Lengths of preperiod and period of a rational number

asked 5 years ago

geroyx
127 ●5 ●8 ●16

updated 5 years ago

FrédéricC

5426 ●4 ●44 ●121

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I should get something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print(float(a))
print('a.period() gives me the length of the period:', a.period())
print('a.preperiod() does not work', #a.preperiod())

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Comments

It's really not well-defined. For all you know, $0.142857142857=\frac{142857142857}{1000000000000}$ , in which case the decimal expansion should be extended with $0$ , so the period will be $1$ and the preperiod will be the place of the last non-zero digit.

nbruin ( 5 years ago )

@nbruin I posted an answer. ;)

geroyx ( 5 years ago )

Ah, I see period is a method on a rational number, not on a float. That makes perfect sense then. Apologies for the noise.

nbruin ( 5 years ago )

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Length of Preperiod and Period

a = 1 b = 2*7 u = denominator(a/b).valuation(2) v = denominator(a/b).valuation(5) def PreperiodLength(a,b): return max(u,v) m = denominator(a/b) / (2^u*5^v) y = 1/m def PeriodLength(a,b): if m == 1: return 0 else: return y.period() #print('m = {}'.format(m)) #print('{} / {} = {}'.format(a, b, float(a/b))) print('{} / {} = {}'.format(a, b, numerical_approx(a/b, digits=PreperiodLength(a,b) + PeriodLength(a,b)))) print('PreperiodLength: {}'.format(PreperiodLength(a,b))) print('PeriodLength: {}'.format(PeriodLength(a,b)))

Out:

1 / 14 = 0.07142857 PreperiodLength: 1 PeriodLength: 6

Comments

Please use python3 syntax (sage 9.0) in your answers

FrédéricC ( 5 years ago )

@FrédéricC I fear this is over my knowledge. What do I have to correct?

geroyx ( 5 years ago )

@geroyx. I edited the lines with print.

slelievre ( 5 years ago )

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