Calculate square root of base 60 numbers

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Bomkem

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Feb 16, 2021
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Hello.

I am wondering if there is any method to calculate square root of Babylonian sexagesimal number (base 60) without converting it to base 10 and using Babylonian method for finding square roots.

Example is 3; 0, 56, 15

Thanks for your time reading and responding to my question and look forward to hearing back from you!

BomKem

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The below is normal steps is to figure out square root of base 60 number

3; 0, 56, 15 = ((15/60+56)/60+0)/60 + 3 = (56.25/60 + 0)/60 +3 = (0.9735+0)/60+3 = 3.015625 -------This is a base 60 to base 10 conversion

then using the Babylonian Method of finding square root, one can find the sqrt(3.015625 ) = 1.736555499

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Bomkem said:
Hello.

I am wondering if there is any method to calculate square root of Babylonian sexagesimal number (base 60) without converting it to base 10 and using Babylonian method for finding square roots.

Example is 3; 0, 56, 15

Thanks for your time reading and responding to my question and look forward to hearing back from you!

BomKem

-----------------------------------------------------------------

The below is normal steps is to figure out square root of base 60 number

3; 0, 56, 15 = ((15/60+56)/60+0)/60 + 3 = (56.25/60 + 0)/60 +3 = (0.9735+0)/60+3 = 3.015625 -------This is a base 60 to base 10 conversion

then using the Babylonian Method of finding square root, one can find the sqrt(3.015625 ) = 1.736555499

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Click to expand...
Is this a class-assignment or an idle curiosity?
 
I am curious to know if there is such a direct way to perform square root of a base 60 number
 
Bomkem said:
Hello.

I am wondering if there is any method to calculate square root of Babylonian sexagesimal number (base 60) without converting it to base 10 and using Babylonian method for finding square roots.

Example is 3; 0, 56, 15

Thanks for your time reading and responding to my question and look forward to hearing back from you!

BomKem

-----------------------------------------------------------------

The below is normal steps is to figure out square root of base 60 number

3; 0, 56, 15 = ((15/60+56)/60+0)/60 + 3 = (56.25/60 + 0)/60 +3 = (0.9735+0)/60+3 = 3.015625 -------This is a base 60 to base 10 conversion

then using the Babylonian Method of finding square root, one can find the sqrt(3.015625 ) = 1.736555499

-----------------------------------------------------------------
Click to expand...
Are you asking whether the Babylonians were able to use the Babylonian Method themselves, using Babylonian numbers? One would imagine so, given its name. But it turns out that we don't know for sure that they used the method at all: https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method

On the other hand, since the method requires only multiplication and division, and they could multiply and divide using their numbers, the answer has to be yes.

Have you tried doing it?
 
This problem is solved.... using the following

1- Babylonian Method for finding square root by hand
2- Sexagesimal reciprocals
3- Babylonian multiplication and addition

Thank you for reading and responding to my post.
 
One method to find a square root is:
To find \(\displaystyle \sqrt{a}\) use the fact that if \(\displaystyle x^2= a\) then \(\displaystyle \frac{a}{x}= x\).

Pick some simple starting value, say x= a/2. Then \(\displaystyle \frac{a}{x}= \frac{a}{\frac{a}{2}}= \frac{2a}{a}= 2\). Now if x= 2 (so that a= 4) we are done. If not this new value is either higher or lower than the previous value of x. If it is higher, so that \(\displaystyle \frac{a}{x}> x\), then \(\displaystyle x^2< a\) so the true square root of a is higher. If it is lower then the true square root is lower. In other words, it lies between the two values of x. We don't know exactly where but it is simplest to choose half way between. The do it again.

For example, to find \(\displaystyle \sqrt{6}\), half of 6 is x= 3. 6/3= 2, not 3. So choose the next value of x half way between 2 and 3, 2.5.

Now repeat: if x= 2.5 then 6/x= 6/2.5= 2.4. Again they are not equal so try half way between 2.5 and 2.4, x= 2.45.
If x= 2.45 then 6/x= 6/2.45= 2.449 (to three decimal places). Half way between 2.45 and 2.449 is 2.4495.

And, in fact, a calculator gives the square root of 6, to three decimal places, as 2.449.

This can be done in any numeration system.
 
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