Roots decimals

How are you guys with mentally calculating roots with decimal places?

Because I don’t know any multiplication tables, calculating roots with decimal places is not as easy as multiplying, addition, division etc. It’s a check-and-proceed -procedure. This morning I calculated the square root of 4692 to 3 decimal places; 68,498. In order for me to calculate this I need to actually concentrate and see the numbers as clearly as I can because I have to hold more and more numbers in my mind very quickly compared to other types of arithmetic problems. I can’t brute force this either. Roots have always been something uncomfortable for me because from my perspective, it is sort of a guess work. Roots with integer results are easier than roots with decimals. I have a much easier time calculating the cube root of 851,971,392 than the square root of 4692.

But what about you? How do you guys calculate the decimals of a root? I’ve seen Shakuntala devi do it so quickly to more than 3 decimal places and I wonder how she did that, perhaps you guys have some ideas?

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Quick reply.
By far the fastest method is called the duplex method. I have written 2 posts about it with detailed instructions and examples.

Do a search.
Let me know if that worked for you.

The duplex method seems really fast but also surprisingly easy from what I read. I probably could be very quick with it as being capable of seeing a lot of numbers at once could be very advantageous with this method but I can’t use it unfortunately.

Shakuntala devi probably used this method too then. I always thought that she was a natural human calculator like me but I know how difficult some calculations can be and I wondered;“if she is like me then how can she do it so quickly?”. Now that I know about the duplex method, it is another sign that she probably wasn’t.

You say it is easy and fast, and I cannot understand why you cannot use it then.

Because I want to stay natural. I don’t use techniques or methods for anything. I was just interested how others handle roots with decimals.

И моят син,с аутизъм,на 8 год,се справя с кубичните корени по-бързо от квадратните,до четвърти знак след запетая,как,незнам…и най лошото е ,че не може да обясни и той.Сега ще ходи на световното в Германия.

via Google Translate:

And my son, with autism, at 8, copes with cubic roots faster than square ones, up to a fourth decimal point, how, I don’t know … and the worst part is that he can’t explain it either. Now he’s going to the world in Germany.


Wow! Can you describe your steps please.

You could use the following excellent post by @Kinma as a model:

The post number is #24

BTW: I can add two numbers in my head at blinding speed, as long as each number contains less than two digits. So @Kinma’s description blew me away.

I enjoyed working through post #24 in the same way as I would enjoy replaying a famous chess game, or a World Championship bridge hand.


Alright, this is how I calculated the square root of 4692.

First I tried to calculate to the closest large square number that is below 4692.

68x68 = 4624.
4692 - 4624 = 68.

the next step is to calculate the first decimal place.

68 / 10 = 6,8.

The first decimal can’t be 5 or higher because:

6,85 x 5= 34,25
34,25 x 2 = 68,50.
68,50 is bigger than 68 so the first decimal is smaller than 5.

6,84 x 4 = 27,36.
27,36 x 2 = 54,72.

The first decimal is thus 4. Up to the next decimal.

68 - 54,72 = 13,28.

13 is close to 6,84 times 2 but just below it so the first number we can try is 9.

0,6849 x 9 = 6,1641.
6,1641 x 2 = 12,3282.

because 9 is the highest number possible as a decimal and 12,3282 is the closest to 13,28, we now know that the second decimal is 9. Now the third decimal.

I couldn’t do the same here as with the other two decimals. The third decimal I got by just checking and multiplying.

I checked 68,499^2:

68,499 x 68,499 = 4692,113001.

68,499 x 68,499 is too big so I guessed that the third decimal would be 8 and it was.

The square root of 4692 was ~68,498.

I don’t know another normal way of calculating the third digit of a square root so I just crunch the numbers until I get the third decimal which takes a lot of time and effort.


What exactly do you consider natural in this context? I don’t understand what makes your algorithm more natural than the duplex algorithm that @Kinma mentioned above.

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I haven’t memorized anything or use any sort of technique, shortcut or (duplex) method. When I see that 68 x 68 is 4624 it is because I calculated it the way you are tought. If you were to ask me 10 years ago, when I was 11, what 68 x 68 is, I would’ve responded just as quickly as now that the answer 4624 is. Most people have to memorize shortcuts, certain squares, tables etc but I don’t.

I find it odd that you do not want to admit to the basic properties of numbers; Distributive, Associative, Commutative for addition and multiplication in your calculations but I salute your determination. I strongly suspect you are alone in your decision to choose one specific method as more “natural” than others. Having been taught a method first does not make it take any natural precedence. I do not say this to change your plan as I rather enjoy your brute force approach and as you have said you have had success with it. I will suggest however that if you try to compare yourself one-to-one with talented folk that accept the properties of numbers as “natural” that you are fighting with one hand tied behind your back.

I wish I had a portion of your will power and talent for number and enjoy hearing about your successes.


It is not necessarily the method but the brainpower. The thing that makes it the way I do it natural is because my brain already has the capacity to do calculations like 4387 x 6793 mentally without training, techniques, shortcuts, certain methods etc.

The reason why I don’t use any form of aid is because my talent won’t be just talent anymore. As soon as I, for example, start practising with techniques, my credibility goes down. I don’t want to end up like Daniel Tammet where a lot of people suspect or think you are fake (I used to think that Daniel Tammet was a natural calculator but when I read that he once entered a competition under a different name, that thought went away).

One day I might consider training my gifts and become the first (or one of the few) legit prodigious savant to enter a mental calculation/memory competition but that would only be in the distant future. For now, I need to, in dbz terms, stay in my base form :slight_smile:

So a hypothetical Michael Jordan that never made it to the NBA is somehow a better thing than a Michael Jordan that did practice on top of just having talent? I can’t really follow your reasoning here.

Why would that be the case? Credibility in terms of what… “legit prodigious savant”?

So, is it your goal to end up being somewhere mid-field but a “legit prodigious savant” rather than placing closer to the top but not being considered a savant because you use techniques that other savants allegedly don’t?

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This is how I do it, taking the root of 4692.
For me and my brain, this feels completely natural.

And btw, it is not the duplex method. That one is great for generating digit by digit btw, but I do it differently.

70^2 = 4,900
65^2 = 4,225

4692 is in the middle of 4225 and 4900, so either try 67 or 68.
68^2 = 4624
(I square 68 using 70-2 and then do an 11 proof: 2x2 = 4 and 4624 - 4400 = 224. 224 - 220 = 4. So 4 and 4. Checks out).

4692 - 4624 = 68.
68 is approximately 1.5% of 4692. A little less, but who’s counting? Wait; I am :wink:
So let’s take 1.4%.
1.4% divided by 2 is 0.7%.
So add 0.7% to 68 to get 68.5.

I usually stop here since irl this is enough data for me.
But if I want to be precise I do another round of this:

I square 68.5 using the data I already calculated:
68.5^2 = 68^2 (=4624) + 2 * 0.5 * 68 + 0.5^2 = 4692.25
0.25 (the amount we have too much off) divided by 4692.25 is approximately one in 19,000 (parts).
1/19,000 divided by 2 is 1/38,000.

Roughly I say 0.25 / 4692.25 is approximately 1/(4700*4). So I do a quick 4 times 4700 = 18,800. I round this to 19,000 and double that to 38,000

Next step: 68.5 / 38,000. I forget the zero’s for now and do 68.5/38:

68.5 - 38 = 30.5 => first digit is 1.
30.5/38 is approximately 0.8 (0.8 * 38 = 24 + 6.4 = 30.4, so very close. Use 0.8)

So I take 1.8 as my answer knowing this is not exact - but very close - and subtract that from 68.5, while moving “1.8” 3 places since we are not dividing by 38, but by 38,000: 0.0018
68.5 - 0.0018 = 68.4982

A calculator gives: 68.49817, so difference is 0.00003.

If I want to do another round I need to calculate 2 * 68.5 * 0.0018 (roughly a little over 0.24) and see how far from 4692.25 I am off.
However; from experience I know that I am now very close. Also; demising returns.

So I do an estimate, square of estimate, see how far off I am to find a better estimate, rinse, repeat.

This is true and there is a large thread on this forum about it.

excellent! He will meet some excellent calculators there and I hope make friendships for life since there are not many people on this planet who can do these things.

Thank you! Great to hear. This is what I am doing this for.

I think I understand. Or at least try to.
You want a way of working that is optimally suited to your brain.
You realise you have a gift and want to make the most of that gift.

If that is your way of working I respect that.

As a side note I highly encourage you and everybody on this part of the forum to read this book:


And that’s where I’m not sure what “natural” means… can’t really be the way it was first taught in school because different countries teach things differently all the time; just compare long division US vs France vs Germany.

I for one would use 50 instead of 70 for 68 as describes in this post:

If you’re curious… same steps as in the example but with a delta of 18 (68-50) for a lhs of 25+18 and a rhs of 18 squared.

I don’t consider this a shortcut or anything that wouldn’t be a “natural” way of calculating this because it builds on top of the up to 25 symmetry for the squares… you can read the rest of the linked post for details.

However, I wouldn’t insist on this method if we were talking about 65 instead of 68. In this case the tenth digits are the same and the unit digits add to 10, so you just multiply the unit digits for the rhs (5x5=25) and multiply the tenth digit with itself plus one (6x7=42) for the lhs… 4225. Essentially, my method of choice for any 2-digit square that ends in 5.

Maybe it’s just me but I like to use a solution that fits the problem… surely you can hammer a screw into the wall almost as easily as a nail, but for me, a screw asks for a screwdriver not a hammer.

Nobody is talking about being better than anyone here, I am sorry if my writing reflected that. I am not the best human calculator in the world and I don’t claim to be, I don’t even want to be, I have other dreams. All that I am saying is that I cannot use any form of aid until I get classified as a savant or something else.

I don’t want people to have doubts about my abilities. There is quite a difference in answering the question:“How do you such large calculations?” With “I practiced a lot and was good with numbers” or “I could always do them, I was just good with numbers”. The difference is that the first answer can give people doubts about your talent because you said you practiced a lot.

I don’t think you actually know the numbers. I can do 5 by 5 digit multiplications in my head without any form of aid, that already puts me in the top 99,9% of people. If I were to do an official IQ test right now then my abilities would put me in the top 99,99% on some subtests because my abilities are innate. Somebody who practised a lot or uses techniques for his or hers abilities, won’t even have his or hers abilities accepted on an IQ test because it isn’t innate.

This is also the difference between a natural human calculator and an “artificial” human calculator. They are both human calculators but the first one’s abilities are innate. The artificial can of course be much faster but it isn’t about being better here.

I hope you understand it better now. All I want is to get classified as a savant or something and until then I cannot do anything that might make people think my abilities aren’t innate/natural such as practising, using techniques, methods, shortcuts etc. I am on this forum to share my experience with people who have knowledge in mental calculations and memory feats. Who knows, perhaps somebody else shows up one day who has similar abilities. @Nagime comes pretty close and I was impressed with his abilities:) Heck, I am impressed by everyone here because I am not alone with my abilities, you guys can keep up and even surpass me by a mile. :smiley:

The most obvious difference is, of course, that savants, by definition, have their special skill or skills in spite of some basic mental disability, generally with low IQ scores overall, while prodigies are persons also with special skills or abilities but without such mental disabilities who generally function at a normal or very high level overall and whose IQ scores generally reflect that level of function.

Do you maybe mean “prodigy” rather than “savant”?

Well, that is it, I don’t know. I am seeing a psychologist right now and in about a month she will investigate if I have autism, which is very likely since I score very high on a lot of autism tests, even the one she had me fill out, so I could be a savant. I have some other odd talents that I’ve mentioned on other posts that really suggest savant ability like me ability to recall the release year of every movie I have ever seen.

It is estimated that 10% of those with autism have some form of savant abilities.

Aside the fact that 0.1% of the world population is still another 7mio that can do the same… you mentioned “mental calculation/memory competition”. At the mental calculation World Cup it’s 8x8 multiplication and just relying on talent wouldn’t put you very high there.

As far as memory competitions… the top 5 (not just the world record) do 500+ digits in 5 minutes when it comes to speed numbers, so there too, just relying on talent wouldn’t be anywhere near the top.

Excellent observation.
However; you could deduct more out of this, imho.

Choosing the last digit as a 5, leads you to comparing 68.5 with 68.
68 being the difference that we are looking for and 68.5 being (upper limit to) the amount that will be added to 6224 by choosing 5 as the next digit.

However; 68.5 is so much closer to 68 compared to the 54.72 (on taking 4 as the next digit).

I would rather overshoot by a half (in the case of taking a 5) than undershoot by 13 (by takng a 4).

In other words; taking a 5 imho is much preferred compared to taking a 4.

Let’s say you would take the 5 instead of the 4, can you work with small negative differences to keep your mind going forward? You might speed up your way of taking a root this way!