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Get the precise value of 2^0.3

yongran kim
Posted 12 years ago
I am using Mathematica 8.0.1.
I learned Mathematica by myself, and till yesterday I didn't need any feature of Mathematica
that is related with precision. But today I have to calculate some complicated formula... and the result should be
quite precise.
I knew that N[Pi, 40] gives the value of Pi upto 40 digits.
So we get 3.141592653589793238462643383279502884197
I wanted to get the value of 2^0.3, upto 40 digits. So I typed
N[2^0.3, 40]
But the result was not satisfactory. It just gave me
1.23114
Only 6 digits below the point.
I empirically know that if I copy a numerical value and paste it on a blank space in Mathematica
one can get better result.... I copied 1.23114 and paste in on a blank space then I get
1.2311444133449163`
But it too is far from 40 correct digits..
How can I get the precise value upto 40 digits ?
And what is the reason that N[2^0.3, 40] does not give 40 digits ?
Thank you very much!
POSTED BY: yongran kim
9 Replies
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Hans Milton
Hans Milton
Posted 12 years ago
Found the Preferences thing:
POSTED BY: Hans Milton
Hans Milton
Hans Milton
Posted 12 years ago
I have not been able to find the setting in Preferences which sets the number of digits to be shown when computing with approximate real numbers.
But it can be done through Option Inspector:
POSTED BY: Hans Milton
For your question,
    I want to know that if there is a command such that if I type the command in Mathematica, then from the moment, every calculation is correct up to 30 digits.

There isn't. 
It can  be immitated with some programming.   $Pre could take a function that makes all numbers 30-digit Reals. 
POSTED BY: Bruce Miller
Peter Fleck
Peter Fleck, WRI
Posted 12 years ago
the 0.3 in your expression is assumed to have machine precision unless you indicate anything else, e.g. as 0.3`40. Recommendation: make it exact, then have N display as many digits as you want:
 In[1]:= N[2^(3/10), 500]
 
 Out[1]= 1.\
 2311444133449162844993930691677431098761377611008177943370655382461007\
 1971935845840402274965089414163879015210459029977791740537864662955570\
 2613997691724733231558814007022109972283688802144111013437208936735367\
 5027231625935739266842338009115208074805416548016365456080998956820503\
 7313855248205883337415538175614392814574320010634650787696020195377044\
 7150320962601865505947644110111023427053849800553471650411634631084313\
9001571098113390373882353914706287757320079410633747638134547549531222\
793642897
POSTED BY: Peter Fleck
yongran kim
yongran kim
Posted 12 years ago
I am using Mathematica 8.0.1.
I learned Mathematica by myself, and till yesterday I didn't need any feature of Mathematica
that is related with precision. But today I have to calculate some complicated formula... and the result should be
quite precise.
I knew that N[Pi, 40] gives the value of Pi upto 40 digits.
So we get 3.141592653589793238462643383279502884197
I wanted to get the value of 2^0.3, upto 40 digits. So I typed
N[2^0.3, 40]
But the result was not satisfactory. It just gave me
1.23114
Only 6 digits below the point.
I empirically know that if I copy a numerical value and paste it on a blank space in Mathematica
one can get better result.... I copied 1.23114 and paste in on a blank space then I get
1.2311444133449163`
But it too is far from 40 correct digits..
How can I get the precise value upto 40 digits ?
And what is the reason that N[2^0.3, 40] does not give 40 digits ?
Thank you very much!
POSTED BY: yongran kim
Maybe you wish to use SetPrecision[2^0.3, 40]. Detailed information can be found in the tutorial for Arbitrary Precision Numbers and references to Precision & Accuracy Control.
POSTED BY: Isaac Abraham
N[2^0.3,40] first finds 2^0.3 accurate to about 6 digits
A very minor remark, actually 2^0.3 is computed with full machine precision, see for example InputForm[2^0.3]. There is a setting in Preferences, which controls how many digits are displayed, and that defaults to 6.
POSTED BY: Ilian Gachevski
Bill Simpson
Bill Simpson
Posted 12 years ago
Calculate the result exactly without using any decimal points and then ask for 40 digit approximation.
In[1]:= N[2^(3/10), 40]

Out[1]= 1.231144413344916284499393069167743109876

N[2^0.3,40] first finds 2^0.3 accurate to about 6 digits and then gives this result to N[6digitvalue,40] which cannot give you any more than the 6 digit value.
POSTED BY: Bill Simpson
yongran kim
yongran kim
Posted 12 years ago
Also I want to know that if there is a command such that if I type the command in Mathematica, then from the moment, every calculation is correct up to 30 digits. (Maybe the calculated result will not be correct up to 30 digits, because if there are too many or complex calculations like exponentiations. But I mean I do not mind this. What I only need is in every single calculation Mathematica does not round off up to 30 digits.)
POSTED BY: yongran kim
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