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Issues with .99999999999...

asked 2016-12-09 05:00:48 -0500

mf gravatar image

updated 2017-04-10 20:13:53 -0500

tmonteil gravatar image

Here is a result that I found surprising and I don't understand completely what corercian is causing it.

sage: int(.9999999999999999)
sage: int(.99999999999999999)
sage: int(0.99999999999999999)
sage: int(0.9999999999999999)
sage: int(0.999999999999999999999999999999999)

What is going on?

More to play with:

sage: a=.99999999999999999; b=0.999999999999999999999999999999999
sage: a<b
sage: int(a)<int(b)


sage: a=.999999999999999990; b=0.999999999999999999999999999999999
sage: int(a)<int(b)
sage: a<b
sage: a=.999999999999999997
sage: a<b
sage: a=.999999999999999996
sage: a<b
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I believe this is due to number of precision It might be a good idea to work with RealField

A.Alharbi gravatar imageA.Alharbi ( 2016-12-09 05:09:57 -0500 )edit

@A.Alharbi they are already elements of some RealField (with various precisions).

tmonteil gravatar imagetmonteil ( 2016-12-09 06:20:05 -0500 )edit

1 answer

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answered 2016-12-09 06:28:12 -0500

tmonteil gravatar image

updated 2017-04-10 20:18:27 -0500

When you write:

sage: a=.99999999999999999; b=0.999999999999999999999999999999999

You should understand that, while they have an exact decimal representation, those numbers can not be written in binary form exactly (i.e. there are no integers p and q such that a=p/2^q). So, when Sage transforms them into floating-point numbers, it has to round them. Since the rounding is done toward the closest floating-point number (with some given precision), sompetimes it is rounded from below, sometimes from above. When two numbers a<b are converted to floating point with the same precision, of course the corresponding floating-point numbers are ordered in a monotonic way. There is no reason why it should be the case when the numbers are rounded with different precisions, since increasing the precision adds some new floating-point numbers inbetween to which b will be rounded.

Here is some Sage code to understand your issue:

sage: a.parent()
Real Field with 54 bits of precision
sage: b.parent()
Real Field with 110 bits of precision
sage: a.sign_mantissa_exponent()
(1, 9007199254740992, -53)
sage: a.exact_rational()
sage: b.sign_mantissa_exponent()
(1, 1298074214633706907132624082305023, -110)
sage: b.exact_rational()

EDIT: It turns out that there was a bug with the way Sage currently deals with .999... vs 0.999..., see!msg...

Without this weird behaviour, things are ordered correctly:

sage: all(RealNumber('0.'+'9'*i) < RealNumber('0.'+'9'*(i+1)) for i in range(3,1000))
sage: all(RealNumber('0.'+'9'*i) < RealNumber('0.'+'9'*j) for i in range(3,100) for j in range(i+1,100))
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In complement to @tmonteil's answer, you may have a look at what Sage's is doing when you enter a=.99999999999999999: it converts the string ".99999999999999999" to a real number of finite precision, the latter being determined by the length of the string. Indeed Sage's preparser is invoking the function RealNumber:

sage: preparse("a=.99999999999999999")

and you can have a look at what this function is doing via

sage: RealNumber??
eric_g gravatar imageeric_g ( 2016-12-09 06:47:02 -0500 )edit

Besides, the example

sage: a=.99999999999999999; b=0.999999999999999999999999999999999
sage: a<b

is not so surprising if you print out a-1 and b-1:

sage: a-1 
sage: b-1 

It is then mathematically correct that a<b.

eric_g gravatar imageeric_g ( 2016-12-09 06:57:02 -0500 )edit

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Asked: 2016-12-09 05:00:48 -0500

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Last updated: Apr 10