[HN Gopher] Estimating square roots in your head
___________________________________________________________________
Estimating square roots in your head
Author : alexmolas
Score : 148 points
Date : 2023-02-02 11:29 UTC (11 hours ago)
(HTM) web link (gregorygundersen.com)
(TXT) w3m dump (gregorygundersen.com)
| paulpauper wrote:
| that is really cool
|
| it's interesting how it's a direct application of calculus. was
| not expecting that
| sfpotter wrote:
| Herons's method is just Newton's method applied to the problem of
| computing a square root recast as a rootfinding problem.
| BitwiseFool wrote:
| 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.
| thomasmg wrote:
| 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").
| zabzonk wrote:
| 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.
| JohnFen wrote:
| 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."
| mapierce2 wrote:
| 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.
| delecti wrote:
| 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.
| phkahler wrote:
| 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 ;-)
| ubicomp wrote:
| 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!
| [deleted]
| peplee wrote:
| 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!
| doubled112 wrote:
| 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.
| klyrs wrote:
| I always had a calculator. If they taught me shortcuts, I
| didn't listen. But, I was not a good student in primary school.
| bee_rider wrote:
| 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.
| JohnFen wrote:
| 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.
| kadoban wrote:
| 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.
| LanceH wrote:
| 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.
| kadoban wrote:
| 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).
| euroderf wrote:
| Isn't this just Newton's method ?
| madcaptenor wrote:
| yes, but the special case that Heron did predates Newton.
| super256 wrote:
| 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
| eklitzke wrote:
| 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.
| dylan604 wrote:
| 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.
| bee_rider wrote:
| 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.
| dylan604 wrote:
| 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.
| sidlls wrote:
| 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.
| hgsgm wrote:
| Does the bad grading hurt? If the kid figured out their
| own way, great! _They don 't need the lesson_!
| dylan604 wrote:
| 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?
| mattmaroon wrote:
| 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.
| hgsgm wrote:
| 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.)
| dylan604 wrote:
| > 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.
| btilly wrote:
| 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.
| hgsgm wrote:
| > 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.
| JohnFen wrote:
| 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.
| jacquesm wrote:
| Let's see what the expert teacher has to say about New Math:
|
| https://www.youtube.com/watch?v=UIKGV2cTgqA
| kayodelycaon wrote:
| 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.
| knaik94 wrote:
| 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, 2 _16_ 1000 (almost 1024), so 1.5ish _4_ 32 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.
| enriquto wrote:
| 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 comite_ 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.
| com2kid wrote:
| 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.
| gumby wrote:
| 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. But that case
| might 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.
| acchow wrote:
| > 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
| [deleted]
| btilly wrote:
| 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.
| scythe wrote:
| 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).
| thehappypm wrote:
| 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!
| marmetio wrote:
| 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
| barbazoo wrote:
| > 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.
| jethro_tell wrote:
| 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.
| nicoburns wrote:
| > 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.
| lcnPylGDnU4H9OF wrote:
| > 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.
| lurquer wrote:
| Btw, doing long division in your head (during sex) is a good
| way to increase endurance...
| SpaceManNabs wrote:
| this person's blog is consistently good! i immediately noticed
| the URL from their post on the reparametrization trick.
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