Significant Figures using Python

Ch. 1, Problems 1, 2, and 3 - Data Reduction and Error Analysis for the Physical Sciences by Philip Bevington:

  1. How many significant figures are there in the following numbers?
    1. 976.45
    2. 84,000
    3. 0.0094
    4. 301.07
    5. 4.000
    6. 10
    7. 5280
    8. 400
  2. What is the most significant figure in each of the numbers? What is the least significant?
  3. Rround off each of the numbers above to two significant digits.

Solution:

As always I prefer to write a piece of code to complete the exercises. Shown below is a python module that has implementations of the necessary functionality.

'''
Implementation of significant digits and associated concepts.
'''

import math

def isNumeric(x):

    '''Checks to see if x represents a numeric value by converting
    into unicode and utilizing the isnumeric() method.'''

    # first convert the number into a string
    strRep = str(x)

    # first make a unicode version so we can ensure we're dealing with
    # something that represents a numeric value:
    uRep = unicode(strRep)
    if ('.' in uRep) and all([x.isnumeric() for x in uRep.split('.')]):
        return True # there's a decimal and everything to the right
                    # and left of it is numeric
    else:
        return uRep.isnumeric()

def mostSigDigit(x):

    '''Returns the most significant digit in x.'''

    assert isNumeric(x), 'x must be numeric!'

    # number the digits:
    enumeratedChars = list(enumerate(str(x)))

    nonZeroChars = [x for x in enumeratedChars if (x[1] != '0') and (x[1] != '.')]
    return nonZeroChars[0][1]

def leastSigDigit(x):

    '''Returns the least significant significant digit in x.'''

    assert isNumeric(x), 'x must be numeric!'

    # number the digits:
    enumeratedChars = list(enumerate(str(x)))

    nonZeroChars = [x for x in enumeratedChars if (x[1] != '0') and (x[1] != '.')]
    mostSignificantDigit = nonZeroChars[0]

    leastSignificantDigit = None
    if '.' in [x[1] for x in enumeratedChars]:
        leastSignificantDigit = enumeratedChars[-1]
    else:
        leastSignificantDigit = nonZeroChars[-1]
    
    # here we have a pair so just return the value:
    return leastSignificantDigit[1]

def numSigDigits(x):

    '''Returns the number of significant digits in x.'''

    assert isNumeric(x), 'x must be numeric!'

    # number the digits:
    enumeratedChars = list(enumerate(str(x)))

    nonZeroChars = [x for x in enumeratedChars if (x[1] != '0') and (x[1] != '.')]
    mostSignificantDigit = nonZeroChars[0]

    leastSignificantDigit = None
    if '.' in [x[1] for x in enumeratedChars]:
        leastSignificantDigit = enumeratedChars[-1]
    else:
        leastSignificantDigit = nonZeroChars[-1]

    enumedSignificantDigits = [x for x in enumeratedChars[mostSignificantDigit[0]:leastSignificantDigit[0] + 1]]

    numDigits = len(enumedSignificantDigits)
    if '.' in [x[1] for x in enumeratedChars]:
        numDigits -= 1
    
    return numDigits

def round_sigfigs(num, sig_figs):

    """Round to specified number of sigfigs.

    >>> round_sigfigs(0, sig_figs=4)
    0
    >>> int(round_sigfigs(12345, sig_figs=2))
    12000
    >>> int(round_sigfigs(-12345, sig_figs=2))
    -12000
    >>> int(round_sigfigs(1, sig_figs=2))
    1
    >>> '{0:.3}'.format(round_sigfigs(3.1415, sig_figs=2))
    '3.1'
    >>> '{0:.3}'.format(round_sigfigs(-3.1415, sig_figs=2))
    '-3.1'
    >>> '{0:.5}'.format(round_sigfigs(0.00098765, sig_figs=2))
    '0.00099'
    >>> '{0:.6}'.format(round_sigfigs(0.00098765, sig_figs=3))
    '0.000988'
    """
    assert isNumeric(num), 'x must be numeric!'

    if num != 0:
        return round(num, -int(math.floor(math.log10(abs(num))) - (sig_figs - 1)))
    else:
        return 0  # Can't take the log of 0

if __name__ == '__main__':

    import decimal
    numberList = ['976.45', '84000', '0.0094', '301.07', '4.000', '10', '5280', '400']

    for eachNum in map(lambda x: decimal.Decimal(str(x)), numberList):

        originalNumStr = str(eachNum)
        nsdStr = ":".join(['numSigDigits', str(numSigDigits(eachNum))])
        msdStr = ":".join(['mostSigDigit', str(mostSigDigit(eachNum))])
        lsdStr = ":".join(['leastSigDigit', str(leastSigDigit(eachNum))])
        roundedNumStr = ":".join(['rounded', str(round_sigfigs(eachNum, 2))])

        resultStr = "; ".join([originalNumStr, nsdStr, msdStr, lsdStr, roundedNumStr])

        print resultStr

When you run this you should get something like the following:

$ python significant.py
976.45; numSigDigits:5; mostSigDigit:9; leastSigDigit:5; rounded:980.0
84000; numSigDigits:2; mostSigDigit:8; leastSigDigit:4; rounded:84000.0
0.0094; numSigDigits:1; mostSigDigit:9; leastSigDigit:4; rounded:0.0094
301.07; numSigDigits:5; mostSigDigit:3; leastSigDigit:7; rounded:300.0
4.000; numSigDigits:4; mostSigDigit:4; leastSigDigit:0; rounded:4.0
10; numSigDigits:1; mostSigDigit:1; leastSigDigit:1; rounded:10.0
5280; numSigDigits:3; mostSigDigit:5; leastSigDigit:8; rounded:5300.0
400; numSigDigits:1; mostSigDigit:4; leastSigDigit:4; rounded:400.0

And we are done.

Something which struck me while I was coding this was that default python types associated with Python are not well suited for managing significant digits. They don't have a notion of significance built in. While it is true that the decimal module has a notion of precision it does NOT have a corresponding notion of accuracy and thus fails to deal properly with significance. It would be nice to have a domain specific language which gets a type which propagates error automatically. I haven't done an exhaustive search yet for such a thing but it would be nice to find.

Also carefully examine the second argument in the call to round. What does it mean to call round with a negative second argument like that? I did a little futzing around with ipython:

$ ipython
Python 2.7.1 (r271:86832, Feb 13 2012, 05:08:31) 
Type "copyright", "credits" or "license" for more information.

IPython 0.13 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: round(234, 2)
Out[1]: 234.0

In [2]: round(234, -2)
Out[2]: 200.0

In [3]: round?
Type:       builtin_function_or_method
String Form:
Namespace:  Python builtin
Docstring:
round(number[, ndigits]) -> floating point number

Round a number to a given precision in decimal digits (default 0 digits).
This always returns a floating point number.  Precision may be negative.

In [4]: round(234, -1)
Out[4]: 230.0

In [5]: round(234, -2)
Out[5]: 200.0

In [6]: round(234345.345, -2)
Out[6]: 234300.0

In [7]: round(234345.345, -1)
Out[7]: 234350.0

The examples above give you a sense of how that second argument works when you use negative values. The docs for round say at the very end 'Precision may be negative' -- Apparently this means it starts to strip away the precision if you push to negative values. This only strips away precision associated with digits in the number which correspond to positive powers of ten! The last two examples should make this concrete. Note how you lose everything to the right of the decimal! Tricky... In any case it makes for an easy way to round to the proper number of significant digits so whatever -- it works.

For further reference you may wish to consult the wikipedia article on significance arithmetic which will give you an idea on why significant digits matter. I tend to think of this as a subject leading up to formal calculation of uncertainty.

VietNam Journeys

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Visitors from the following countries do not require a visa and can stay for the following number of days.
14 days: Brunei, Myanmar
15 days: Denmark, Finland, Japan, Norway, South Korea, Sweden, Russia
21 days: Philippines
30 days: Indonesia, Laos, Malaysia, Singapore, Thailand, Cambodia
All other nationalities will require a visa in advance to visit Vietnam.
In order to boost tourism, the Vietnamese government has made the island of Phu Quoc a visa-free zone. Those flying there through Ho Chi Minh City or arriving by boat will not need to apply for a visa beforehand. This is regardless of your nationality. Travelers are given 15 days to spend on the island. Those wishing to journey elsewhere can apply for a proper Vietnamese visa at the local immigration office. All passports should be valid for at least 45 days when arriving in Phu Quoc.
Visas can be applied for at most Vietnamese embassies and consulates. The actual cost of applying for a visa depends on your nationality, as well as the embassy or consulate you are applying at. Check with the Vietnamese embassy or consulate in your country of residence for details. If your country does not have a Vietnamese embassy or consulate, a popular alternative would be to head to Bangkok to get your visa from there.
Some Vietnamese Embassies offer a "While you wait service" (May 2008), where a single entry visa can be gained in 15 minutes. This service costs USD92, but is approved instantly. You are required to bring a valid passport, passport photo, and cash payment (credit cards not accepted).
Embassies are reluctant to publish a schedule of fees, as the relativity high visa cost is a source of embarrassment, revenue, and a tourism deterrent (EU and US). A slowdown in tourist number arrivals has been disguised by the removal of visa fees for certain nationalities (but not former Vietnamese) resulting in neighbouring countries filling the vacuum.
Foreign citizens of Vietnamese origin can apply for visa exemption that allows multiple entry for 3 months at a time which is valid for the duration of the passport.
An increasingly popular alternative is to arrange a visa on arrival, which is not only considerably cheaper but also alleviates the need for passports to be posted to the Vietnamese Embassy in the country of origin.

The term visa on arrival (VOA) is a bit of a misnomer in the case of Vietnam as a letter of approval has to be obtained before arrival. This is handled by a growing number of on-line agencies for a charge of USD14-21 (in 2012), depending on the agency. Most agencies accept payment by credit card. Some accept payment by Western Union.
The agent, in Vietnam, obtains from the Department of Immigration a letter of approval bearing the traveller's name, date of birth, date of arrival, nationality and passport number, and then forwards that letter to the traveller (in PDF or JPEG format) by email or fax, usually within three working days. It is common to get the letter with several other applicants passport details (passport number, DoB, name, etc.). You might share your personal information with up to 10-30 other applicants on the same letter(s). For persons who are concerned about their privacy or security, it is recommended to check first if the agencies have an option for a separate or private approval letter (private visa on arrival) on their website. Very few online agencies have this option. Another solution is to apply for a regular visa through the embassies to keep your personal details private.
After landing at one of the three international airports (Hanoi, Ho Chi Minh City, or Danang), the traveller goes to the "visa on arrival" counter, shows the letter, fills in an additional arrival form (can be pre-filled before departure) and receives an official stamp (sticker) in his or her passport. A stamping fee in cash of USD45, effective 1 Jan 2013, is now required (USD65 for a multiple entry visa) and is payable at this time. Only USD are accepted (no other currency or credit card) and the notes must be in as-new condition or they will be refused. Two passport photos are also required.
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Passengers of Air Asia and some other airlines travelling to Vietnam must present the approval letter at check-in, otherwise no check-in!
Vietnam has moved away from arrival/departure cards.
Depending on the present level of SARS, avian flu you may be subjected to a so-called health-check. There is no examination, though, but yet another form to fill in and, of course, another fee. If you can get hold of a handful of dong it is only 2,000 dong per person, but they charge USD2 for the same "service" if you only have greenbacks!

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