Estimating square roots in your head | Hacker News Reader

Estimating square roots in your head

gregorygundersen.com

309 pointsalexmolas111 comments

I have a better proposal than the last line. Just repeat Heron's method, using the approximate square root.

In the article they find that sqrt(33) is approximately 5.75 = 5 3/4. Square that and we get 25 + 2 * 15/4 + 9/16 = 33 1/16 = g.

Now we want n / g = 33 / 5.75 = 33 / (23 / 4) = 132 / 23 = 5 17/23.

Average that with g and we get (5 3/4 + 5 17/23)/2 = (5 69/92 + 5 68/92)/2 = 5 137/184 = 5.74456217... The actual answer is 5.74456264...

What's going on here is that we're using Newton's method, and doubling the number of digits of accuracy every time. By contrast the Taylor series is only adding a fixed number of digits for each term. So "do the simple stupid thing, take feedback, iterate" is far better "do the complex thing right the first time".

That's a lesson from math that is widely applicable everywhere. Such as for running startups.

The slightly easier version is to use Heron's method, but know the squares of half-integers. They are: 2.25, 6.25, 12.25, 20.25, 30.25, 42.25, 56.25, 72.25, 90.25. You might notice there is a similar pattern to these as the familiar odd-number increments in the sequence of integer squares. It also happens that (n + 1/2)^2 = n(n + 1) + 0.25

The iterated method tends to double the number of bits of accuracy at each iteration. Moving from integers to half-integers gives you one more bit of accuracy on average. When you apply the first iteration, this becomes two extra bits.

This already gives accuracy within 0.2% in one step for any integer (except 3 and 5, with errors of 1.0104% and 0.6231% respectively).

You don't actually even have to know the squares of half integers, though.

Like, if you're looking for sqrt(30) and you're trying to decide 'should I use 5 or 6 as my first estimate?'... you don't actually care, just pick 5.5. You're looking for a decent estimate, and in that situation you can tell 5.5 is going to be a better one than either 5 or 6, without actually knowing or caring what 5.5^2 actually is.

Likewise if you're going for sqrt(33) you could take a reasonable guess that 5.75 will be a better starting estimate than either 5.5 or 6, just by gauging roughly where 33 sits on the logarithmic scale between 25 and 36.

The point of Heron's method, though, is that no matter your starting estimate, it is guaranteed to be even closer to the right answer. So even if your mental logarithmic estimate is really well attuned and you think 'yeah, 33 looks like about 5.74 squared', Heron can refine that to 5.7446.

I‘m sorry, can you show an example of where this helps in the usage of Heron‘s Method?

Here are the results for numbers up to 50:

http://codepad.org/NBmpZGpT

The advantage simply comes from getting a better initial guess in some cases. When the test number is halfway between two perfect squares, the advantage is maximized -- for 30 we get 5.5 for the one method and 5.47727~ for the other, with a true value of 5.47722~. For some numbers, like 35, the integer square is closer than the half-integer, so the results are the same, but these are also the numbers with the best approximations to start with.

I once met a guy who could compute logarithms in his head, as well as exponentials (or antilogarithms, as he called them). He did that slowly but steadily, at about one decimal per second, to an arbitrary precision. He could start giving the answer before you finished reciting your question. That's actually easy with logarithms, because the logarithm is essentially the number of digits.

As a byproduct, he could compute square roots easily, and also cubic roots and tenth roots. The easiest case for him, he said, because it was just "a shift".

I cannot recall his name, he was a kind of "showman" that did his tricks to a large audience. From Colombia or Peru, I think. He was invited to our math department, and after his show, he explained in petit comité some of the secret sauce. We were a bit surprised to hear that he essentially memorized a large part of a table of logarithms. But maybe he was just trolling us.

> As a byproduct, he could compute square roots easily, and also cubic roots and tenth roots. The easiest case for him, he said, because it was just "a shift".

https://en.wikipedia.org/wiki/Shifting_nth_root_algorithm

> We were a bit surprised to hear that he essentially memorized a large part of a table of logarithms. But maybe he was just trolling us.

I am not familiar with that particular demonstration but having studied similar things, they are all a trade-off between doing computation cleverly in your mind and memorizing information. The more you memorize and the less you have to compute on the fly, increasing speed (step 1 of practicing that type of show is to learn mnemotechnic thoroughly).

I used to be able to do logarithms in my head, not to arbitrary precision though, I'd just get 2 or 3 decimal places.

Square roots, same thing, take enough math classes and don't use a calculator to do the arithmetic, and after a few quarters you can get down to a couple of digits pretty easily.

If you used a slide rule you got a feel for what the answer will be, so in that case this guy's skill might be less than a superpower and more like good intuition. More likely though that case would not apply as slide rules have been dead for half a century.

I'm not old enough to have used a slide rule "professionally" (i.e. for work or even school) but I did use my dad's old engineering slide rule for a while in high school physics out of sheer orneriness and desire to be weird.

Once I got used to it I often had the answer (to a decimal place or two) faster than my classmates using calculators, when doing problems as the teacher was working them out on the board in class. Of course I had some advantages they did not, for example I only needed one or two digits of significance for problems like that, and if I got the power of 10 wrong it was immediately obvious.

This is nice, but I personally just do a rough estimate based on known perfect squares and the idea of graph in my head. If I need anything more accurate than +-0.5 for small numbers, and +-10 for large, I'd just take out a calculator. Small numbers are much easier to estimate accurately than large numbers.

Everyone generally knows the perfect squares up to at least 12, and then for bigger values, you can use even powers of 2, which I assume people also know. The useful trick is remembering the square root product/dividend property.

Some examples, for a small number, 33, that's less than 36 but more than 25 and it's a lot closer to 36, so I'd guess 5.8 or 5.7 (actual value 5.74..). Halfway between the numbers wouldn't necessarily mean halfway between the known factors. For a bigger number like 1076, you can use 1024, and since it grows more slowly as the factors get bigger, I'd assume something like 32.5 (actual value 32.74..). For a number like 34128, 34128 is ~ 32000, 2 * 16000, 2161000 (almost 1024), so 1.5ish432 so 192ish [1.5*128=128+64] (actual value 184.74).

Heron's method is much better in terms of error though. On the other hand, the method I use is better for larger numbers. I thought about trying Heron's after breaking down the number into smaller factors, but there's something about the division operator with decimals that feels exhausting to even think about.

BTW there’s a formatting issue with your last example. 2161000 should be 2 * 16 * 1000

In the movie "Gifted" a girl calculates the approximate square root of a 4-digit number in her head. Here the scene on YouTube: https://www.youtube.com/watch?v=37meAwQqPsE

Later, it is said she is using the Trachtenberg system: https://en.wikipedia.org/wiki/Trachtenberg_system

The square root method seems to be described in the book "The Trachtenberg speed system of basic mathematics", the text is available on archive.org: https://archive.org/stream/TheTrachtenbergSpeedSystemOfBasic... (search for "square root").

There's a lot of these math "tricks" or "shortcuts" that sound crazy at first listen, but then with practice, they turn out to be quite useful. I forget what the most recent brouhaha was called, but when I finally read up on what this "new" math being taught was, I just rolled my eyes. The problem with the recent teaching kids shortcuts to me was that they were seemingly only teaching the shortcut rather than teaching the long way so there's a proper understanding before teaching the shortcut.

In high school, I participated in an event called Number Sense. 10 minutes to answer up to 80 questions. Catch was no scratch paper, no errant marks, no erasing, no modifications for anything. If you tried to turn a 7 into a 9, it was marked wrong. squares and roots were common. 3 digit numbers multiplied by 3 digit numbers. lots of things that once you knew the shortcuts made it very possible to do this.

Did they only teach the shortcuts? It seems just as likely to me that they taught all the way through, but the parents only started complaining because, without seeing the whole development, they weren’t able to come up with the shortcuts on their own.

The solution of course is for the parent to read, like, a paragraph from their kid’s textbook.

They did not only teach short cuts. New math actually does a much better job of helping kids understand how it works rather than memorize rules and it has significant data to back that up. It’s basically better in every way except that it is unfamiliar to parents.

People are just averse to change. I can’t tell you how many math-illiterate middle aged people I know who have said something like “If it ain’t broke don’t fix it” about new math. Then I ask them some simple multiplication problem they can’t do in their head and point out that maybe it is broken and that’s why they’re not good at math.

This is some first class patronizing bullshit.

First of all if it was truly "New Math" that you were talking about, that was generations ago. And the classic that took it down was https://www.amazon.com/Why-Johnny-Cant-Add-Failure/dp/039471... - which was written by a math professor. Almost certainly what you're talking about is Common Core, not New Math.

So let's move on and pretend you talked about what you probably meant to talk about.

It is easy for you to dismiss the concerns of math illiterates whose kids failed to learn. But I've got an advanced math degree, and I assure you my complaints do not come from a lack of comprehension. Please do not dismiss them.

Next, Common Core was multiple things. Officially it was a set of national standards. That set of standards could theoretically have been met by a variety of different programs. But there was also a set of textbooks produced that had Common Core all over the titles, which necessitated extensive retraining of teachers in programs that also had Common Core all over the name. And the entire package - standards, textbooks, training and the changed classroom process - were all generally called Common Core.

I bring this up because I'm going to talk about what actually happened. And I've seen a lot of defenders try to sidestep by pointing to the standards and talking about how many ways that they could have been met. Yes, there is a theory under which it could have been great. But that isn't what happened. And the complaints are about what happened.

What I observed with my own child is this. I don't know how well his 3rd grade teacher understood math in the first place - given how many teachers in practice can't tell you whether 3/5 is larger than 2/3, odds are not great. However her retraining in Common Core apparently left her confused about everything except how to convey a general sense of enthusiasm. Therefore my son got shown 3 ways to do long division, none of which he understood, and I suspect none of which SHE understood. Given the plethora of problems that he had to do (from his point of view) with random techniques, he learned none of them. He managed to still score in the top 5% on state tests, but only because he was good at doing problems in his head. He was missing basic skills like how to write anything down, which I had to fix a couple of years later with extensive tutoring to teach him what school was supposed to.

Talking to other parents, the biggest difference between our experience and theirs is that my son got the tutoring he needed. Common Core was an unmitigated disaster in practice.

Now, you say, these are teething problems and could have been addressed if the program ran on long enough? I disagree. This was an entirely predictable disaster, intentionally created by major players in the education disaster, which is only one of many waves of disasters. From the actual New Math disaster, they learned that there is good money to be made from rewriting all the textbooks, giving expensive training, redoing the tests, and so on. And when you take advantage of a particular reform wave for enthusiasm, you guarantee that the rollout will be bad enough to generate a backlash. A backlash that generates its own reform wave, which all the same institutions fall over backwards to assist, guaranteeing a new set of textbooks, retraining, new tests, and so on. Very profitable for them, and since most parents only get to see 1 or 2 iterations, few put blame where blame belongs for the disaster that kids go through. But if you come from a family with a lot of teachers like I do, you get more perspective.

Anyways, back to what happens. You admit that it is a problem that it is unfamiliar to parents. But that problem is much bigger than you acknowledge. For a variety of societal reasons, schools ignore the general ineffectiveness of homework and assign lots of it. Research shows that this moves the responsibility of teaching from schools to families. (With corresponding impacts on families that lack the skills, but let's not digress.) And so if the parents don't know the techniques taught, the parents can't help. It is essential that either schools not assign homework to 3rd graders, or they assign homework that parents can help with.

And with Common Core, they assigned homework that parents couldn't help with. I know, I tried. My son would come with a worksheet with lots of boxes where you were supposed to write the right thing in each box to practice the technique. The problem was that my son didn't know what technique he was supposed to write down. I looked at it and found at least TWO techniques that could have been used to fill out on that worksheet. I had no idea which one the teacher intended so couldn't help. (Turns out that the teacher intended a third - there are lots of techniques that work.) And so there was absolutely no way that this homework could serve any useful purpose other than performative art.

Moving on, let's discuss the issue of the techniques.

Common Core advocates preached the value of understanding multiple approaches for the same problem - that when you do you understand better. And also pointed out that different students find different approaches click, and so theorized that showing multiple approaches would let students find what worked for them, and create mastery. Indeed each technique was mathematically sound, and each also had some evidence of effectiveness. Plus pilot programs found that people who understood this approach were effective.

What's wrong with this picture?

First, knowing multiple techniques and fluidly switching between them is a result of mastery, it is not a path to it. For absolute beginners it is more important to master one way of doing it, then elaborate. There are many techniques that could work, and which one you pick first doesn't matter as much as that you DO only pick one.

Second, results about what works when experts teach are meaningless. Experts teaching something that they are passionate about do well regardless of what methodology they do or don't use. Therefore their success is both expected, and not a predictor of success when you roll the program out.

Third, the multiple techniques idea is incredibly demanding on the teacher. The teacher has to know all of the techniques well enough to recognize which one a given student is clicking with so that the teacher can focus on what that student needs. Most teachers do not have this level of mastery - my son's clearly did not. And even if the teacher does, this is an incredible level of individual attention to demand when faced with realistic class sizes.

The result is that all techniques got shown to all students, most of whom mastered none of them. And the students failure to master any technique was a predictable disaster. Indeed from my perspective as someone with exposure to the reform cycle, almost certainly an institutionally intended one.

And finally, let's talk about your "significant data to back that up" point about Common Core. To a first approximation, there is zero data to back that up for Common Core as it was implemented. As I already indicated, the kinds of evidence that existed in advance of the standards being finalized are not ones that we rationally should expect to translate to practice in the classroom. Furthermore from first principles we should distrust any big bang, rewrite everything, reform. Changing everything is inherently risky because any mistake cascades. As I noted at https://news.ycombinator.com/item?id=34631838 we should do the simple thing first, get feedback, and iterate.

And if you ARE going to do a big bang upgrade, you should upgrade to something WITH REAL WORLD EVIDENCE OF EFFECTIVENESS! There may still be teething pains. But you've got good reason to believe that there won't be inherently shortcomings in the approach itself.

If they had done that, the single program with the best data that I'm aware of is https://en.wikipedia.org/wiki/Singapore_math. Note the focus on greater mastery of fewer techniques, each of which is mastered through multiple modalities. Yes, the Common Core people said that they included techniques from that, so they were at least as good. But including something plus a lot of other things doesn't actually work when the thing you're including works BECAUSE IT IS SIMPLE. Lose the simple, and you lose what works about it.

But, of course, Singapore math will never be adopted. Why? Because those with political influence in the educational sector wouldn't get to write new textbooks, do retraining, or rewrite tests - those things already exist. Worse yet all evidence suggests that it would work. Which would end the reform gravy train that the industry has depended on for decades.

It is worthy of note that Common Core had both math and English standards. I only talked about math. The English disaster was a little different, but just as predictable. https://www.brookings.edu/blog/brown-center-chalkboard/2021/... goes into this a little bit.

> given how many teachers in practice can't tell you whether 3/5 is larger than 2/3,

Your claim is that math education was better before Common Core, when, as you say, not even teachers understood the fractions they needed to teach?

The Eureka books show how to solve the problems on the page adjacent to the homework. Eureka website has free parent resources on websites. If your teacher is sending home mystery homework, that's just a bad teacher.

Your claim is that math education was better before Common Core, when, as you say, not even teachers understood the fractions they needed to teach?

I made no such claim.

To the contrary I pointed at a 50 year old book on how bad math education was, and said that there have been wave after wave of reforms since, each driven by the failures of the previous one. From this you should infer that I think math education has been broken for a very long time. Only the details of how it is broken change.

Furthermore you point out the past failures of teachers to understand the fractions that they needed to teach. But you haven't said anything to indicate we should believe that it has improved since. I don't believe it has improved. Teachers STILL don't understand fractions properly.

Classic. Just blame it on the teacher.

She didn't pick the textbook, and printed out the prepared homework sets which went with the textbook. She also didn't have a choice about how much homework - that was school policy. As was not having the kids carry their textbookshome.

While there were problems with the teacher, this one really can't be blamed on her.

But it gets better. My son was in this course in 2012. According to https://deutsch29.wordpress.com/2019/10/06/a-great-minds-com..., the contract to create that website was awarded in 2012. We would have needed access to a time travel machine to take advantage of a website that did not yet exist! (I don't remember what textbook we had, but it certainly did not then have a website associated with it.)

So stop blaming the teacher, who was essentially also a victim of the system. Instead read through my explanation of why disaster was guaranteed from the start.

I've said this before, but public education seems to be in a death spiral of:

1. Reduce teacher autonomy

2. A lot of the better teachers dislike this, so move on

3. Decide that the teachers remaining aren't competent enough, so reduce their autonomy

4. GOTO 2

Not being able to decide how much, and what homework to assign is beyond absurd for someone who is so least ostensibly a teacher. If you are dictating what they are doing to that detail, stop requiring a college degree and call them day-care providers.

You're missing the step where Teachers College of Columbia University first hijacked the teaching profession for a specific progressive philosophy that aspired to a scientific approach to education, but went with pseudoscience around things like "learning to learn" instead. This provided fertile ground for fad after fad that later swept teaching. This created grounds to want to restrict teacher autonomy in the first place.

I recently re-read To Kill a Mockingbird and was surprised to find a first hand account of what an initial encounter with this philosophy was like back in the 1930s.

What I'm seeing from your long essay is that the common core curriculum is indeed great but average teachers are too stupid / mathematically illiterate to teach it, so we would be better settling for mediocrity instead.

What I'm seeing from your long essay is that the common core curriculum is indeed great...

That summary is woefully incorrect.

I will grant that Common Core had well-meaning people, some good math, and interesting pedagogical theories behind it. But it was broken on every other level. "Great" is hardly the last adjective that I'd use for it.

I believe that its pedagogical theories don't work in practice for children. The process by which it was created guaranteed that its standards would not be achievable in practice. The process by which the textbooks were created guaranteed that they would mostly consist of rough spots which would need a lot of ironing out for their target audience. The process by which teachers were trained was supported by decades of practice in creating enthusiasm for bad ideas, and the teachers showed it. And the inevitable failures of real teachers to deliver would guarantee that they could be blamed for the eventual failure of the whole endeavor.

...but average teachers are too stupid / mathematically illiterate to teach it...

What did I just say about blaming teachers for a multi-level disaster that was doomed from the start? How convenient of you to provide a demonstration.

...so we would be better settling for mediocrity instead.

HAH!

I discussed Singapore math briefly at the end. According to https://factsmaps.com/pisa-2018-worldwide-ranking-average-sc... Singapore is currently #2 in the world for delivered results on elementary school math education. But the #1 country, China, adopted the same techniques. Individual schools in the USA that adopted that technique have also had great success.

If we care about actual evidence over rhetoric, we should choose what has proven to be excellent in practice. Which is NOT Common Core.

> Then I ask them some simple multiplication problem they can’t do in their head

to be fair, it does require some practice. the stuff i used to do in my head is long since idle. there have been times i've struggled to remember the shortcut to the point i could have done it the long way faster.

the one that gets me is the simple ability (or lack of) to be able to calculate tips and other percentages. regardless of how you feel about tips, it is definitely something we do a lot. except, now, we don't and we have apps to do it for us. we can't figure out quickly what 25% off would make the price. so many day to day things like that is the worrying bit to me. not how quickly can Karen estimate the square root of a 3 digit number, because why would Karen even be in that situation. Karen is interested in 25% off, but can't without her phone. i've already decided the price still isn't worth and a have moved on before she can even unlock her device.

Not being able to figure out a tip does annoy me a bit, because when you know how 10% works then you've got all you need, just halve it and add it back on and you've got your minimum 15%, or double it for 20%.

It's not like that requires practice. Maybe to do it quickly, but moving a decimal place over in your head or halving or doubling is about as easy as it gets.

I don't want to disparage people who aren't good at math (compared to many here I'm like a grade schooler) but how do you make it though school without knowing the basics of percentages?

Are your referring to Eureka Math? (What is usually incorrectly called "Common Core")

New Math is from the 1950s-1970s and almost entirely abandonesld except for some gifted/enrichment programs.

Wikipedia says: > Topics introduced in the New Math include set theory, modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, and abstract algebra.

Eureka/Engage is a very watered down version of that (but still decent.)

Are your referring to Eureka Math? (What is usually incorrectly called "Common Core")

Eureka Math is produced by Great Minds. Until 2015, they were called Common Core Inc. In fact they predated the Common Core Standard, and were sufficiently influential in the creation of the standard that the standard was effectively named after them.

Therefore calling it Common Core IS historically correct. Furthermore I have a strong distaste for their attempt to rebrand themselves to distance themselves from the failures that they were instrumental in creating in the first place.

Again I direct you to https://deutsch29.wordpress.com/2019/10/06/a-great-minds-com... to see more tidbits about their history.

>bases other than 10,

i had forgotten just how impressionable kids can be, especially when they really want to be. i "argued" with my buddy's oldest (a 5th grader) about "1 + 1 = 10". after a few rounds of nuh-uh, yes-huh, i gave a very rough explanation and ending with "in binary". until a few days later, buddy asks me to not be so cavalier with the challenging of the maths. apparently, his daughter had challenged her teacher with 1+1=10, and after a brief back and forth, she stipulated "in binary" to her teacher. the teacher being very young and not familiar with binary, sent the poor kid to the principal's office.

Probably, yes.

This. The complaint from parents seemed to universally be that they didn't understand the "new math". The conclusion they reached was that there was something wrong with the method, rather than the more obvious conclusion that they should have taken a bit of time to learn and understand it.

Right, I watched like a ten minute YouTube video when my nephew’s parents were complaining about it and found it not difficult at all to comprehend. I didn’t exactly struggle with the old math either though, but so many people did that it is clear there is room for improvement.

Maybe it is as you say. I don't have kids in school, so it was all a big nothing burger to me written off as a bunch of Karens needing something complain about for whatever purpose it serves them.

Common Core: "Common core math is a set of national educational standards that push kids to think of math equations differently. With common core math, kids begin questioning the relevance of each equation. Instead of just solving an equation for its sake, common core math makes children deliberate the reason behind the equation."

oooooh, scary. making kids think. me thinks that's the issue.

That's the theory, maybe. I have kids in school. In practice it seems more like simply rote memorization of multiple techniques to solve a problem. It's especially hard when a child figures out his own way, but is graded/marked on doing it the specific way(s) a homework or test question demands.

this is actually something i had to endure as well specifically from knowing these tricks and being well practiced in "doing it in my head". i had to be retrained to show my work. each new teacher would assume i was some how cheating on the homework by just writing the answers. that may be harsh, but that's the way it was always received. if they weren't going to give me the benefit of the doubt, why should i for them?

To this day, I remain unsure of what work I was meant to be showing in 3rd grade to "convert" between ratios and fractions.

Probably just change the colon to a horizontal bar?

Yes, that's all there is to it, which is why I never showed any work and didn't get full marks.

Does the bad grading hurt? If the kid figured out their own way, great! They don't need the lesson!

the thing that convinced me to switch to elaborately showing my work was partial credit. in the competitive tests, if you didn't know the correct answer, you'd look at the next 3 questions to see if you could answer them. a skipped answer was counted as wrong, and points deducted. since the test increased in difficulty as you progressed, you'd have to consider how many questions you were skipping and if it was better to just stop.

in classes, even if the answer was incorrect, if you showed your work you could receive partial credit. if it was clear you were using the right steps to get there but forgot to carry a one or skipped a step somewhere else but it was clear that the basic understanding on how to solve the problem was close, you could avoid a big goose egg for that question. IIRC, the AP exams were like that as well. more about showing understanding of the process than just the correct answer is what they were trying to achieve.

Yes: some kids are quite sensitive to that sort of thing. More importantly, it's not actually teaching them anything except that they must conform.

Let's see what the expert teacher has to say about New Math:

https://www.youtube.com/watch?v=UIKGV2cTgqA

That's a lot easier than the method I was taught. The "old" way always resulted in me trying to carry state in my head, which doesn't work well for ADHD-limited memory.

Lehrer is talking about "new math" from 60 years ago. (it consisted of talking about set theory, union and intersection, also mentioning commutative and distributive princicples, i.e. mentioning some things that are not super necessary for arithmetic but which you will encounter later if you pursue mathematics further. In this way you would hopefully be more effective in contributing to the arms and space races and defeating communism.) https://en.wikipedia.org/wiki/New_Math

not clear that GP is referring to the same thing at all, perhaps this https://www.parents.com/kids/education/math-and-science/new-...

I do Number Sense right now!

I would never do this much work in my head :-)

You can get a decent answer just by glancing at the fixed scales of a slide rule [1]. This is also called a nomogram, and you can make your own custom ones programmatically with PyNomo [2].

[1] https://youtu.be/dT7bSn03lx0?t=11m45s

[2] http://pynomo.org/wiki/index.php/Main_Page

Was anybody else not taught things like this in school? Shortcuts like these where you get pretty close, but not exact are really useful; however, I recall math in school having to be exact or it was wrong. These handy get-you-close-enough tricks are helpful in competitions, standardized tests, real life. Anyways, thanks for sharing!

Lots of approximation algorithms fall out of calculus pretty naturally. So they end up taught as examples or applications in that sort of class, rather as tricks when learning algebra or arithmetic.

I learned them for exams that involved lots of arithmetic. A lot of it was folk knowledge passed along the peer group partly by asking some of the kids who were one year ahead (and closer to the exam). There was some stuff like this but also other “tricks” (in physics problems, approximating sqrt(g) by pi and stuff like that), or calculating the integral of e^ax times sin bx by “completing” it to e^(a+ib)x, or the cube root technique here http://thinkinghard.com/blog/CubeRoots.html that we used mostly for modulo arithmetic.

I’ve forgotten nearly all the useful stuff and now only recall trivia like the base ten logarithm of a few small primes. I wonder if those school kids are still passing on this sort of cobbled together knowledge, now decades past.

I always considered this kind of trick stuff somewhat beneath me, as a child, since it did not discover some underlying truth of the universe like a mathematics proof in number theory would. But that was dumb because in the end I managed neither anything interesting in Number Theory nor complete mastery of thE tricks and at least the latter would have been entertaining to document.

As an aside, there was a kid in our group who could rapidly decompose numbers into sums of powers of two (a skill that turned out useless even in the constrained environment), essentially obtaining the binary representation as soon as you wrote the number out (or showed him the written one). He couldn’t explain how, though, because he was mentally not quite there and would frequently lose verbal skills. After the nth time he stood up on the desk and randomly peed while we were there he was out of the place. Besides it wasn’t that useful and it could just be high skill at the straightforward modulo and shift algorithm. Going the other way was also fast for him but useless for us, though more impressive since adding 2^64 (a number I could still tell you off the top of my head) to some other power of two seems less straightforward.

I wasn't. At least, not officially. I struggled in math, though, and one of my teachers (not my math teacher) taught me several of these tricks on the sly. I remain grateful to him to this day.

I always had a calculator. If they taught me shortcuts, I didn't listen. But, I was not a good student in primary school.

Your solution must be exactly as it is taught, or the solution is worthless.

You're not supposed to think for yourself in school, just do what you are told.

Just tried this in my head with a random n=150.

g = 12, since 12x12 = 144, pretty close to 150.

150/12 = 150/(4x3) = 50/4 = 12.5

Average of 12 and 12.5 is 12.25.

Real answer: 12.247

Amazing!

> We start by finding a number that forms a perfect square that is close to 33

I usually just stop there which works for my use cases. :)

> b=n/g. In practice, computing b in your head may require an approximation

I'm not sure how realistic this is to do in your head at all. Won't you introduce an error similar to the error of g anyway? Maybe "in your head" means without a calculator but on paper using long division.

so, I don't know why this isn't brought up, but doing a lot of this kind of stuff in the field on construction sites, I'd have used 5 x 5 = 25 and 6 x 6 = 36 then split the difference so (36-25)/2 = 11/2 = 5.5, then you split the difference again in this process.

For me, most estimation in the field is finding two known facts above and below and splitting the difference. Depending on your required precision, the above gets you pretty close right off the bat.

Yep, I use this for all sorts of things. Usually two known data points plus close-enough linear interpolation is accurate enough to estimate a value or sanity check a value you’ve been told.

And that’s why the system of inches and fractions is better than it seems. Very easy to divide by two, which is a very common need.

I mean, it's not easier than dividing in base 10.

I've used both extensively and this is a really odd take.

Half of three and 5/8 inches: one and 5/16 plus 8/16: one and 13/16. Boom.

That’s faster than dividing decimals in your head.

It’s a useful system for residential construction, fast and accurate within about 0.031 inch. That kind of construction is done on four foot modules. Everything is marked right on the tape measure.

The only time I use a calculator is to find the spacing to make an integral number of clapboards span a window. And, I have a construction calculator that works in feet, inches, and fractions, if I need to calculate stairs or something fancy.

For cabinetmaking, either system is fine.

For machining, I prefer metric, when the equipment can handle it. Otherwise, just use decimal inches, no feet, and keep your calculator handy.

so, for conversations sake, this is a number that is made to fit nicely with imperial fractions, however, 3.625/2 is 1.5+.3125 or 1.8125.

In reality, the problem would actually be 9.2 or 9.3 CM depending on how you're rounding and what you're building. And that would be 4.6 CM and the simplicity of that isn't even close.

I've run a huge number of construction crews of all stripes and, on one crew over half of my crew came from a metric using country. I found out of about 25 people, it was substantially easier to convert my crew to metric than imperial because it's way simpler.

As for myself, I'll work in what ever tape measure I have on hand but the idea that the fractions are simpler is objectively wrong. It may feel that way because you started there, but having converted more than one crew to metric to cutdown on waste from miscuts and miscommunications I can tell you that attempts to switch crews to imperial for the same reason is pretty rough.

> Maybe "in your head" means without a calculator but on paper using long division.

There's not really any benefit if you have paper available. But it's possible to do long division (and other "paper" techniques) in your head if your short-term memory is good enough.

> But it's possible to do long division (and other "paper" techniques) in your head if your short-term memory is good enough.

That's actually how I learned long division. I didn't want to pay attention to any of these weird methods of arithmetic that require me to write things so I just did it all in my head and this happened to be a method that worked (go figure!). Many of my K12 math teachers understandably didn't like the fact that I would write down the correct answers to things without showing how I got there.

On the topic of this article, I'm actually really glad for this trick. If anything is my hobby it is mental arithmetic.

Btw, doing long division in your head (during sex) is a good way to increase endurance…

Meanwhile, your partner says, “Beige. I’m going to paint the ceiling beige.”

Exactly.

I ended up using fractions in my head, which works pretty well as you essentially defer the division until the end. It’s a lot less state to carry in your head too.

Basically you compute

(n + g^2)/(2g)

33 + 36 = 69 2 * 6 = 12 69/12 = 5+ 9/12 = 5 + 3/4 = 5.75

It converges more slowly, but you can also just binary search for the square root. This is just easier for me to remember how to do.

You can also cheat it a bit by trying to bias towards the way you intuit the answer lies.

If you can avoid floating point division and search for what squares into the number you're looking at, it may converge more quickly per computer cycle.

Heron's method does converge quickly per algorithm cycle, though.

I haven't tried both, but you could definitely do your binary search with only multiplication and bit shifts.

Ok, now I'm looking up number of cycles for division, which is a lot less than I remember. How much has this changed in the last 20 years? any?

Ugh, now I have to try both out tonight. Thanks everyone.

Yeah, my vague sense of low-level stuff these days is that it's moved a lot more towards memory/bus bound, and branch-prediction dependent.

Anything that's in a register already and doesn't branch is about the same cost (probably technically extremely wrong, but that's my rule of thumb).

Herons’s method is just Newton’s method applied to the problem of computing a square root recast as a rootfinding problem.

I got C’s in college math but was a little clever when I was younger. I came up with my own method for estimating square roots, which was not bad (but not nearly as good as Heron’s method).

Find the biggest whole number less than the square root of N, call it A.

Sqrt N =~ A + (N-A^2) / ((A+1)^2-A^2)

e.g.

Sqrt 33 = 5 + 33-25 / 36-25 = 5 8/11 = 5.73

I didn’t explain it to my 5th grade teacher but told my older brother, who was taking calculus. He said I was doing a sort of Taylor expansion.

What I learned later was that I was quick at mental math and got by until I was taking real calculus, then I fell behind as I didn’t know how to learn material because of bad study habits and not being clever enough to intuitively get it (my roommate who was slightly less clever at quick math breezed through the same college math classes and became a physicist).

Basically you're linearly interpolating between, for example, sqrt(25) and sqrt(36) to get sqrt(33). That will work quite well, but it's always an underestimate.

That’s a concise summary, thank you!

i think estimating is one of the most important skills in maths (well, arithmetic/common sense) that is rarely taught in schools. for example, estimating the cost of your supermarket trolley, estimating how much wallpaper you will need to paper a room, etc.

now, obviously you want to get these things accurate, but the estimate tells you when you got the calculation wrong in your measurements, spreadsheet or calculator.

i was guilty of this when i over-valued my late-dad's book collection in a python program. i sort of knew it was wrong but what i didn't realise he had changed his data format halfway the dataset.

You're right. It's hard to teach in schools though because of standardized (often multiple-choice) tests; teachers are incentivized to teach "skills" that have objective correct answers that can be easily tested.

Then by the time students hit college, they're resistant to any sort of mathematics that doesn't have a correct answer. I've been trying to sell it as a means of error-correction, or as a sanity check. If they've gotta answer a question "Joe is 6'4" tall and, at the suggestion of an ergonomist, wants to build a desk 40% of his height. How tall should the desk be?" I urge them to not immediately go into math-brain and just think for a minute, and estimate a reasonable answer to match their calculations against. This mental workflow sticks for a few of them by the end of the class.

Absolutely. The concept of a "sanity check" is so valuable. Thinking about what the answer should "look like", ignoring the precise value, always makes me much more confident in a precise value I've computed. It's the kind of thinking makes math so much more useful in the "real world" too.

It was my shop teacher who taught me the value of estimating. As he put it, "in the real world, you rarely need high precision. Start with the quick-and-dirty, most of the time that's plenty good enough."

We played a game in high school that I called "The Root is Right!" We'd take turns coming up with crazy numbers for each other to estimate in our heads. Then we'd draw little pictures of prizes for the person closest to the real root. We all got super good at estimating roots!

We used a lazy version of Heron’s method, which was just to memorize a bunch of general number roots, and then average our guesses based on closeness to that root. It ended up being as easy as mental math around tipping at a restaurant. It definitely came in handy for engineering classes later on!

Isn't this just Newton's method ?

yes, but the special case that Heron did predates Newton.

I've used this is control systems where I wanted the square root of a signal. Given that the signal should not change quickly (may even have a low pass filter on it) doing one iteration of this is much quicker than doing a regular square root computation each time step. Obviously, the usefulness of an approximation depends on what you're doing, but this does converge to the exact solution over time ;-)

I used to perform a lot of mental arithmetic when I was kid and even read some books about different techniques (Vedic mathematics, Trachtenberg system, etc.). I don't remember the specific techniques from the books these days. Since I can also do multiplication and division in my head, I simply figure out which two consecutive integers when squared bound the number of interest. I then divide the difference between the number of interest and the lower integer by the sum of the two integers and add that fraction to the lower integer to get the square root. For example, if we want to calculate the square root of 1051, I know that 32^2 is 1024 and 33^2 is 1089. Then, 1051^0.5 = 32 + (1051-1024)/(32+33) = 32 + (27/65) ~ 32 + (26/65) + (1/65) ~ 32 + 0.4 + 0.02 ~ 32.42. The square of 32.42 is actually 1051.0564, so that's pretty close.

Knowing powers of two in your head (as I guess many of us do) can helps with root estimates and this algorithm.

Take an example sqrt(5819): That's between two powers of two 2^12 = 4096 and 2^13 = 8192. The square roots of those numbers are easy, since you just half the exponent sqrt(4096) = 2^6 = 64

That gives us our initial estimate g for the algorithm from the article. Now we do: b = n / g = 5819 / 64 To do this division in our head we remember that we already know 4096 is 64 x 64. So we just need the remainder (5819 - 4096) / 64 ~= 1700 / 64. Now that's low enough that I can approximate it in my head 64 * 30 is too high by about 3 * 64 so the answer should be around 27 (actual answer 26.5). So b = 64 + 27 = 91

So now we finish the algorithm, sqrt(5819) ~= g + b / 2 = (64 + 91 / 2) = 77.5.

So, how did we do?

The actual sqrt is 76.28, so seems pretty good!

> We start by finding a number that forms a perfect square that is close to 33. Here, let’s pick g=6, since 6^2=36.

As somebody who is numerically essentially blind I'm not only incredibly jealous and in awe of most of the comments in this thread but also utterly perplexed by the above. How does that even come to mind that 6 is a good starting value? Do you people just intuit this stuff or do you rapidly run through the options? If you're asking me to come up with something that when multiplied by itself is something near 33 then you're in for a long wait.

There's another comment in this thread:

>Everyone generally knows the perfect squares up to at least 12, and then for bigger values, you can use even powers of 2, which I assume people also know.

Do they? Jeez.

In order to do math in your head, you have to have certain basics memorized, and you combine these to come to a solution. It’s pretty common to learn the perfect squares up to 12 or so

1x1 = 1 2x2 = 4 3x3 = 9 4x4 = 16 5x5 = 25 6x6 = 36 7x7 = 49 8x8 = 64 9x9 = 81 10x10 = 100 11x11 = 121 12x12 = 144

If you have that readily accessible in your head, you can simply do a linear scan until you find which two numbers the square is between. In practice you will probably skip a few or just recognize it as being close after some practice, but that’s the general idea.

As an aside the perfect squares are just a portion of the single digit times table that you would want in your head to do multiplication or long division.

Knowing the squares up to 12 isn’t that difficult to memorize and comes in handy… I imagine those who have them memorized did so in K-12 and just never forgot it.

In every elementary school I have encountered in the US, one learns the multiplication tables up to 10 in 3rd grade (age ~8). Some schools go up to 12 (presumably because imperial units are somewhat base 12?).

Edit: On a lark, I just tested this on my wife who is not mathy whatsoever, and she was able to remember them up to n=12 (with a slight hesitation for 7^2 == 49).

For another lark, go to Kahn Academy and check out how he teaches arithmetic up to third grade.

You have to look really hard to find addition and multiplication tables. They’re there, with the rather lame comment that they’re good to know.

But, they are not in his main stream, which depends on gimmicks to figure out the answer, instead of just rote memorization of the tables (which is good for the brain, as well).

As an example, I know my multiplication table just fine through 12. But, something went wrong in the second grade, and my addition was terrible. I depended for years on tricks: to add three, count up very quickly on the three points of the numeral 3. Same for four. For five, count one point twice.

I was in trouble at adding seven, because I couldn’t do something like add three twice!

I was shocked that Kahn uses tricks like that.

It's not worth feeling shocked over. When you are approaching fundamentals, it doesn't matter what method you use. The education system failed you. From my experience teaching kids with math anxiety, I learned it's impossible for people to accurately judge where their weakness at the core. But a pattern I found was that they were scared of doing basic things the "wrong" way. Short-sighted teachers worry about the "wrong" way because they feel like it will slow kids. The proper approach is to let kids use whatever way is fastest for them, and they'll learn other methods on their own as they continue practicing problems. Teachers who force a specific method for arithmetic make the problem worse because the kids end up more worried about doing it the right way rather than worrying about getting the answer right.

We live in an age with calculators, no one cares if you use your fingers to count. I have ADHD and feel limited by my working memory often, using fingers or repeating a number I want to remember over and over feels like having extra RAM. Even the way kids are taught to count is different depending on where you live. Studies show that kids who use fingers are stronger in quantitative reasoning. But growing up, I knew teachers who made fun of students for using fingers to count.

Imagining numbers as dots and counting or breaking a number into smaller numbers to add is not a "trick" it's an algorithm that is as valid as any other. It's counterproductive to associate the word "trick" with "wrong".

For a while I wrote my own system of dots to correspond with numbers, 1, 2, 3 I focus on the end points, 4 (I wrote it open) makes a square with four corners if you ignore the extensions, 5 I count when I change directions and the end points, 6 I imagine dots of a domino tile, 7 is basically two layers a four and then the end points of the character, 8 is similar to six but I count the two circle, and 9 is similar to six but I count circle and then both sides of the bottom curve (a 3x3) grid.

Even if my brain gets tired or distracted, I know I can still add by dots because it's so procedural and I don't need to "think", I just remember the starting digit and then count up as I follow the dots. I use saying the word out like as a form of RAM to this day. Repeating a word, to me, uses a completely different part of my mind, so I free up 100% of working memory and cognition. I have "forgotten" numbers while doing mental math and have reminded myself from hearing myself say it. Describing these techniques, I recognize I sound like a literal computer and almost not human, but it's struggle I learned to work past. It works, I can do relatively more advanced mental arithmetic compared to peer even.

For multiplication, I would recommend Anki. This kind of memorizing is what that entire system excels in.

https://www.theguardian.com/science/blog/2012/jun/26/count-f...

I do the dots thing too. I've never heard anybody else describe it and, to be honest, it's quite comforting to hear I'm not the only one. In fact I don't just imagine dots, I imagine die faces. This obviously gets problematic after six. I too feel limited by my working memory: it makes mental arithmetic of numbers with two digits very very hard without an external store like fingers.

So, our methods for addition are nearly the same. I like your seven. Eight is just add ten and subtract two, and nine is subtract one.

But the thing is, doing addition and subtraction should be as fluid as multiplication. Your dots and my points are slower than having the table memorized, where the answer just appears in your head instantly. And so I made an effort to learn the table later in life. But, my memory isn’t great for that stuff anymore, so it wasn’t easy.

The tricks don’t work very well when you are asked to count backwards from one hundred by sevens. (Which used to be on a standard cognitive test!)

I don’t think the system failed me - it was a good school - so, I think it was a joint effort. I didn’t learn the table properly for an unknown reason, and the school didn’t realize that, because my methods worked and were never tested in competition.

I don’t think the tricks are “wrong”, just suboptimal. That’s why I think Kahn should teach tables, and only fall back on other methods when necessary.

I was brought up in a academically competitive environment, both in school and at home. I also have a natrual affinity towards mathematics and puzzles, but none of that makes a difference. This kind of arithmetic math inutition is nothing more than practice. You might be a little removed from the last time you had to do this level of math, so it's not fresh. I have tutored high school math for the last 10 years, so I built that intution over time checking students' work. But that only makes a difference in speed.

The reason the blog post and I made the assumption about perfect squares up to 12 is because it's a consequence of just knowing multiplication tables up to 12x12. The squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. The most basic approximation of a square root would be rounding to the nearest perfect square. The author didn't pick g=6 because they had some hidden intuition that 6 would be close, they picked 36 instead of 25 and knew that'll be 6. If I wanted to know the square root of 72, that's between 64 and 81. You'd know it's closer to 64 because the difference between 64 and 72 is smaller than 81 and 72. The actual numerical differences aren't that useful, you just need an understanding of what it's close to.

The reason why I assumed most people would know even powers of 2 is because of CS. It's just comes up so often, for example looking at algorithm complexity in relation to simplying with respect to log base 2 or binary representation of integers. The numbers also come up when thinking about primitives conversion such char to integer or how floating points work. As well as understanding amortized memory allocating algorithms, like how much bigger to make a dynamic array when it's filled. Even if you don't explicitly know why, numbers like 4, 16, 64, 256, 1024, 4096 are familiar, which are 2^2, 4^2, 8^2, 16^2, 32^2, 64^2. They are all also powers of 2 and you could write them as 2^2, 2^4, 2^6, 2^8, 2^10, and 2^12. I felt like it's a fair assumption on HN.

When I mentioned visualizing the graph, I just meant the non linear mapping between numbers and their squares. That is more raw intution, but it's not numerical in any way, it's knowing how the graph looks. And that comes from remembering the relationship between algorithm complexities, like log(x) vs sqrt(x) vs x vs x^2 vs 2^x.

An an educator, I strongly disagree with the idea of anyone being numerically "blind". If you struggle with rapidly finding that option, my only advice is brush up on some multiplication tables, and to give yourself time to speed up. Math anxiety is a problem for some, but that's a problem related to fear of failure and not cognitive ability.

I've often wondered how others get the multiplication tables to stick. When I was in school and tasked with learning and being tested on them I remembered them - just about - for long enough to be tested (primarily from a quick reminder on the way to school) and then they were gone.

My overall experience was that they were a very boring song with terrible lyrics. I can't ever say that any meaning clicked especially. The teacher called out the first part of the verse and I "sang" it internally and hopefully got it right. Having gotten through that it was gone by lunchtime in time for a different set of words to the same song next week.

I tried again about 30 years later as a adult and had much the same experience. You might as well have been asking me to remember items on a tray.

To me it's like this:

chicken x tree = rock

brick x kangaroo = Susan

boat x walnut = dinosaur

Now imagine you have to remember 288 of those (because you might be asked to produce either side of the equals sign) and somebody asking you to recall one arbitrarily.

How do you get your students to get them to stick?

https://www.mathmammoth.com/practice/multiplication

https://www.mathmammoth.com/practice/multiplication-table

It's all about practice. For some, it's easier to create something competitive from it. Some of my students really enjoy challenging me, because they are at that age in life where they want to stick it to "the man". So we do 60 second times competition to see who can finish more.

At its core, learning this is boring. Having a benchmark that you work to improve, gamifying it, is the fastest way to improve. It's like typing or exercise, the fundamentals aren't cognitively taxing. I think it's a little harder to practice after you leave school, because there's no math or science class where you would immediately apply and practice what you learned.

There's no general solution that would work for every student, and not every student has the same weakness. The website I linked is genuinely one of my favorites, it's like duolingo for math, for free with no account needed. I think the second link will help build intuition behind the relationship between the numbers. All of the quizzes/games on this site are fantastic.

https://www.mathmammoth.com/practice/sieve-of-eratosthenes

This is fantastic to understand numbers and primes. Start with multiplication/division. Then work with GCF, which is greatest common factor between two or more numbers, for example 24 and 36, 6 * 4 or 12 * 2 == 24 ; 6 * 6 = 36, 12 * 3; so the GCF of 24 and 36 is 12. GCF is a way to practice multiplication/division naturally.

Arithmetic puzzles are also a good way to get some practice where it's not just boring. Fermi questions are as well. There's more advanced algorithms for multiplying things like large numbers in your head, but in my opinion, an estimate is good enough for 99% of real life situations. Fermi questions help build an intuitive math sense that will be applicable to real life situation. For memorizing, using "tricks" is fine, like the 9 multiples all add up, looking at digits, to 9. 9 * 8 is 72 which is 7 plus 2. Multiples below 10 also start with 1 less than the digit that is not 9, so 8 - 1 = 7, and 7_ and the second digit adds up to 9, so 2, 72. I'll leave the rest as an exercise for reader. Good luck.

I learned this method of calculating square roots to arbitrary precision and could probably easily produce 5-6 significant digits in my head if I concentrated enough:

       5. 7  4  4  5  6
      33.00 00 00 00 00
      25 = 5 * 5
       8 00
       7 49 = 10 7 * 7
         51 00 
         45 76 = 114 4 * 4
          5 24 00
          4 59 36 = 1148 4 * 4
            64 64 00
            57 44 25 = 11488 5 * 5
             7 19 75 00
             6 89 34 36 = 114890 6 * 6
               30 40 64

I can't make sense of this.

Look for "square root by long division". That's what it seems to be called these days, but it was just taught as "the square root algorithm" when I learned it many decades ago.

Who want to calculate a fraction in their head? My method for this:

* Pick the lower integer square root (5)

* The distance from 25 to 33 is over half the distance from 25 to 36.

* Eyeball that that little over half on a mental picture of sqrt(x) between 0 and 1. Guesstimate 0.7.

* 5.7 is the guess. You can multiply that out on your head and similarly iterate on the error if you want closer.

I estimate square roots by just imagining a square. The area of tyhe square is the thing you are square rooting, the sides of the square are the roots. Eg, wahts the square root of 2345. Well roughly imagine a square of Area 2300, the sides are bigger than 100 (1/2 as big), but not more than 200 which would be 2x too big, so its probably around 150.

Your method apparently isn't very good seeing as a square with area 2300 has sides of length ~48

Well, considering 100 * 100 is clearly more than 4 times 2300 its more an error of my basic mulitplication than my method of estimating squares. Was just focused on something else at the time of writing (as indicated by the spelling). Considering that a square root by definition is the side of the square described, visualising it sure doesn't hurt.

If I tried again I'd say 100* 100 = 10,000, 1010 = 100, 50 50 = 2500, so that going to be pretty close. 48 is pretty close to 50.

Trivia: The guy who wrote QOI and QOA, @phoboslab, had a phone interview with facebook a decade ago where he was asked to implement sqrt() in js.

https://twitter.com/phoboslab/status/1554127643011850242

I was given a similar interview problem in one of my first job interviews (I think at Google?) 15 or so years ago. I don't remember the exact question, but I do remember that I needed to use Newton's method to solve it. However I was a math major in university, and I remember the interviewer pointing that out, so I felt like it was fair game.

One I learned recently:

E.g. sqrt(7) = 3 x sqrt(1-2/9)

~= 3 x (1 - (1/2)(2/9))

= 3 x 8/9 = 2.6667

Where the second line uses the first term of the binomial series, i.e. (1+x)^(1/2) ~= 1+ x/2

Real answer is 2.65 in this case, obviously picking a factor to pull out that makes x as small as possible works best

It dawns on me that I was never taught how to calculate a square root manually during all my schooling. We used them all the time in Algebra II and Calculus, but we were never shown a procedure like with multiplication or long division.

oh I used to derive this in high school by Newton's method: x[k] = x[k - 1] - x[k-1]^2/2x[k-1] = x[k] / 2 + a / 2x[k]; base case x[0] = a reasonable guess of the initial square root.

Do a few iterations and you will get a close estimate of the square root.

In fact the babylonians found it first by intuition a few thousand years ago it was Newton and Ralpson who proved and generalized it to all derivable functions I think?

So the big advantage of this is that it is easily applicable to say like cubic root or quintic root, although the difficulty goes harder and harder.

Just use:

https://en.wikipedia.org/wiki/Fast_inverse_square_root

and invert :)

Reminds me of the rule of 72.

Which also happens to be a second order approximation based on Taylor series.

Is that a trend? Second order is the sweet spot?

Anyone can do this with anything. Just how accurate your estimations are, well, remains to be seen

That's pretty neat. I'll never need to use it, but it's pretty neat.

this person's blog is consistently good! i immediately noticed the URL from their post on the reparametrization trick.

that is really cool

it's interesting how it's a direct application of calculus. was not expecting that

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