenever you are close to a perfect square from both sides, is N+R/(2N+1) instead of N+R/(2N). This works because the distance between two consecutive perfect squares N^2 and (N+1)^2 is 2N+1. Taking the residual over 2N+1 would be basically looking at a weighted interpolation between the following and previous perfect square. Graphically, what this would be doing is basically approximating the parabola X^2 with a series of segments connecting all perfect squares. So the result is close to perfect if the number is close to a perfect square, and far away if it is in the middle between two perfect squares.
I guess the "heavy lifting" of the trick is the first step, where you figure out the rounded-down square, and if you know the rounded-down and rounded-up square then you might as well just make a rough guess.
You can actually invert this method to go off of the next perfect square, by inverting your addition/subtraction operations. i.e. √95 -> 10=√100, 100-95=5, 10-(5/20)=9.75 √95=9.747 Using the normal method gives you 9+(14/18)=9.777, which is clearly less accurate. You'll find that the method that uses the smaller numerator (aka the one that uses the closer perfect square) will be the more accurate method.
Dude, this is so good. Square roots have always been difficult for me. I love math and I'm not bad at it at all. It's the class I did best in all through school. Every year I'd get high 90's in math while I tested very poorly in most other subjects. I never fully memorized my multiplication tables though because my brain just works differently, probably partly because I'm dyslexic with numbers, so that made everything else a little harder, especially square numbers. I never would've thought of a trick like this but once you explained how it worked it instantly made perfect sense. I feel like this trick just gave me a better sort of understanding of, or intuitive feel for, square numbers and their roots, like over all, as a concept.
Thanks a ton, you're a lifesaver! My statistics teacher made this way more complicated by doing it through long division and I was lost. Now I'm an ace in finding square roots!
I was never taught how to do this, but this method is how ive always done approximation in my own head, Seeing it done and on board makes the process understandable so much easier. Thank you!
Me after a couple of math vids on RUclips: Cube that address Square that license plate Find food at whole foods with perfect square prices Evaluate the sale and maximise profit -Gotta love RUclips
Wow, the trick at the end was super helpful. I always knew my time’s tables/perfect squares, but estimating the tenths and hundredths digit was always a rough estimation. Thank you.
This is really interesting and helped me a lot!!! Learning about square roots for قدرات. Watched this while waiting for dad to come home to go to the beach
Thank you so much! I didn't know this trick back in college. Without a calculator, I've been using differential calculus approach just to find the approximate square root of a number.
U just helped me to prepare for a test i have on predicting square roots tommorow!!!!! Wish me luck!!! EDIT: By the way i surprisingly got 100% the most in the class!!!!! (Gr.9 quiz)
Giselle P. Wrong. According to your statement: 2+2=4 -1=3 -1 never equals 3 A more correct statement would be 2 plus 2 equals 4 4 minus 1 equals 3. Curb your memes.
Well to be technical, the square root would be much easier when comparing the remainder number to the next closest perfect square. So 36 squared is 6, but we have 3 left over from the 39, the next closest square is 49, the difference betwen 36 and 49 is 13, so we have our remainder over the difference. The square root of 39 is 6 and 3/13ths.
my Math teacher was so mad at me because when she ask me about square roots I didn’t know nothing but after I watched this video I was able to do any kind of square roots😊😎 Thank you!!!🙃
My past significant other had a daughter who was 5 at the time....I taught her the way to show the radical of 49 with her fingers....it was a huge hit!!
I love math tricks and such, and this absolutely blew my mind. I saw the title and it immediately reminded me of the multiplication by 11 trick I memorized a while ago. Such a great video
I would adjust a little bit : if the digits after decimal points are >25 and 50 and 75 remove 0.03 for instance 190: 13²=169, 13(21/26)=13.81. 81>75 so remove 0.03: 13.78, which is the perfect result
I hated mathematics until yesterday. Yesterday I found your channel and I love it. I feel so good when I do the example right within seconds and all in my head! Thank you so much!
The fact that these are so close to the real answers doesn’t seem like a coincidence. Perhaps there’s another step we don’t know about yet in order to get the exact answer.
It's actually relying on a process learned in calculus called linearization. So while there are other techniques that can be used to get more precise values the technique used here is kind of a one shot deal. It gives a fairly precise answer that can be easily calculated by hand.
It’s not a coincidence at all! It comes from the following fact: For some number N and a smaller number a, √(N² + a) = N + a/(2N) - a²/(8N³) + a³/(16N^5) - 5a^4/(128N^7) + ... Where the sum technically goes on forever. If you’d like to google around about this, search for the “Taylor series of square root”, which is the technical mathematical name for this fact. Anyways, the reason this trick works is that the later numbers are so small. Let’s use the √27 calculation from the video as an example of how this works. N = 5 and a = 2, because 5² + 2 = 27. Replacing N and a in the sum, we get √(5² + 2) = 5 + 2/(2×5) - 2²/(8×5³) + 2³/(16×5^5) - 5×2^4/(128×5^7) + ... Then if we want a better guess we can just add more terms: Using the first two terms, which is what we did in the video, we get 5 + 2/10 = 5.2. Using the first three terms, we get 5 + 2/10 - 4/1000 = 5.196. Using the first four terms, we get 5 + 2/10 - 4/1000 + 8/50,000 = 5.19616. I won’t bother doing this with five terms, because we’ve gotta stop somewhere. The actual value of √27 is 5.196152... which is REALLY close to our guesses! But, as we keep getting more accurate, the calculations get harder, and the numbers we added got smaller and less important. Eventually, calculating more terms becomes pointless. If you want an error of ±0.02, the first two terms (what’s shown in the video) will work as long as the number you’re square-rooting is at least 34. TL;DR: getting a better answer is possible and we know how to do it, but it’s not really worth the effort.
TheBasikShow basically explained (quite nicely) how most calculators work - using Taylor series to get a closer and closer approximation as the number of iterations increases - calculators can only do arithmetic - so anything complex or irrational (i, e, pi, sqrt, etc.) must use a series approximation to get enough digits of resolution to satisfy the display capabilities. Even the TI-84+ Silver uses simple arithmetic. But SOME calculators (like my TI-89 Titanium 😎🏎) have a symbolic equation solver (Computer Algebra System) and can give irrational solutions, solve symbolic equations as well as run code, graph in 3 dimensions & parametrics, and more.
If you want to make it even more accurate, you can subtract the difference squared divided by 8 times the cube of the perfect root. Yes, you're all secretly marvelling at the power of calculus :-) *Edit:* I can probably use the magic of Unicode to turn this all into a formula. So, let's define some notation: we want to calculate _√x,_ where for example, _x = 87._ Let the closest perfect square to _x_ be _y,_ so _y = 81._ The video shows you that _√x ≈ √y + (x - y)/(2√y)_ which, in our example, is _√87 ≈ √81 + (87 - 81)/(2√81) = 9 + 6/18 = 9.33..._ I'm saying you can add an extra term to the approximation: _√x ≈ √y + (x - y)/(2√y) - (x - y)²/(8√y³)_ which, in our example, is _√87 ≈ √81 + (87 - 81)/(2√81) - (87 - 81)²/(8√81³) = 9 + 6/18 - 36/5832 = 9.3272..._ Since the actual square root of 87 is _√87 = 9.3274...,_ that extra term of _-(x - y)²/(8√y³)_ gained us about 0.01 units of absolute accuracy. The next term would be _+ (x - y)³/(16√y⁵),_ but that gets tough by hand.
@@hunterblacc4336 You mean you didn't take calculus? This is an application of calculus (Taylor's approximation theorem using derivatives), but not calculus itself.
Wow this is great, it’s fascinating because most tricks I see decrease in accuracy for larger numbers but this one actually increases, not only that but the error margin decreases pretty rapidly. The highest error point is at 3.99999999999 with 25% off of the actual answer. It resets to 0% at perfect squares before rising to only 8.333% off at 8.99999999999 and then less than 5 there after, after 64 it’s permanently less than 1% off.
Also interesting: If you accidentally pick the wrong perfect square, it still kind of works... Sqrt(155) Based on 12^2=144, with a difference of 11, gives 12+11/24 = 12.4853 Based on 11^2=121, with a difference of 24, gives 11+24/22 = 12.0909 Correct answer would be 12.4499. This error self correction works better the higher the number you're taking the square root of.
Little tip: when you add the fraction, instead of using 2n as denominator try with 2n+1, as it is more precise. The reason 2n is recommended is because it usually gives an easier fraction to deal with, but if you happen to find one where using 2n+1 is easy, then it's better. But, after all, the difference only matters with small numbers, as increasing the value minimises the error anyway. For example sqrt(22): the real value is 4.69, with the "2n" method it's 4+6/8=4.75 (+0.06), with the "2n+1" method it's 4+6/9=4.667 (-0.03). Another example is sqrt(34): the real value is 5.83, with the "2n" method it's 5+9/10=5.9 (+0.07), with the "2n+1" method it's 5+9/11=5.82 (-0.01). But, again, the difference is so small that, for what it's worth, you can just choose the easiest among the two.
youre right 2n+1 gives a better approximation for these cases but 2n isnt reccomended because it gives an easier fraction. 2n is used because it is a tangent line approximation. 2n+1 gives a better approximation because the function sqrt(x) is concave down so this method will always overestimate the square root
@@howareyou4400 I didn't say it's always better, but that it is overall more precise. 2n+1 is always slightly wrong, while 2n is a few times slightly better but most times a lot worse.
Why is this fun for meeeeee. Thanks for sharing!!!! So much less stressful learning this way when it's just in my free time and not like school at all. 😭😭😭
So, now after 60+ years out of school I can finally calculate square roots mentally.
never too late to learn! that is what i love about the maths - i just learned a shortcut on subtracting fractions from a 2nd grader the other day!
Wait, you’re that old?
@@CTGReviews Yeah, but what's the square root of that?
@@ryangauthier3957 What is it? Cross multiplication?
@@want-diversecontent3887 actually it was subtracting mixed numbers... stack them up like a normal subtraction problem and "borrow" a whole fraction from the ones column in order to help subtract a larger fraction on the bottom. i was used/trained to find a common denom and turn them both into improper fractions, subtract the numerators and convert the answer back into a mixed number... this was kinda mind blowing to me after a couple decades of practice one way!
This guy is like the backdoor dealer of mathematics
FR LMFAOOO
Yo kid, you want some roots?
@@neelparmar6690 oOhHhH i WoUlD lIkE sOmE rOoOoOtSsSs
@TIV67 Playz ?????
Lol, that was a hilariously mathematical comment.
This is actually impressively precise. For my own enjoyment I wrote a script that used this method for the root of every number up to a million. The highest margin of error was a little over a fifth for the root of three. With precision trending to increase proportional to the size of the number. By the time you reach thirty the number is accurate to +/-0.1
Did this method ever guess low? Seemed like all the examples in the video slightly overestimated the true value.
It gets better if you adjust it a but, if the number is closer to the next highest square rather than the previous lower square. You use that higher square and subtract much like he adds. So 34 would be 6 - 2/12 = 5.83. But who wants to do subtraction, amirite
@@WhatsAGoodName42069 I noticed that as well and came to the same conclusion. Considering I was happy with the root of 87 is nine and a bit before this video. I'll take accurate to a tenth and not do subtraction.
But if you got a really big number like 938993 how do you know what the nearest whole square root number is
@@Geweldenaar that's memorization my friend. Personally I only know up to 20squared off the dome. And that's fine by me I give up on mental arithmetic in the mid 3 digits anyway.
The trick is actually based in calculus. It’s the Taylor series approximation using two terms for the square root function. Without reading through the hundreds of comments already made I’m sure several other people have probably mentioned this. Just shows you that calculus can be really useful!
What’s really neat is that if you use a Taylor series approximation with three terms, its even closer to the exact value of the square root.
how would you do it with three terms?
Tell us the three term trick.
Wasn't able to take calc in high school.
It just finds the slope of y = sqrt(x) at the perfect square, then does a y = mx + b using that slope. I'm not sure how this is using a Taylor series
Taylor series of 3 terms: f(a)+f'(a)/1!(x-a)+f''(a)/2!(x-a)^2
f(a)=a^(1/2)
f'(a)=1/2a^(-1/2)=1/(2√a) (the 1st derivative, which is the slope at point a)
f''(a)=-1/4a^(-3/2)=-1/(4√a^3) (the 2nd derivative, which is slope of the slope at point a)
x is the point you want to find, and a is the closest number (or any number, but the accuracy won't be the same)
Then you have to expand it and find the cube then bla bla bla congratulations you're accurate for maybe the 3rd digit but we will round the first digit anyways
@@ncpolley Basically the trick in this video is sqrt(x^2 + y) is approximately equal to x + y/(2*x). The original square root can be rewritten to be x * sqrt(1 + y/x^2). The Taylor series for something in the form sqrt(1 + z) is 1 + z/2 - z^2/8 + z^3/16 +... Using that Taylor series, the 3 term approximation would be x + y/(2*x) - y^2/(8*x^3) which is a little rough for mental math but doable. The easier way to improve the accuracy would be to use the closest square (rather than the square below) since y in the original square root can be negative or positive, and the accuracy of the Taylor series approximation is better the smaller the "z" term is.
"I'll give you acouple seconds to work that out"
Me: Sweating
A 30 second add: "I got you"
@TIV67 Playz dafuq
@TIV67 Playz Amen... B-Men, C-Men... X-Men
If jesus can walk on water can he swim on land
Demon too, I must say.
@TIV67 Playz well and good, will it work on how to find square root?
Note : this trick is actually quite good in competitive exams where approximations work,,, thanks
BHAI BANKING ASPIRANT HAI KYA?
@@kankanmalakar no bro
@Ronak Bhadarka anywhere where calculator isn't allowed
maybe in your exams.... in mine, you really had to give the correct answer. Ah, but I am from a developed country, not from india....
@@mikatu and that's why u don't know the jugaad technology we use for the optimum utilisation of the resources available to us😉
I can't believe how much an 8 minute video expanded my brain by
Lmao
It came from Binomial theorem....
It is Approximation of infinite series.....
Lol. I mean its only useful for lower grades like grade 8 because in higher grades u would prob need more exact numbers
@@duk8227 bro, its from Binomial Theorem .... If at higher level you need more digits... Them Expand it more and more..... Its infinite series.......
You can get whatever amt u need...
@@greatmnn3868 u can just type in a calculator and get the correct result. The trick might be interesting in some cases to get the first decimal number but if you are able to estimate the root by taking the root of the next smaller quadratic number you have a reasonable estimation already. In other words, this is useless on any level.
I gained 7 new wrinkles in my brain just by watching this video
How engineers find square roots instantly:
Step 1: find the closest perfect square
Done
As an engineer I can say that it's either this or "Ok, someone boot up MATLAB"
@@finnrock5558 personally I prefer to use simple tools such as calculators and physicists.
@@Consul99 ah yes, physicists. The best tools
@@finnrock5558 @Herp Derp You two must work for good companies, we're not allowed to claim for physicists.
I love this level of humor
Here's how this works algebraically:
Let √c = a + b, where a represents the whole number component of the answer, a ∈ Z, and b represents the decimal component, 0 < b < 1
c = (a + b)²
c = a² + 2ab + b²
Now solve for b:
2ab = c - a² - b²
b = (c - a² - b²)/2a
b² (the decimal component squared) will be very small, so we can ignore it
therefore, b ≈ (c - a²)/2a
in other words, b ≈ the original number, c, minus the nearest perfect square below it (the whole number component squared), all over 2a: which is the formula given in the video for working out the non-whole number part of the answer
e.g. √27 = 5 +b
27 = (5 + b)²
27 = 25 + 10b + b²
10b = 27 - 25 - b²
b = (27 - 25 - b²)/10
b ≈ (27 - 25)/10
b ≈ 0.2
√27 ≈ 5 + 0.2 = 5.2
As further proof, the approximate answer will always be out from the actual answer by a margin of b²/2a:
5.2 - 5.196... = (0.196...)²/10
Should have more likes. Thanks for the explanation.
My brain has not developed that kind of math yet
woah! that’s interesting, but i sure don’t care!
*sweats tears*
So simple
Never expected Bruce from Finding Nemo to be teaching me math tricks at 3am yet here I am
3:38am right now for me too lmao
4:09 AM
@@muhammadtroll2835 yo another Muhammad
Can you explain me...i don't get it.
@@DestinyatTube same voice
This is actually perfect for me, I recently learned about square roots this year and this will be very useful for the years to come. Thank you
If anyone wants to know how this works, it essentially uses the fact (x+a)² = x²+2ax+a²
For example,
√(37) is a little more than 6 so call that little extra bit a
Then √(37) = 6+a
Squaring both sides gets
37 = 36 + 12a + a²
1 = 12a + a²
Since a is small, a² is small and can be ignored thus a ≈ 1/12 so √(37) ≈ 6+1/12
Approximations are better when a is smallest (the number being rooted is large and close to a perfect square)
You can get better approximations by then saying √(37) = (73/12 + a) and repeating the process
You guys are nerds
Watching the video at 2am, couldnt sleep bc the interest was carving inside of me, thnks for saving my sleep.
@@undertraper same. here its now 3am ^^"
i dont understand...cause im still in elementry school...
This can be improved slightly by choosing the CLOSEST perfect square as opposed to just picking the one below. For the sqrt 138 example, choose 12^2 = 144, then SUBTRACT 6/24 (which is the same as 1/4) from 12, leaving 11.75 as the answer, which is closer than the 11.77 obtained using the original method.
Thank you I'll remember this
Yeah it can be done for any power if you subtract your real number from the nearest perfect and take the derivative of the constant where the number is x and adding that to the perfect power
I’m sorry that sounded very confusing pretty much f(x)+f’(x)(a-x)
It also gets improved just by adding 1 (or subtracting 1 if calculating from the perfect square above) to the denominators, such that sqrt(138) = about 11 & 17/23, which is about 11.739 (compared to the real answer of about 11.747).
It works better this way, since (n+1)^2 - n^2 = 2n + 1 (or in this case, 12^2 - 11^2 = 11*2+1 or 144 - 121 = 23) rather than 2n.
Thanks!
Why does this get recommended the day after my test
And why does this get recommended on day when my friend said this is magic and i will never get it on internet
Badluck I guess ?.....
This is RUclips's idea of a joke.
0g1f I want to pass the test
Because you were probably talking a lot about your test near your phone/ Siri / Alexa or texting or messaging about it. Your phone and apps listen to you and analyse your messages and then adverts and marketing appear to suit what your talking about.
This is great! I always thought I was 'bad at math' until I took a VERY basic course in my 40s and discovered that, not only can I do math well enough to get by, but it's actually fun.
Your video presentation is super clear and friendly. I was able to do all the problems the first time through; going to check out your other videos now. Thank you!
Can u give link for that course
Do you have a link to the course? 🥺
@@Derpalerpa Hi, unfortunately I don't have a link, but it was an in-person class called "Math Modules" at Berkshire Community College in Massachusetts. The textbook is "Introductory Algebra" by Marvin Bittinger - publisher is "Pearson."
Last year I picked up the textbook again and started studying on my own. That's going really well, mostly because the book is so well written. Hope this helps!
Math is fun when your “life” doesn’t depend on it. Aka grades.
Edit: 13k likes!? Holy... Thank you, everyone.
Also, there's a fight in the comment section lmfao. I'm very sorry I caused this chaos.
Grades is fun. They motivate you to discipline yourself into the mentality to work hard and eventually success and master it.
Good for people who can't motivate themselves without them, I guess.
@@alexanderkaiser89 you get judged with grades your entire life, it's hard to keep focusing always giving your best.
@@SapphireeCos As I’ve been taught and what I have researched on my own in this very topic, it firmly depends on discipline and willpower. I’ve you’re being judged by your grades, it is partially up to you, to change that very situation and your surroundings.
idk man it's still not fun for me even after school and not needing to worry about grades. After all I really only use the stuff from grade school in my day to day.
For anyone who's curious:
This is based on a linear Taylor Series approximation of the square root function. For example-
Sqrt(x + dx) ~ sqrt (x) + (1/2)(dx)(1/sqrt(x))
And this is exactly what he's doing
neat
There is also an elementary reason for this, for non-calculus students.
@@aSeaofTroubles then tell about it please
@@JonathanBartlesSWBGaming the formula in the video comes out from the taylor series with an error of the first degree. So they are using the same way to calculate it.
Thank you, for doing this. It saves me the trouble.
I trust this man with my life.
@TIV67 Playz no
@TIV67 Playz Jesus is coming indeed bro!
Ooof…I misread that you trust this man with your wife…
@@CH-yt6qi same 🤡🤒
@TIV67 Playz these guys are getting annoying
I was never really good at math. Learning this method actually piqued my interest with numbers. I enjoyed the challenge. Thank you!
I made a quick excel spreadsheet and graphed the percent errors of this method. The %error curve is a sawtooth pattern, but the peaks look to be decaying exponentially. This is very interesting, thanks for sharing.
-Can you show us an image of the graph?-
@@Blitztein_beta how would they do that 🤣
Yeah its because the taylor series is just the sum of the derivatives devided by the faculty of the index (nth derivative/n!).
As we are only using the first summand of that series, the rest of the series is our error.
Now this sum for the sqrt of x looks like an exponential of 1/x, if you look at the series formulation of exp.
Thus you get the exponential decrease in the peaks.
P.S. thought about it again. It is just wrong.
@@TheMrk790 you know what's insane, anyone that understood what you said knows you're just bloviating and anyone who doesn't, doesn't care about what you're saying.
@@pato1541 what if you’re too dumb to know what he’s saying but still smart enough to know he’s just stroking his own ego
the way this came out 3 hours before my scholarship exam starts thank you
U should thank the youtube algorithm.
@@safwanshahriar4108 yeah. lol
Hey, how'd it go?
@@sseblante it went great! i actually passed and now im a scholar! 👍
@@nicoletion1729 Congratulations 😁
This is a good place to get my brain cells back after seeing tiktoks
LmaoXD
Just took a devious lick of knowledge 😜
@@Nerbto cringe tiktok 0 braincells user
@@mo-vz2xk thank you
@@Nerbto I have never felt more disgusted just by reading words
0:58 “this is fairly accurate but not %100”. because thats how math works lol
Actually said “YOOOO” and had to walk around for a second because the math hit too good LMAO
WTF SAME
I guess it works for people who can convert fractions into decimals in an instant, but since I am not one of those people, this method doesn't help me too much.
@@crimsonneko fractions to percentage is harder
@@BingQiLing. not really. Once you have the decimal you just move the decimal place 2 times to the right and then you have your percent.
He did not tell you that in the video, but in case the sqrt is closest to a full number higher than itself, you should use that instead, and your second part of the answer (that comes after the point) will be negative. You'll do the division normally, and then subtract from the big number (i.e. closest full number that stays in front) and you'll get your answer. Example: sqrt of 24, you'll go for 25, and the subtraction will lead to 5 -1/10, which is 4.90. The actual answer is 4.89
Thanks
no the answer actually is 4.9 when you round it up
@@mike_slav0477 the point of this method is that it's a simpler version of an approximation equation. You'll never get the exact same number unless you round it up. I simply wanted to show that you should use the closer number even if it's a higher one. If you were to use ✓16 it wouldn't work. That's all there is to it
thanks lov
Thank you a lot
I stumbled upon something similar to this a few years ago and presented my technique to the math class. Teacher was convinced enough to let us use the technique over the slower "multiply 'til you find it" method in the curriculum.
Reminds me of Richard Feynman when an abacus salesman challenged him to a math contest. Ended in a square root contest. Feynman won because the number was close to a perfect square and he could quickly refine the number. That and doing square roots on an abacus is very labor intensive.
Alternate title: Thor teaches you how to solve square roots :)
BRUH I DIDN'T NOTICE TILL NOW
@TIV67 Playz Does Jesus love Gidon?
@TIV67 Playz amen
Hes austrailan tho
@@extestential6638 One clearly does not understand what a joke is
It is also possible to start with the closest square root above and subtract the fraction as shown. Example root138 = 12 - (6/24). Root 138 = 11.75. Good luck!
you can also apply this same step to your won approximation. It will be a littler harder to calculate, but it will improve your results :)
e.g. use 138-11.75², and then 11.75 + difference / (2*11.75)
Because it's Newtons Method, and you can use as many iterations, as you like :)
I wonder if it becomes more accurate to do it to the closest suqare, since in that example you were spot on while the video was off by 0.02.
@@GahloWake Yes, it would be more accurate to do it to the closest square. How the trick works is it uses the common school formula (a+b)^2 = a^2+2ab+b^2, sets a as the lower perfect square so that 0 < b < 1, and then assumes that b^2 is small enough to ignore. If you go to the closest square (sometimes above), what you are doing is reducing the size of b^2, in other words reducing the error of the trick
@@AntonioNoack use normal english. im mere mortal i do not speak the language of the gods.
This actually makes the trick more accurate, trying for root 15 by the addition method will give 4, but the subtraction method you suggested (essentially follow the closest perfect square and do accordingly) gave 3.875, which is very very close.
I can’t believe how simple this is. Definitely wish I knew about this 10 years ago.
I was stunned at how stupidly easy it is
Fr
Yeah but on a test basically all those decimals are wrong so you would have the wrong answer
@@feeldog1019 I think it helps to see the number in a "normal" form so its easier to understand
@@feeldog1019 depends on the instructor. I had a chemistry teacher that would allow you to approximate your answer if you could prove your answer was reasonable. Usually this was only on tests where calculators weren’t allowed, but it’s worth noting. Plus with this method you could trivialize multiple choice questions by going with whatever’s closest to your result.
I totally LOVE this method! THANK YOU!!!
it's difficult to find the nearest perfect square if the number is very large.
But only needing to know a nearby square to estimate a root can really help at times. :D
It's difficult to do any operation when numbers are very large.
Its difficult to find larger numbers but its closer the larger you go.
@@redsgxd Esitmating the sqrt of something large is still still relativly easy using this method, your error just gets bigger. Just Divide your number by 100 until it is smaller than 100. ex 32767=100*100*3.2767 sqrt using the above method (That I will modefy slightly for increased accuracy) 4 (closest square) -3.2767= .7233 so you get 2-(.7233/((2-1)*2))=1.819 then cuz we divided by 100 (or 10^2) twice multiply back by 10*10 and your final answer is 181.9 sqrt(32767) is exactly 181.1..... so still reasonably accurate
That's the disadvantage of it,it kind of breaks with bigger numbers
Less than 1 month remaining for my competitive exam, and i have found the channel. I'm feeling blessed
@You dont know me I'm from Bangladesh, and its my undergraduate admission exam!
@@hasiburrahman5637 how’s you do
@@hasiburrahman5637 how’d u do
update
Poke update
How to get answers wrong.. but CLOSE!
From all the sqrt approximation calculations I've seen so far this one and Andrew's are the most reliable.
VF7 how to estimate without a calculator dingbat
@nano bot27 It's called approximation.
It's called aproximation. My astrophysics teacher always says: "ah well it's the same power of 10 right?". It's just a matter of how accurate you require your result to be whether you want to use this or not :P
@@paulie5B it's actually possible to calculate more accurately but you need a lot more space for paper
You actually don't understand how much this is gonna help me. I'm in Algebra 2, and every other concept is pretty easy for me to understand, but I never did pick up radicals and stuff like that, so they keep moving on expecting that everyone already knows how to do this. This is genuinely gonna make my life so much easier.
Can u tell how to do the square root of 48
@@Nectrik I got ~7 by going down from √48 to √36, which is 6, and then the difference between 48 & 36 is 12, so that's the numerator, and twice 6 is 12, so it's the denominator. 12/12 is one, which would make the 6 into an approximate 7. I checked my answer with the calculator and the answer was something like 6.93, which is close enough in my opinion.
Why didn’t I get taught this in school? They ask about estimating square roots on the gre and other advanced tests, this would have been so helpful.
Cause School doesnt teach you anything
@@absalon1992 not even a little smidgy widgy?
@@spoopyscaryskelebones3846 What
@@spoopyscaryskelebones3846 nah the smidgy widgy is taught only by the ancient ones.
@@spoopyscaryskelebones3846 Smidgy Widgy is an infamous art. You think schools are going to teach that? You have to travel out to the himalayas and speak to master Wu to learn Smidgy Widgy.
For smaller numbers, you can get a much better approximation if you calculate a square root of a number that's 100 bigger and then divide the result by 10, so sqrt(2) = sqrt(200)/10. This is because the higher the number is, the better the approximation will be.
Great tip deep in the comments!
This should be pinned
Great tip.
Yeah ! but finding the required intuitive square root of 8700 (93) is far more complicated than 87 (9). The final fraction becomes also more challenging (51 / 186). You get of course your loved 93.27 / 10 = 9.327.
In the end, it would be far better to just take a few minutes to learn to do the real method for finding square roots to as many digits as you desire. You are not gaining anything at all by trying to learn these "tricks"
What I am suggesting is that you have no business learning "tricks" to do something, until after you have actually learned a functional method to do that something.
I love how he keeps asking how you did, it's one of the many things he does to make this video nice and engaging
Thanks for sharing this, I realized this pattern some years ago and i have always found it really interesting
For 138, we can also do square root of 144 minus the difference/double of the √144
=12- 6/24
=12-.25
=11.75
Square root *
THX
The smaller the difference is, the closer you get the answer
Maybe even if its lesser or greater, atleast the most closest will be more accurate.
more closer
This guy's like the breaking bad of mathematics. This is methmatics. lol
i applaud you i am so proud
3:33 how does its 6.25 I cant understand
@@e_hland i think it's 6.25 because 3/12= 0.25 and he added that to 6.
@@e_hland 3 is 25% of 12 thats why bro
and 25% equals 0.25
Honestly I stumbled across this while I was half drunk.😂😂 And I was somehow able to follow along and get them all right! I even kept the sheet to prove I get it! This taught me better than school did! Props to you mate! This is awesome!! I hated math but getting these answers right boosted the hell out of my confidence!!
lol, me? graduated HS in 1979; never went any further than gen math. FFWD to 2006 I found myself in college earning a BCS/IT which required algebra 1&2 and Geometry; needless to say I did not do so well with sqrt's ;) FFWD-again, to tonight and I find this video while having drank 1/2 bottle of 10 year old single malt Irish whiskey in celebration of my 61st birthday.... WHAM! 'I now can do sqrt's in my head! I'm not fast (yet). Thanks, just doesn't cut it.
@@marriedwithchildren-familylife the square root of half a bottle proves the Irish reverse-discovered North America
Yea i looked at this at 10 pm and followed along fully, this guys a legend!
Are you laughing at yourself for being half drunk and scrolling through RUclips to watch math videos? Fckng boring af bro
I used to have a shot of Scotch before every calculus test! LOL
After I saw you do the fist one, I did one of my own which is 149, I got 12 5/24. Came back, did all the ones you have before you and got all correct. Thank you sooooo much this was extremely helpful and effective. What good would it do to my test scores if I didn’t subscribe? None. Glad to be apart of the team.
i don’t need sleep, I need answers.
it essentially uses the fact (x+a)² = x²+2ax+a²
For example,
√(37) is a little more than 6 so call that little extra bit a
Then √(37) = 6+a
Squaring both sides gets
37 = 36 + 12a + a²
1 = 12a + a²
Since a is small, a² is small and can be ignored thus a ≈ 1/12 so √(37) ≈ 6+1/12
Approximations are better when a is smallest (the number being rooted is large and close to a perfect square)
(copied off another comment)
@@lolzhunter lol dude get a life
@@RajSinghTanwar_ Somehow I find it less lively to boast ignorance than to share knowledge. Might just be me though.
@@magnusanderson6681 stfu you aint got no life so how can you boast cuz u aint got nothin to boast bout lol
@@RajSinghTanwar_ Clearly, the guy mentioned that he copied the comment. I would recommend you to stop being so ignorant and a recalcitrant person as he was only helping the other person. If you're a person who can only say to other people to "go get a life" - then I say you're the one who needs to get a life.
Thank you :)
Note: pls spread happiness and harmony. It's not hard
I’ve never seen this before, but it’s actually pretty easy to see what that trick is really doing mathematically. All it’s doing is converting the continuous function root(x) and converting it to a piece wise function by connecting the points at perfect squares with strait lines. It works because the inverse function of x^2 has a derivative of 2x, so the slope of each new line is 2x. It’s a pretty neat trick.
Exactly...
This also means that if n is the root of the next smaller perfect square the error will never exceed 1/(2n) which is really neat.
Yeah dude I was totally thinking the same thing!
Can you explain why it doesn’t grab an exact number?
@@jameyfrey7445 What do you mean by "grab an exact number"?
One improvement: Add 1 to the doubled number in the denominator. So, instead of calculating the square root of 27 as 5 + 2/10, do it as 5 + 2/11. On average, this cuts the amount that any given solution is off from the actual amount by half.
genius.
yeh but kinda a hard fraction to work out mentally
@@txtce In this case, definitely, but if we were doing, say, the square root of 94, which would be 9 + 13/18 with their method, and 9 + 13/19 with mine, I wouldn't say that 13/19 is that much harder than 13/18.
How about difference in squares?
For some reason I cant wrap my head around doubling the square root, i dont understand where that part comes from.
But what about this method. I just thought of this but im positive someone else has invented it.I
87 is between 81 and 100 (9 and 10 squared). The differece is 19 and you have 6 remainder. 6/19 is around .315.
I know its not as accurate but it makes more sense to me.
Also 650.
625=25
676=26
676-625=51
650-625=25 remainder
25/51= .49ish
So 650squard root is 25.49
@@tacobender1643 Yep, that is actually the basis behind what I mentioned. The 'doubling of the square root' comes from the fact that the difference between the squares will always be halfway between twice the lower square root, and twice the higher square root. So, in your first example, 2 x 9 = 18, and 2 x 10 = 20. The actual difference is 19, as you noted. So, if you add 1 to double the square root of the lower number, you get the difference between the squares.
this is absolutely awesome, went form completely guessing and taking a good 30 seconds before watching this to being able to calculate roots in like 2 seconds. in 8 minutes of my time. absolutely worth it
“Yeah, you should be able to work that out by now”
*starts sweating profoundly*
Right
*profusely
Na
This was genuinely interesting. I wish you were my teacher, I’d listen with all ears because your teaching methods are actually enjoyable
I see why you have a million subs. How you teach is simple, clear, and direct. Thank you so much for your lessons!
It’s nice when teachers can be teachers and not forced into some regimented way of explaining things handed down by an out of touch and overly academic administration.
This is amazing. I believe it’s called “Approximating Irrational Numbers” and this has been the easiest way to find the square root of “imperfect squares” that I have ever seen. I’m super grateful ❤
They should literally teach this in schools. I'd say 7th grade would be a good time to learn this. (Or maybe don't teach it, just so your viewers have an edge over the average person :))
But this is learned at school, infinite series approximation
No they shouldn’t, he is doing this through a method of calculus, which is hard to understand that early
@Milo Dog mabye, but either way this isnt rly useful cause u usually can just use a calculator.
@@eitankalfa7836 Nahh this is extremely useful for competitive exams where the calculator isn't allowed
@@eitankalfa7836 calculators should never be allowed in schools. It's good at making you bad at math and nothing more. Everything should be done manually. There are more ways to calculate square root manually than this. They could teach those if this is too difficult.
He's talking like there's someone sleeping in the next room
Or the same room:)
Shhhhh!
An SJW.
The sleeping elephant holding its ears
Cause he's an aussie
For those who finished calculus 2: this is the first order term of a Taylor expansion of the square root function. Nice explanation.
Or simply, the tangent line approximation
Exactly. This is essentially how we did distance approximations in 16-bit video games back in the 90s.
Very nice and helpful sir! This is very useful for estimating radicals. Thanks again!
I got curious and tested out this aproximation method in excel against excel's internal square root number.
I calculated the percentual difference between the aproximation and the actual sqrt given by excel.
i worked it out from 1 up to 1600
So it happens that the diference (as expected) resets to zero whenever you get to a perfect square (like 36 or 100...), and then increases as the number climb to the next perfect square.
When you plot it the graph looks like the teeth of a saw.
What's cool is the fact that as the numbers go up the teeth go wider and shallower, so the aproximation get's better and better.
The absolute worst case is number 3 that as a square root 1.73205 and the aproximation yelds 2, thus giving a percentual difference of 13.397%.
But, for instance, 1598 has a square root 39.97499 an the aproximation yelds 39.98718. That's 0.03048% difference. (at 1600 it resets the differences as it is the square of 40)
+Nuno Gonçalves
Thanks for sharing your findings! BTW your descriptions bring so much more to this method. Thanks again.
+tecmath I should be the one doing the thanking for the brain candy, so...
Thank you.
...Keep them coming.
BTW, if you want, i can send you the excel file.
Regards
Percentual is not a word. "Percent" difference.
+Danana J:
Thanks. It's always good to know.
Allow me to excuse myself, English is not my native language and i'm not using a spelling checker.
Most words that exist in my native language have an English counterpart, so i tend to apply them.
Sometimes, such as this one, that is not the case and the word doesn't exist in English (rare).
Needless to say we have both "Percent" and "Percentual" (the later meaning that we are talking about something with percent-like caracteristics).
Regards.
+Nuno Gonçalves Yes i undertood your use of "percentual". Btw this "amazing" trick done by tecmath is just a Taylor series approximation, which is most mathematicians use to do these "party" tricks.
The material in this video is application of the binomial approximation:
sqrt(1 + x) = (1 + x)^(1 / 2) ~ 1 + x / 2.
Let the value under the square root be written as a + b, where a is the closest perfect square and b is the remainder (difference from the perfect square).
You then have
sqrt(a + b) = sqrt(a) * sqrt(1 + b / a) ~ sqrt(a) * (1 + b / (2 * a)) = sqrt(a) + b / (2 * sqrt(a)).
I like your funny words, magic man
I came here looking for this, thanks a lot! I think this should have been added to the video.
If you say so
Well that's a whole lot less fun
Burn the witch 🔥
And just like that... 32 years into my life hating math... I now know what a square root is and was able to pause the video and complete each of these problems. Thank you! That was surprisingly fun!
I have had this video in my watch later for approximately 5 years, I always knew I would need it. now im watching after it got recommended to me for searching up a similar topic
It would be nice if you could show why this is true?
Differential calculus. So we should thank Newton for this :P
Say you need to calculate root(x+ dx) where dx is much smaller than x
Here x is the nearest perfect square, and dx is the extra number
We say let y = root(x) and y+dy = root(x+dx)
Our goal is to find dy, we can differentiate y
dy/dx = 1/2root(x)
Or dy = dx/2root(x)
Hence our final answer is
root(x+dx) = root(x) + dx/2 root(x)
You can now understand why he is taking that extra number, and dividing by the double of root of the nearest perfect square.
This also explains, why you cannot use this method for squaring numbers, where the extra number is comparable to the nearest perfect square, the differential approximation works best when dx is much smaller than x (atleast 1 order of magnitude, else you get horrible answers :P)
Thank you for this. :)
It feels good to know why certain simple methods work.
you can also try looking up "binomial theorem" if you are not that familiar with calculus. this approximation is the first two terms in the expansion of (a^2 + b)^(1/2). you can also use (a^2 - b)^(1/2) as others have pointed out, if it gives you a smaller "b". the smaller the value of "b" gets compared to "a", the better the approximation gets. this is because it makes the terms you are ignoring in the binomial expansion after the first and second become smaller faster. this is why he says to use the perfect square closest to the number you are taking the root of.
You mean the Newton-Raphson method?
no science here
true
I've noticed that the closer you are to a perfect square, the more accurate the answer gets. When I say closer I don't mean sqrt 2 is very accurate bc it's one away for sqrt 1, I mean fraction wise of the number, 2 to 1 is a big jump of 1/2, so it's not that accurate. But 27 to 25 is a much smaller fractional jump, which is why it's so close to exact.
Actually, you get good results when you are just above a perfect square, but very bad if you are just below. Sqrt(24) -> 4 + 8/8 =5. A better formula, that indeed works well whenever you are close to a perfect square from both sides, is N+R/(2N+1) instead of N+R/(2N).
This works because the distance between two consecutive perfect squares N^2 and (N+1)^2 is 2N+1. Taking the residual over 2N+1 would be basically looking at a weighted interpolation between the following and previous perfect square.
Graphically, what this would be doing is basically approximating the parabola X^2 with a series of segments connecting all perfect squares. So the result is close to perfect if the number is close to a perfect square, and far away if it is in the middle between two perfect squares.
yeah of course because x has to be way greater than dx (you ve already seen the formula above in the comments.
@@joebangles9951 RUclips doesn't allow links
I guess the "heavy lifting" of the trick is the first step, where you figure out the rounded-down square, and if you know the rounded-down and rounded-up square then you might as well just make a rough guess.
You can actually invert this method to go off of the next perfect square, by inverting your addition/subtraction operations.
i.e. √95 -> 10=√100, 100-95=5, 10-(5/20)=9.75
√95=9.747
Using the normal method gives you 9+(14/18)=9.777, which is clearly less accurate.
You'll find that the method that uses the smaller numerator (aka the one that uses the closer perfect square) will be the more accurate method.
Dude, this is so good. Square roots have always been difficult for me. I love math and I'm not bad at it at all. It's the class I did best in all through school. Every year I'd get high 90's in math while I tested very poorly in most other subjects. I never fully memorized my multiplication tables though because my brain just works differently, probably partly because I'm dyslexic with numbers, so that made everything else a little harder, especially square numbers.
I never would've thought of a trick like this but once you explained how it worked it instantly made perfect sense. I feel like this trick just gave me a better sort of understanding of, or intuitive feel for, square numbers and their roots, like over all, as a concept.
What about integration and calculas? You were good at that too?
this is the best trick i've seen so far! thankyou very much for this
Thanks a ton, you're a lifesaver! My statistics teacher made this way more complicated by doing it through long division and I was lost. Now I'm an ace in finding square roots!
I was never taught how to do this, but this method is how ive always done approximation in my own head, Seeing it done and on board makes the process understandable so much easier. Thank you!
Me at the begening : lol he is just gonna say « use a calculator »
At the end : yep, pretty accurate
How
Beginning*
No it’s not.
What are you talking about. He didn't say anything remotely similar to that.
@@bengraham9448 biggenengue*
Awesome video. I'm currently working with finding the distance between coordinates and this is a life saver. Thanks!
Me after a couple of math vids on RUclips:
Cube that address
Square that license plate
Find food at whole foods with perfect square prices
Evaluate the sale and maximise profit
-Gotta love RUclips
Get out more! 😉
Next shall be Multiplying Taxes
😂😂😂
RUclips loves you too - it knows more about you than an experimenter knows about his lab rats
E
Chinese adults: *this 2nd grade in china*
Indian kid: Hold my Mathmatics Notebook 😂😂
anand kashyap OMG RIGHT 😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂👳🏾
Unless you’re Chinese, that’s very racist :))
edit: I'm chinese, btw
TheAnimationGalaxy it’s not VERY racist but it is a stereotype and can offend some people
Nice job conforming to be in a rasict who uses hurtful stereotypes
Wow, the trick at the end was super helpful. I always knew my time’s tables/perfect squares, but estimating the tenths and hundredths digit was always a rough estimation. Thank you.
This is really interesting and helped me a lot!!! Learning about square roots for قدرات. Watched this while waiting for dad to come home to go to the beach
Thank you so much! I didn't know this trick back in college. Without a calculator, I've been using differential calculus approach just to find the approximate square root of a number.
Surprisingly helpful and easy! Strange I wasn't taught this earlier!
U just helped me to prepare for a test i have on predicting square roots tommorow!!!!! Wish me luck!!!
EDIT: By the way i surprisingly got 100% the most in the class!!!!! (Gr.9 quiz)
Good luck!
You are under legal obligation to inform us of the outcome of your test
@@tecmath thank u
Congrats bro what did the teacher say?
@@liamdatta6888 the teacher:StOp ChEAtINg
So excited to be watching this on a Saturday morning.
THE BEAST HAS AWOKEN
AFTER THE 11 MONTH SLUMBER
TO GIFT UNTO US
SQRT
Yeah!Finally a video!!!!
illdiewithoutpi unto? dunno that word
LOL! TAU!
thats old english. search it on the net
TAU! vihart would like you lol
2+2 is 4
-1 is 3
Quick maths
God left the server.
Giselle P. Wrong.
According to your statement:
2+2=4
-1=3
-1 never equals 3
A more correct statement would be
2 plus 2 equals 4
4 minus 1 equals 3.
Curb your memes.
Jenshie 02 yeah but he forgot to put 4-1, he just said -1 is 3.
Curb your memes.
Jenshie 02 oh and he misspelled "maths" it's mafs
acrilico 3451 ah you clearly don't understand the meme "curb your memes" it's a fkn joke.
R/WHOOSH
Well to be technical, the square root would be much easier when comparing the remainder number to the next closest perfect square. So 36 squared is 6, but we have 3 left over from the 39, the next closest square is 49, the difference betwen 36 and 49 is 13, so we have our remainder over the difference. The square root of 39 is 6 and 3/13ths.
my Math teacher was so mad at me because when she ask me about square roots I didn’t know nothing but after I watched this video I was able to do any kind of square roots😊😎 Thank you!!!🙃
My past significant other had a daughter who was 5 at the time....I taught her the way to show the radical of 49 with her fingers....it was a huge hit!!
I would like to know more.
Please elaborate
Would you like to continue
Explain
I love math tricks and such, and this absolutely blew my mind. I saw the title and it immediately reminded me of the multiplication by 11 trick I memorized a while ago. Such a great video
Exact method I used all through school - excellent for multiple choice where you dont need to show your work and approximate answers suffice.
As always, our brothers from down under keep the content Crisp and Clear with a minimum of fuss and fluff.
Thanks for sharing 👍
I would adjust a little bit : if the digits after decimal points are >25 and 50 and 75 remove 0.03
for instance 190: 13²=169, 13(21/26)=13.81. 81>75 so remove 0.03: 13.78, which is the perfect result
Nice way to demonstrate the Taylor series works!
√(N^2 ± M) = N ± M/(2N) ∓ M^2/(8N^3) .... where N is an integer and M an integer smaller than N.
This is crazy after only the first problem I was able to do them in seconds thank you so much dude!
THIS method I find easy and useful. Of course, I might forget every single step, but a few second's refresher will quickly "bring me up to speed."
These RUclips type channels could help provide free education to the masses
I hated mathematics until yesterday. Yesterday I found your channel and I love it. I feel so good when I do the example right within seconds and all in my head! Thank you so much!
The fact that these are so close to the real answers doesn’t seem like a coincidence. Perhaps there’s another step we don’t know about yet in order to get the exact answer.
It's actually relying on a process learned in calculus called linearization. So while there are other techniques that can be used to get more precise values the technique used here is kind of a one shot deal. It gives a fairly precise answer that can be easily calculated by hand.
It’s not a coincidence at all! It comes from the following fact:
For some number N and a smaller number a,
√(N² + a) = N + a/(2N) - a²/(8N³) + a³/(16N^5) - 5a^4/(128N^7) + ...
Where the sum technically goes on forever. If you’d like to google around about this, search for the “Taylor series of square root”, which is the technical mathematical name for this fact.
Anyways, the reason this trick works is that the later numbers are so small. Let’s use the √27 calculation from the video as an example of how this works. N = 5 and a = 2, because 5² + 2 = 27. Replacing N and a in the sum, we get
√(5² + 2) = 5 + 2/(2×5) - 2²/(8×5³) + 2³/(16×5^5) - 5×2^4/(128×5^7) + ...
Then if we want a better guess we can just add more terms:
Using the first two terms, which is what we did in the video, we get 5 + 2/10 = 5.2.
Using the first three terms, we get 5 + 2/10 - 4/1000 = 5.196.
Using the first four terms, we get 5 + 2/10 - 4/1000 + 8/50,000 = 5.19616.
I won’t bother doing this with five terms, because we’ve gotta stop somewhere. The actual value of √27 is 5.196152... which is REALLY close to our guesses! But, as we keep getting more accurate, the calculations get harder, and the numbers we added got smaller and less important. Eventually, calculating more terms becomes pointless. If you want an error of ±0.02, the first two terms (what’s shown in the video) will work as long as the number you’re square-rooting is at least 34.
TL;DR: getting a better answer is possible and we know how to do it, but it’s not really worth the effort.
TheBasikShow basically explained (quite nicely) how most calculators work - using Taylor series to get a closer and closer approximation as the number of iterations increases - calculators can only do arithmetic - so anything complex or irrational (i, e, pi, sqrt, etc.) must use a series approximation to get enough digits of resolution to satisfy the display capabilities. Even the TI-84+ Silver uses simple arithmetic. But SOME calculators (like my TI-89 Titanium 😎🏎) have a symbolic equation solver (Computer Algebra System) and can give irrational solutions, solve symbolic equations as well as run code, graph in 3 dimensions & parametrics, and more.
@@sinistersparky9657 i have never seen a guy flexing his calculator. Lmao
Rahul Singh, You should see the people in my (Calculus III) class at UofT
I tried this on a few numbers and checked them with a calculator. Pretty darn close. It's kinda like using 22/7 for pi. I'm impressed.
If you want to make it even more accurate, you can subtract the difference squared divided by 8 times the cube of the perfect root. Yes, you're all secretly marvelling at the power of calculus :-)
*Edit:* I can probably use the magic of Unicode to turn this all into a formula. So, let's define some notation: we want to calculate _√x,_ where for example, _x = 87._ Let the closest perfect square to _x_ be _y,_ so _y = 81._
The video shows you that _√x ≈ √y + (x - y)/(2√y)_ which, in our example, is _√87 ≈ √81 + (87 - 81)/(2√81) = 9 + 6/18 = 9.33..._
I'm saying you can add an extra term to the approximation: _√x ≈ √y + (x - y)/(2√y) - (x - y)²/(8√y³)_ which, in our example, is _√87 ≈ √81 + (87 - 81)/(2√81) - (87 - 81)²/(8√81³) = 9 + 6/18 - 36/5832 = 9.3272..._
Since the actual square root of 87 is _√87 = 9.3274...,_ that extra term of _-(x - y)²/(8√y³)_ gained us about 0.01 units of absolute accuracy. The next term would be _+ (x - y)³/(16√y⁵),_ but that gets tough by hand.
I’m having a stroke reading this it’s probably really simple but can you explain it simpler
@@dragondrawings3144 I hope your brain is okay. I have edited my comment with symbols.
So this is what I missed out on. Honestly doesn’t seemed too much more difficult compared to the concepts in algebra…🤷♀️
@@hunterblacc4336 You mean you didn't take calculus? This is an application of calculus (Taylor's approximation theorem using derivatives), but not calculus itself.
@@Mew__ thank you much
Thank you! If only I found this before my exam- this is actually pretty accurate! I tried it with 104 and it worked a charm!
Wow this is great, it’s fascinating because most tricks I see decrease in accuracy for larger numbers but this one actually increases, not only that but the error margin decreases pretty rapidly.
The highest error point is at 3.99999999999 with 25% off of the actual answer. It resets to 0% at perfect squares before rising to only 8.333% off at 8.99999999999 and then less than 5 there after, after 64 it’s permanently less than 1% off.
Also interesting:
If you accidentally pick the wrong perfect square, it still kind of works...
Sqrt(155)
Based on 12^2=144, with a difference of 11, gives 12+11/24 = 12.4853
Based on 11^2=121, with a difference of 24, gives 11+24/22 = 12.0909
Correct answer would be 12.4499.
This error self correction works better the higher the number you're taking the square root of.
I think it works better the closer the perfect square is to the number
Wouldn't the difference between 121 and 155 be 34 and not 24? In which case it would be 11 + 34/22 which is approx. 12.54
Little tip: when you add the fraction, instead of using 2n as denominator try with 2n+1, as it is more precise. The reason 2n is recommended is because it usually gives an easier fraction to deal with, but if you happen to find one where using 2n+1 is easy, then it's better. But, after all, the difference only matters with small numbers, as increasing the value minimises the error anyway.
For example sqrt(22): the real value is 4.69, with the "2n" method it's 4+6/8=4.75 (+0.06), with the "2n+1" method it's 4+6/9=4.667 (-0.03).
Another example is sqrt(34): the real value is 5.83, with the "2n" method it's 5+9/10=5.9 (+0.07), with the "2n+1" method it's 5+9/11=5.82 (-0.01).
But, again, the difference is so small that, for what it's worth, you can just choose the easiest among the two.
youre right 2n+1 gives a better approximation for these cases but 2n isnt reccomended because it gives an easier fraction. 2n is used because it is a tangent line approximation. 2n+1 gives a better approximation because the function sqrt(x) is concave down so this method will always overestimate the square root
It's better to know both.
2N+1 does not necessarily give better result. For example:
sqrt(100001) is closer to 100+1/200 than 100 + 1/201
@@howareyou4400 I didn't say it's always better, but that it is overall more precise.
2n+1 is always slightly wrong, while 2n is a few times slightly better but most times a lot worse.
Interesting but if I really want accuracy I'll just use a calculator.
Me after a long day of work: Finally, now I will watch something interesting and non-work related
ends up watching this lol
I’ve searched everywhere for the right explanation and you just did it thank you
Why is this fun for meeeeee. Thanks for sharing!!!! So much less stressful learning this way when it's just in my free time and not like school at all. 😭😭😭