Re: [Jprogramming] Finding Irrational Numbers
My first reaction to Bo's remarks (far below) was "well said and very
sensible" although my instinct was to use 1e8 near half the precision of
floats. I started playing with tacit versions and got to observe what I
thought was a surprising number of false positives.
First some tacit fun:
rat1=:13 : '1e9>1{"1]2 x: y'
rat2=:1e9 > 1 {"1 (2:)
rat3=:1e9 > 1 {"1 x:~&2
rat1
10 > 1 {"1 [: ] 2 x: ]
rat2
10 > 1 ({"1) 2:
rat3
10 > 1 {"1 x:~&2
ts=:6!:2 , 7!:2@]
100 ts 'rat1 1r3^~i.1'
0.118844 1.00918e7
100 ts 'rat2 1r3^~i.1'
0.117574 1.02228e7
100 ts 'rat3 1r3^~i.1'
0.117814 1.02228e7
All very close in performance. I think rat2 is the most readable but
rat3 is a character shorter. Now some experiments where I had some
expectations.
+/rat3 1r3^~i.1
109
I expected around
3%:1-1
21.5436
poke around for a small false positive
i#~rat3 1r3^~i=:i.1000
0 1 8 27 64 89 125 216 343 441 512 607 729
89^1r3
4.46475
x: 89^1r3
1344576733r301154199
rat3 89^1r3
1
How about using 1e5 instead of 1e9:
rat4=:1e5 > 1 {"1 x:~&2
+/rat4 1r3^~i.1
26
+/rat4 1r2^~i.1
100
+/rat4 o.i.1000
4
And a couple more experiments
+/rat3 o.i.1000
289
+/rat3 0.01*1r3^~i.1
28
So, as previously noted by others, beware. But if you are going to use a
test for rational, you might do better with something less than 1e9.
On 6/14/2018 7:00 PM, 'Bo Jacoby' via Programming wrote:
I completely agree with Raul. We all agree that the problem of identifying
irrational floating point numbers is unsolvable, simply because any floating
point number is rational. Critics need not show that a proposed algorithm is
imperfect, but rather suggest improvements.
rational=.3 : '1e9>1{"1]2 x: y'
rational 10^_8 _9 NB. useless result
1 0
rational (%3)^~i.9 NB. useful result
1 1 0 0 0 0 0 0 1
/Bo..
Den 16:37 torsdag den 14. juni 2018 skrev Raul Miller
:
Or, put different, it's a heuristic:
It's likely to be right significantly more often than it's wrong for a
variety of likely contexts.
But it's going to be wrong some of the time, also, and it's useful to
understand why that would be.
Thanks,
--
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Re: [Jprogramming] Finding Irrational Numbers
I completely agree with Raul. We all agree that the problem of identifying
irrational floating point numbers is unsolvable, simply because any floating
point number is rational. Critics need not show that a proposed algorithm is
imperfect, but rather suggest improvements.
rational=.3 : '1e9>1{"1]2 x: y'
rational 10^_8 _9 NB. useless result
1 0
rational (%3)^~i.9 NB. useful result
1 1 0 0 0 0 0 0 1
/Bo..
Den 16:37 torsdag den 14. juni 2018 skrev Raul Miller
:
Or, put different, it's a heuristic:
It's likely to be right significantly more often than it's wrong for a
variety of likely contexts.
But it's going to be wrong some of the time, also, and it's useful to
understand why that would be.
Thanks,
--
Raul
--
For information about J forums see http://www.jsoftware.com/forums.htm
--
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Re: [Jprogramming] Finding Irrational Numbers
Or, put different, it's a heuristic: It's likely to be right significantly more often than it's wrong for a variety of likely contexts. But it's going to be wrong some of the time, also, and it's useful to understand why that would be. Thanks, -- Raul -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
What Bo means is that if you execute a =: irrational_number and then convert the approximation a to rational, the denominator will be large. Not necessarily true: first of all, x: uses tolerant comparison: x: 1.001 1 so to look at all the bits of a you need (x:!.0) 1.001 900719925474100r900719925474099 [and note: that was a perfectly respectable rational number, but it comes out with a large denominator] Moreover, x: 1.00014973485793 NB. It's an irrational number, I just got tired of typing 1 Henry Rich On 6/14/2018 10:12 AM, William Tanksley, Jr wrote: I have no idea why you're saying that. Irrationals don't have denominators; they cannot be written as fractions. And no floating point numbers are irrationals. On Thu, Jun 14, 2018 at 12:18 AM 'Bo Jacoby' via Programming < programm...@jsoftware.com> wrote: If a floating point number (a) , is irrational, then (x:a) has a denominator greater than 1e10. Some rational and irrational numbers are: x:%%:>:i.5 1 676982533219r957397879968 9943229281777r17222178307344 1r2 8728368286235r19517224820674 The denominators are: 1 957397879968 17222178307344 2 19517224820674 1 9.57398e11 1.7e13 2 1.95172e13 So the problem of identifying irrational numbers is reduced to the problem of finding the denominator in x:a /Bo. Den 23:42 onsdag den 13. juni 2018 skrev Linda Alvord < lindaalvor...@outlook.com>: I wrote a definition with only one argument. Linda Get Outlook for Android<https://aka.ms/ghei36> From: Programming on behalf of Raul Miller Sent: Wednesday, June 13, 2018 4:36:31 PM To: Programming forum Subject: Re: [Jprogramming] Finding Irrational Numbers So, something like this? fe=:4 :0 p=. 9!:10'' vi=. _. ve=. y try. 9!:11 x ve=. ":&.>y 9!:11]15 vi=. ":&.>y catch. end. 9!:11 p ve=vi ) Thanks, -- Raul On Wed, Jun 13, 2018 at 4:22 PM Skip Cave wrote: So I'm thinking something like this, to find exact numbers in a floating point array: pps 15 NB. Set the print precision. Create a mixed floating point vector b: ]b=:(,3? 10^1+i.4),%a=:1+i.20 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 0.2 0.167 0.142857142857143 0.125 0.111 0.1 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 0.0667 0.0625 0.0588235294117... Create a "find exact" verb 'fe' where x=the number of digits max that can be exact, and y is the vector to be tested (boolean result): fe=. 4 : 'x> ># each ": each <"1,.y' ]c=.9 fe b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 Show exact: c#b 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 0.05 Show inexact: (-.c)#b 0.333 0.167 0.142857142857143 0.111 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 0.0667 0.0588235294117647 0.0556 0.0526315789473684 #b 32 #c#b 20 I'm sure this scheme has some holes, but it is a reasonable first cut. Skip -- For information about J forums see https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm --- This email has been checked for viruses by AVG. https://www.avg.com -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
I have no idea why you're saying that. Irrationals don't have denominators; they cannot be written as fractions. And no floating point numbers are irrationals. On Thu, Jun 14, 2018 at 12:18 AM 'Bo Jacoby' via Programming < programm...@jsoftware.com> wrote: > If a floating point number (a) , is irrational, then (x:a) has a > denominator greater than 1e10. > Some rational and irrational numbers are: >x:%%:>:i.5 > 1 676982533219r957397879968 9943229281777r17222178307344 1r2 > 8728368286235r19517224820674 > The denominators are: > 1 957397879968 17222178307344 2 19517224820674 > 1 9.57398e11 1.7e13 2 1.95172e13 > So the problem of identifying irrational numbers is reduced to the problem > of finding the denominator in x:a > /Bo. > > Den 23:42 onsdag den 13. juni 2018 skrev Linda Alvord < > lindaalvor...@outlook.com>: > > > > I wrote a definition with only one argument. > > Linda > > Get Outlook for Android<https://aka.ms/ghei36> > > > From: Programming on behalf of > Raul Miller > Sent: Wednesday, June 13, 2018 4:36:31 PM > To: Programming forum > Subject: Re: [Jprogramming] Finding Irrational Numbers > > So, something like this? > > fe=:4 :0 > p=. 9!:10'' > vi=. _. > ve=. y > try. > 9!:11 x > ve=. ":&.>y > 9!:11]15 > vi=. ":&.>y > catch. > end. > 9!:11 p > ve=vi > ) > > Thanks, > > -- > Raul > On Wed, Jun 13, 2018 at 4:22 PM Skip Cave wrote: > > > > So I'm thinking something like this, to find exact numbers in a floating > > point array: > > > > > > pps 15 NB. Set the print precision. > > > > > > Create a mixed floating point vector b: > > > > > >]b=:(,3? 10^1+i.4),%a=:1+i.20 > > > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 > 0.2 > > 0.167 0.142857142857143 0.125 0.111 0.1 > > 0.0909090909090909 0.0833 0.0769230769230769 > 0.0714285714285714 > > 0.0667 0.0625 0.0588235294117... > > > > > > Create a "find exact" verb 'fe' where x=the number of digits max that can > > be exact, and y is the vector to be tested (boolean result): > > > > > >fe=. 4 : 'x> ># each ": each <"1,.y' > > > > > >]c=.9 fe b > > > > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 > > > > > > Show exact: > > > >c#b > > > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 > > 0.05 > > > > > > Show inexact: > > > >(-.c)#b > > > > 0.333 0.167 0.142857142857143 0.111 > > 0.0909090909090909 0.0833 0.0769230769230769 > 0.0714285714285714 > > 0.0667 0.0588235294117647 0.0556 > 0.0526315789473684 > > > >#b > > > > 32 > > > >#c#b > > > > 20 > > > > > > I'm sure this scheme has some holes, but it is a reasonable first cut. > > > > > > Skip > > -- > > For information about J forums see > https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 > -- > For information about J forums see > https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 > -- > For information about J forums see http://www.jsoftware.com/forums.htm > > > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
1e9<1{"1]2 x: %:i.26
0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0
1{ is better than }. .
1{"1]2 x: %:14
6286063111
This denominator is <1e10
/Bo.
Den 11:44 torsdag den 14. juni 2018 skrev Skip Cave
:
OK. Thanks to Bo, sizing the denominators looks like the way to go. Here's
a verb that looks at the denominators of extended/rational numbers and
determine if the denominator is too big, and thus the number is inexact:
NB. First generate some exact & inexact numbers:
]t=.x:%%:>:i.5
1 676982533219r957397879968 9943229281777r17222178307344 1r2
8728368286235r19517224820674
NB. Now get the denominators, & check the size - too big=0, small enough -
1:
-.,1e10<}."1(2:)t
1 0 0 1 0
NB. Use the selection vector to pick out the exact values:
t#~-.,1e10<}."1(2:)t
1 1r2
]b=.x:%%:>:i.15
1 676982533219r957397879968 9943229281777r17222178307344 1r2
8728368286235r19517224820674 250258529119r613005700122
12895459543967r34118178995231 1532495590117r4334552095641 1r3
16884524975r53393556131 535761049157r1776918377341
3208086930518r313911750...
b#~ -.,1e10<}."1(2:)b
1 1r2 1r3
Skip
On Thu, Jun 14, 2018 at 2:18 AM 'Bo Jacoby' via Programming <
programm...@jsoftware.com> wrote:
> If a floating point number (a) , is irrational, then (x:a) has a
> denominator greater than 1e10.
> Some rational and irrational numbers are:
> x:%%:>:i.5
> 1 676982533219r957397879968 9943229281777r17222178307344 1r2
> 8728368286235r19517224820674
> The denominators are:
> 1 957397879968 17222178307344 2 19517224820674
> 1 9.57398e11 1.7e13 2 1.95172e13
> So the problem of identifying irrational numbers is reduced to the problem
> of finding the denominator in x:a
> /Bo.
>
>
>
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Re: [Jprogramming] Finding Irrational Numbers
OK. Thanks to Bo, sizing the denominators looks like the way to go. Here's a verb that looks at the denominators of extended/rational numbers and determine if the denominator is too big, and thus the number is inexact: NB. First generate some exact & inexact numbers: ]t=.x:%%:>:i.5 1 676982533219r957397879968 9943229281777r17222178307344 1r2 8728368286235r19517224820674 NB. Now get the denominators, & check the size - too big=0, small enough - 1: -.,1e10<}."1(2:)t 1 0 0 1 0 NB. Use the selection vector to pick out the exact values: t#~-.,1e10<}."1(2:)t 1 1r2 ]b=.x:%%:>:i.15 1 676982533219r957397879968 9943229281777r17222178307344 1r2 8728368286235r19517224820674 250258529119r613005700122 12895459543967r34118178995231 1532495590117r4334552095641 1r3 16884524975r53393556131 535761049157r1776918377341 3208086930518r313911750... b#~ -.,1e10<}."1(2:)b 1 1r2 1r3 Skip On Thu, Jun 14, 2018 at 2:18 AM 'Bo Jacoby' via Programming < programm...@jsoftware.com> wrote: > If a floating point number (a) , is irrational, then (x:a) has a > denominator greater than 1e10. > Some rational and irrational numbers are: >x:%%:>:i.5 > 1 676982533219r957397879968 9943229281777r17222178307344 1r2 > 8728368286235r19517224820674 > The denominators are: > 1 957397879968 17222178307344 2 19517224820674 > 1 9.57398e11 1.7e13 2 1.95172e13 > So the problem of identifying irrational numbers is reduced to the problem > of finding the denominator in x:a > /Bo. > > > -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
If a floating point number (a) , is irrational, then (x:a) has a denominator greater than 1e10. Some rational and irrational numbers are: x:%%:>:i.5 1 676982533219r957397879968 9943229281777r17222178307344 1r2 8728368286235r19517224820674 The denominators are: 1 957397879968 17222178307344 2 19517224820674 1 9.57398e11 1.7e13 2 1.95172e13 So the problem of identifying irrational numbers is reduced to the problem of finding the denominator in x:a /Bo. Den 23:42 onsdag den 13. juni 2018 skrev Linda Alvord : I wrote a definition with only one argument. Linda Get Outlook for Android<https://aka.ms/ghei36> From: Programming on behalf of Raul Miller Sent: Wednesday, June 13, 2018 4:36:31 PM To: Programming forum Subject: Re: [Jprogramming] Finding Irrational Numbers So, something like this? fe=:4 :0 p=. 9!:10'' vi=. _. ve=. y try. 9!:11 x ve=. ":&.>y 9!:11]15 vi=. ":&.>y catch. end. 9!:11 p ve=vi ) Thanks, -- Raul On Wed, Jun 13, 2018 at 4:22 PM Skip Cave wrote: > > So I'm thinking something like this, to find exact numbers in a floating > point array: > > > pps 15 NB. Set the print precision. > > > Create a mixed floating point vector b: > > > ]b=:(,3? 10^1+i.4),%a=:1+i.20 > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 0.2 > 0.167 0.142857142857143 0.125 0.111 0.1 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0625 0.0588235294117... > > > Create a "find exact" verb 'fe' where x=the number of digits max that can > be exact, and y is the vector to be tested (boolean result): > > > fe=. 4 : 'x> ># each ": each <"1,.y' > > > ]c=.9 fe b > > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 > > > Show exact: > > c#b > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 > 0.05 > > > Show inexact: > > (-.c)#b > > 0.333 0.167 0.142857142857143 0.111 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0588235294117647 0.0556 0.0526315789473684 > > #b > > 32 > > #c#b > > 20 > > > I'm sure this scheme has some holes, but it is a reasonable first cut. > > > Skip > -- > For information about J forums see > https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
I wrote a definition with only one argument. Linda Get Outlook for Android<https://aka.ms/ghei36> From: Programming on behalf of Raul Miller Sent: Wednesday, June 13, 2018 4:36:31 PM To: Programming forum Subject: Re: [Jprogramming] Finding Irrational Numbers So, something like this? fe=:4 :0 p=. 9!:10'' vi=. _. ve=. y try. 9!:11 x ve=. ":&.>y 9!:11]15 vi=. ":&.>y catch. end. 9!:11 p ve=vi ) Thanks, -- Raul On Wed, Jun 13, 2018 at 4:22 PM Skip Cave wrote: > > So I'm thinking something like this, to find exact numbers in a floating > point array: > > > pps 15 NB. Set the print precision. > > > Create a mixed floating point vector b: > > > ]b=:(,3? 10^1+i.4),%a=:1+i.20 > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 0.2 > 0.167 0.142857142857143 0.125 0.111 0.1 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0625 0.0588235294117... > > > Create a "find exact" verb 'fe' where x=the number of digits max that can > be exact, and y is the vector to be tested (boolean result): > > >fe=. 4 : 'x> ># each ": each <"1,.y' > > >]c=.9 fe b > > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 > > > Show exact: > >c#b > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 > 0.05 > > > Show inexact: > >(-.c)#b > > 0.333 0.167 0.142857142857143 0.111 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0588235294117647 0.0556 0.0526315789473684 > >#b > > 32 > >#c#b > > 20 > > > I'm sure this scheme has some holes, but it is a reasonable first cut. > > > Skip > -- > For information about J forums see > https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm=02%7C01%7C%7C41ac47c72e1441cc1f0b08d5d16d6b12%7C84df9e7fe9f640afb435%7C1%7C0%7C636645190246397362=C%2B8YZ2EAW533k4brRJj7gZXQzxUXVbnCthCXYM1YZ1I%3D=0 -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
So, something like this? fe=:4 :0 p=. 9!:10'' vi=. _. ve=. y try. 9!:11 x ve=. ":&.>y 9!:11]15 vi=. ":&.>y catch. end. 9!:11 p ve=vi ) Thanks, -- Raul On Wed, Jun 13, 2018 at 4:22 PM Skip Cave wrote: > > So I'm thinking something like this, to find exact numbers in a floating > point array: > > > pps 15 NB. Set the print precision. > > > Create a mixed floating point vector b: > > > ]b=:(,3? 10^1+i.4),%a=:1+i.20 > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 0.2 > 0.167 0.142857142857143 0.125 0.111 0.1 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0625 0.0588235294117... > > > Create a "find exact" verb 'fe' where x=the number of digits max that can > be exact, and y is the vector to be tested (boolean result): > > >fe=. 4 : 'x> ># each ": each <"1,.y' > > >]c=.9 fe b > > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 > > > Show exact: > >c#b > > 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 > 0.05 > > > Show inexact: > >(-.c)#b > > 0.333 0.167 0.142857142857143 0.111 > 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 > 0.0667 0.0588235294117647 0.0556 0.0526315789473684 > >#b > > 32 > >#c#b > > 20 > > > I'm sure this scheme has some holes, but it is a reasonable first cut. > > > Skip > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
Oh, I see - yes, as long as the values are represented precisely, yes.
Basically:
terminates=:4 :0
(q:x) -:&(*./) x ,: {:2 x:y
)
60 terminates 1r120
1
60 terminates 1r121
0
Thanks,
--
Raul
On Wed, Jun 13, 2018 at 3:51 PM William Tanksley, Jr
wrote:
>
> No need to specify the precision if the input is actually rational numbers,
> not floating point numbers.
>
> On Wed, Jun 13, 2018 at 12:47 PM Raul Miller wrote:
>
> > You would also have to specify the precision of the result.
> >
> > Put different: if that specification is too precise (pushing the
> > limits of what floating point numbers can represent), the algorithm
> > would degrade to meaninglessness because of floating point
> > inaccuracies. And, if that specification was not precise enough then
> > you wouldn't be able to represent many values.
> >
> > Something like six digits of decimal (base 10 (base 10 (base 10
> > (... precision might be a sweet spot.
> >
> > Thanks,
> >
> > --
> > Raul
> >
> > On Wed, Jun 13, 2018 at 3:34 PM William Tanksley, Jr
> > wrote:
> > >
> > > You could do this with rational numbers, since whether or not they
> > > terminate IS an interesting puzzle. Of course, you have to specify a
> > > "decimal" base -- 1/3 doesn't terminate base 10, but does terminate base
> > 60.
> > >
> > > On Wed, Jun 13, 2018 at 12:26 PM Raul Miller
> > wrote:
> > >
> > > > Floating point numbers implicitly terminate in infinitely repeating
> > > > zeros after the 52 expressed bits of mantissa.
> > > >
> > > > Or, put differently, when we need to represent numbers which have
> > > > non-zero bits that can't be represented, we approximate. Or, another
> > > > view of floating point numbers is that they each represent an infinity
> > > > of values which divide the number line between the preceding and
> > > > following values (with a few special cases, like the ininities).
> > > >
> > > > I hope this helps in your efforts to express what you are looking
> > for...
> > > >
> > > > Thanks,
> > > >
> > > > --
> > > > Raul
> > > > On Wed, Jun 13, 2018 at 3:23 PM Skip Cave
> > wrote:
> > > > >
> > > > > Ok. Then I redefine my question:
> > > > >
> > > > > Given the vector a:
> > > > >
> > > > > ]a =. % 1+i.20
> > > > >
> > > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1
> > 0.0909091
> > > > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > > > > 0.0526316 0.05
> > > > >
> > > > >
> > > > > Define a verb that will find all the floating-point numbers in a that
> > > > will
> > > > > eventually terminate in infinitely repeating zeros.
> > > > >
> > > > >
> > > > > Skip
> > > > >
> > > > >
> > > > > On Wed, Jun 13, 2018 at 2:12 PM Henry Rich
> > wrote:
> > > > >
> > > > > > The trailing 0 repeats forever.
> > > > > >
> > > > > > Henry Rich
> > > > > >
> > > > > >
> > > > > >
> > > > >
> > --
> > > > > For information about J forums see
> > http://www.jsoftware.com/forums.htm
> > > > --
> > > > For information about J forums see http://www.jsoftware.com/forums.htm
> > > --
> > > For information about J forums see http://www.jsoftware.com/forums.htm
> > --
> > For information about J forums see http://www.jsoftware.com/forums.htm
> --
> For information about J forums see http://www.jsoftware.com/forums.htm
--
For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
So I'm thinking something like this, to find exact numbers in a floating point array: pps 15 NB. Set the print precision. Create a mixed floating point vector b: ]b=:(,3? 10^1+i.4),%a=:1+i.20 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.333 0.25 0.2 0.167 0.142857142857143 0.125 0.111 0.1 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 0.0667 0.0625 0.0588235294117... Create a "find exact" verb 'fe' where x=the number of digits max that can be exact, and y is the vector to be tested (boolean result): fe=. 4 : 'x> ># each ": each <"1,.y' ]c=.9 fe b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 Show exact: c#b 0 2 7 36 93 57 808 875 559 4674 8635 6196 1 0.5 0.25 0.2 0.125 0.1 0.0625 0.05 Show inexact: (-.c)#b 0.333 0.167 0.142857142857143 0.111 0.0909090909090909 0.0833 0.0769230769230769 0.0714285714285714 0.0667 0.0588235294117647 0.0556 0.0526315789473684 #b 32 #c#b 20 I'm sure this scheme has some holes, but it is a reasonable first cut. Skip -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
No need to specify the precision if the input is actually rational numbers, not floating point numbers. On Wed, Jun 13, 2018 at 12:47 PM Raul Miller wrote: > You would also have to specify the precision of the result. > > Put different: if that specification is too precise (pushing the > limits of what floating point numbers can represent), the algorithm > would degrade to meaninglessness because of floating point > inaccuracies. And, if that specification was not precise enough then > you wouldn't be able to represent many values. > > Something like six digits of decimal (base 10 (base 10 (base 10 > (... precision might be a sweet spot. > > Thanks, > > -- > Raul > > On Wed, Jun 13, 2018 at 3:34 PM William Tanksley, Jr > wrote: > > > > You could do this with rational numbers, since whether or not they > > terminate IS an interesting puzzle. Of course, you have to specify a > > "decimal" base -- 1/3 doesn't terminate base 10, but does terminate base > 60. > > > > On Wed, Jun 13, 2018 at 12:26 PM Raul Miller > wrote: > > > > > Floating point numbers implicitly terminate in infinitely repeating > > > zeros after the 52 expressed bits of mantissa. > > > > > > Or, put differently, when we need to represent numbers which have > > > non-zero bits that can't be represented, we approximate. Or, another > > > view of floating point numbers is that they each represent an infinity > > > of values which divide the number line between the preceding and > > > following values (with a few special cases, like the ininities). > > > > > > I hope this helps in your efforts to express what you are looking > for... > > > > > > Thanks, > > > > > > -- > > > Raul > > > On Wed, Jun 13, 2018 at 3:23 PM Skip Cave > wrote: > > > > > > > > Ok. Then I redefine my question: > > > > > > > > Given the vector a: > > > > > > > > ]a =. % 1+i.20 > > > > > > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 > 0.0909091 > > > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > > > > 0.0526316 0.05 > > > > > > > > > > > > Define a verb that will find all the floating-point numbers in a that > > > will > > > > eventually terminate in infinitely repeating zeros. > > > > > > > > > > > > Skip > > > > > > > > > > > > On Wed, Jun 13, 2018 at 2:12 PM Henry Rich > wrote: > > > > > > > > > The trailing 0 repeats forever. > > > > > > > > > > Henry Rich > > > > > > > > > > > > > > > > > > > > -- > > > > For information about J forums see > http://www.jsoftware.com/forums.htm > > > -- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > -- > > For information about J forums see http://www.jsoftware.com/forums.htm > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
You would also have to specify the precision of the result. Put different: if that specification is too precise (pushing the limits of what floating point numbers can represent), the algorithm would degrade to meaninglessness because of floating point inaccuracies. And, if that specification was not precise enough then you wouldn't be able to represent many values. Something like six digits of decimal (base 10 (base 10 (base 10 (... precision might be a sweet spot. Thanks, -- Raul On Wed, Jun 13, 2018 at 3:34 PM William Tanksley, Jr wrote: > > You could do this with rational numbers, since whether or not they > terminate IS an interesting puzzle. Of course, you have to specify a > "decimal" base -- 1/3 doesn't terminate base 10, but does terminate base 60. > > On Wed, Jun 13, 2018 at 12:26 PM Raul Miller wrote: > > > Floating point numbers implicitly terminate in infinitely repeating > > zeros after the 52 expressed bits of mantissa. > > > > Or, put differently, when we need to represent numbers which have > > non-zero bits that can't be represented, we approximate. Or, another > > view of floating point numbers is that they each represent an infinity > > of values which divide the number line between the preceding and > > following values (with a few special cases, like the ininities). > > > > I hope this helps in your efforts to express what you are looking for... > > > > Thanks, > > > > -- > > Raul > > On Wed, Jun 13, 2018 at 3:23 PM Skip Cave wrote: > > > > > > Ok. Then I redefine my question: > > > > > > Given the vector a: > > > > > > ]a =. % 1+i.20 > > > > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 > > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > > > 0.0526316 0.05 > > > > > > > > > Define a verb that will find all the floating-point numbers in a that > > will > > > eventually terminate in infinitely repeating zeros. > > > > > > > > > Skip > > > > > > > > > On Wed, Jun 13, 2018 at 2:12 PM Henry Rich wrote: > > > > > > > The trailing 0 repeats forever. > > > > > > > > Henry Rich > > > > > > > > > > > > > > > -- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > -- > > For information about J forums see http://www.jsoftware.com/forums.htm > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
You could do this with rational numbers, since whether or not they terminate IS an interesting puzzle. Of course, you have to specify a "decimal" base -- 1/3 doesn't terminate base 10, but does terminate base 60. On Wed, Jun 13, 2018 at 12:26 PM Raul Miller wrote: > Floating point numbers implicitly terminate in infinitely repeating > zeros after the 52 expressed bits of mantissa. > > Or, put differently, when we need to represent numbers which have > non-zero bits that can't be represented, we approximate. Or, another > view of floating point numbers is that they each represent an infinity > of values which divide the number line between the preceding and > following values (with a few special cases, like the ininities). > > I hope this helps in your efforts to express what you are looking for... > > Thanks, > > -- > Raul > On Wed, Jun 13, 2018 at 3:23 PM Skip Cave wrote: > > > > Ok. Then I redefine my question: > > > > Given the vector a: > > > > ]a =. % 1+i.20 > > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > > 0.0526316 0.05 > > > > > > Define a verb that will find all the floating-point numbers in a that > will > > eventually terminate in infinitely repeating zeros. > > > > > > Skip > > > > > > On Wed, Jun 13, 2018 at 2:12 PM Henry Rich wrote: > > > > > The trailing 0 repeats forever. > > > > > > Henry Rich > > > > > > > > > > > -- > > For information about J forums see http://www.jsoftware.com/forums.htm > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
Floating point numbers implicitly terminate in infinitely repeating zeros after the 52 expressed bits of mantissa. Or, put differently, when we need to represent numbers which have non-zero bits that can't be represented, we approximate. Or, another view of floating point numbers is that they each represent an infinity of values which divide the number line between the preceding and following values (with a few special cases, like the ininities). I hope this helps in your efforts to express what you are looking for... Thanks, -- Raul On Wed, Jun 13, 2018 at 3:23 PM Skip Cave wrote: > > Ok. Then I redefine my question: > > Given the vector a: > > ]a =. % 1+i.20 > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > 0.0526316 0.05 > > > Define a verb that will find all the floating-point numbers in a that will > eventually terminate in infinitely repeating zeros. > > > Skip > > > On Wed, Jun 13, 2018 at 2:12 PM Henry Rich wrote: > > > The trailing 0 repeats forever. > > > > Henry Rich > > > > > > > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
To see the literal structure of a floating point number, you can use:
fpdigits=:3 :'(1 j.0 11 e.~i.64)#'' ''-.~":,(8#2)#:a.i.|._8{.3!:1 y+1.1-1.1'"0
NB. sign (1 is negative), mantissa+1023, exponent with implied leading 1
Here, I'm going to use it to illustrate two issues:
a) floating point representation is not a decimal notation.
fpdigits 0.2
0 000 1001100110011001100110011001100110011001100110011010
b) as Henry Rich pointed out, the zeros repeat for otherwise exact
binary fractions:
fpdigits 0.5
0 010
There are a few other tricks and issues related to floating point
numbers, but this should help?
Thanks,
--
Raul
On Wed, Jun 13, 2018 at 3:06 PM Skip Cave wrote:
>
> Skip said:
> Given the vector a:
>
> ]a =. % 1+i.20
>
> 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091
> 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> 0.0526316 0.05
>
> Roger said:
> Your list a are reciprocals of integers and so are all rational.
>
> Roger also said:
> All rational fractions result in infinitely repeating floating point
> numbers. They are the same sets.
>
> Skip says:
> So all the numbers in a are rational, and infinitely repeating? Like 0.5,
> 0.25. 0.2, 0.1? I'm confused. These examples don't seem to have infinitely
> repeating decimals.
>
> Skip
>
>
> On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
> wrote:
>
> > What's an irrational number in this context? Your list a are reciprocals
> > of integers and so are all rational. On the other hand, going just by the
> > display, 0.5 is a rational number (1%2), but since the display is to 6
> > significant digits, for all you know 0.5 could be
> > 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
> >
> >
> > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave
> > wrote:
> >
> > > Here's another problem similar to my previous one about finding integers
> > in
> > > a floating point array:
> > >
> > > Find the irrational numbers in a floating-point array:
> > >
> > > Given the vector a:
> > >
> > > ]a =. % 1+i.20
> > >
> > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091
> > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > > 0.0526316 0.05
> > >
> > >
> > > Create a function that will generate a boolean array indicating the
> > > locations of the irrational numbers in a.
> > >
> > >
> > > Skip
> > > --
> > > For information about J forums see http://www.jsoftware.com/forums.htm
> > --
> > For information about J forums see http://www.jsoftware.com/forums.htm
> --
> For information about J forums see http://www.jsoftware.com/forums.htm
--
For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
Ok. Then I redefine my question: Given the vector a: ]a =. % 1+i.20 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 0.0526316 0.05 Define a verb that will find all the floating-point numbers in a that will eventually terminate in infinitely repeating zeros. Skip On Wed, Jun 13, 2018 at 2:12 PM Henry Rich wrote: > The trailing 0 repeats forever. > > Henry Rich > > > -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
The trailing 0 repeats forever. Henry Rich On 6/13/2018 3:05 PM, Skip Cave wrote: Skip said: Given the vector a: ]a =. % 1+i.20 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 0.0526316 0.05 Roger said: Your list a are reciprocals of integers and so are all rational. Roger also said: All rational fractions result in infinitely repeating floating point numbers. They are the same sets. Skip says: So all the numbers in a are rational, and infinitely repeating? Like 0.5, 0.25. 0.2, 0.1? I'm confused. These examples don't seem to have infinitely repeating decimals. Skip On Wed, Jun 13, 2018 at 12:44 PM Roger Hui wrote: What's an irrational number in this context? Your list a are reciprocals of integers and so are all rational. On the other hand, going just by the display, 0.5 is a rational number (1%2), but since the display is to 6 significant digits, for all you know 0.5 could be 0.50314159265358979... (0.5+ pi*1e_7) and irrational. On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave wrote: Here's another problem similar to my previous one about finding integers in a floating point array: Find the irrational numbers in a floating-point array: Given the vector a: ]a =. % 1+i.20 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 0.0526316 0.05 Create a function that will generate a boolean array indicating the locations of the irrational numbers in a. Skip -- For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm --- This email has been checked for viruses by AVG. https://www.avg.com -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
Skip said: Given the vector a: ]a =. % 1+i.20 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 0.0526316 0.05 Roger said: Your list a are reciprocals of integers and so are all rational. Roger also said: All rational fractions result in infinitely repeating floating point numbers. They are the same sets. Skip says: So all the numbers in a are rational, and infinitely repeating? Like 0.5, 0.25. 0.2, 0.1? I'm confused. These examples don't seem to have infinitely repeating decimals. Skip On Wed, Jun 13, 2018 at 12:44 PM Roger Hui wrote: > What's an irrational number in this context? Your list a are reciprocals > of integers and so are all rational. On the other hand, going just by the > display, 0.5 is a rational number (1%2), but since the display is to 6 > significant digits, for all you know 0.5 could be > 0.50314159265358979... (0.5+ pi*1e_7) and irrational. > > > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave > wrote: > > > Here's another problem similar to my previous one about finding integers > in > > a floating point array: > > > > Find the irrational numbers in a floating-point array: > > > > Given the vector a: > > > > ]a =. % 1+i.20 > > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > > 0.0526316 0.05 > > > > > > Create a function that will generate a boolean array indicating the > > locations of the irrational numbers in a. > > > > > > Skip > > -- > > For information about J forums see http://www.jsoftware.com/forums.htm > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
> On a similar note, is there a way to detect which numbers in a vector of
> rational fractions will result in infinitely repeating floating-point
> numbers?
All rational fractions result in infinitely repeating floating point
numbers. They are the same sets.
On Wed, Jun 13, 2018 at 11:44 AM, Skip Cave wrote:
> Ok. I see.
>
> We know pi is an irrational number. However, in J:
>
> x: o.1
>
> 1285290289249r409120605684
>
>
> J converts pi to a rational fraction approximation of pi. So I'm not sure
> how to generate a vector of truly irrational numbers in J. Can it be done,
> or is there no way to define/create true irrational numbers in J? I expest
> it would be hard to support irrational numbers on a finite word-size
> machine, but then I also thought extended integers would be impossible. I'm
> guessing you might have to create a new noun type - irrational?
>
>
> On a similar note, is there a way to detect which numbers in a vector of
> rational fractions will result in infinitely repeating floating-point
> numbers?
>
>
> Skip
>
>
> On Wed, Jun 13, 2018 at 1:17 PM Roger Hui
> wrote:
>
> > Mathematically, all finite precision floating point numbers (e.g. 64-bit
> > floats) are rational since they are all ratios of integers. You have to
> > specify something (size of repeating pattern? size of denominator?
> relative
> > size of numerator/denominator? ??) before the question can be answered.
> >
> > For example, with your second vector a,
> >
> >]a=.(20?.20){(100*%1+i.10),(10?.20)*o.1
> > 15.708 18.8496 37.6991 9.42478 16.6667 50 25 100 14.2857 50.2655 10
> 11.
> > 3.14159 0 33. 43.9823 12.5 20 40.8407 56.5487
> >
> >x: a
> > 5646741500662r359482728877 4414041858589r234172193603
> > 376716722090r9992721411 4414041858589r468344387206 50r3 50 25 100 100r7
> > 1681140150209r33445220617 10 100r9 1285290289249r409120605684 0 100r3
> > 10299431003375r234172193603 25r2 20 15619071123724r382438827...
> >
> >x: 4 5$a
> > 5646741500662r359482728877 4414041858589r234172193603
> > 376716722090r9992721411 4414041858589r468344387206
> > 50r3
> > 50 25
> > 100 100r7 1681140150209r33445220617
> > 10 100r9
> > 1285290289249r409120605684 0
> > 100r3
> > 10299431003375r234172193603 25r2
> >20 15619071123724r382438827053 188358361045r3330907137
> >
> >x:!.0 ]4 5$a
> > 4421398595017775r281474976710656 2652839157010665r140737488355328
> > 2652839157010665r70368744177664 2652839157010665r281474976710656
> > 2345624805922133r140737488355328
> > 50 25
> >100100r7
> > 884279719003555r17592186044416
> > 10100r9
> > 884279719003555r2814749767106560
> > 2345624805922133r70368744177664
> > 3094979016512443r70368744177664 25r2
> > 20 1436954543380777r35184372088832
> > 1989629367757999r35184372088832
> >
> > Some are "obviously" rational and some aren't but all are mathematically
> > rational.
> >
> >
> >
> > On Wed, Jun 13, 2018 at 11:06 AM, Skip Cave
> > wrote:
> >
> > > Roger,
> > > You're right. The array i generated was all rational numbers. I'll try
> > > again:
> > >
> > > ]a=.(20?20){(100*%1+i.10),(10?20)*o.1
> > >
> > > 6.28319 10 11. 16.6667 25.1327 100 0 21.9911 33. 56.5487 12.5
> > > 31.4159 53.4071 14.2857 25 43.9823 50 12.5664 59.6903 20
> > >
> > >
> > > Is that a better combination of rational and irrational numbers?
> > >
> > >
> > > I think I was originally thinking that floating-point numbers that have
> > > infinitely repeating patterns after the decimal are irrational. but
> that
> > is
> > > not the correct definition of irrational. However, a verb that could
> find
> > > those kinds of numbers (infinitely repeating pattern) in a vector of
> > > floating point numbers could be useful.
> > >
> > >
> > > Skip
> > >
> > >
> > >
> > >
> > > On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
> > > wrote:
> > >
> > > > What's an irrational number in this context? Your list a are
> > reciprocals
> > > > of integers and so are all rational. On the other hand, going just
> by
> > > the
> > > > display, 0.5 is a rational number (1%2), but since the display is to
> 6
> > > > significant digits, for all you know 0.5 could be
> > > > 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
> > > >
> > > >
> > > > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave >
> > > > wrote:
> > > >
> > > > > Here's another problem similar to my previous one about finding
> > > integers
> > > > in
> > > > > a floating point array:
> > > > >
> > > > > Find the irrational numbers in a floating-point array:
> > > > >
> > > > > Given the vector a:
> > > > >
> > > > > ]a =. % 1+i.20
> > > > >
> > >
Re: [Jprogramming] Finding Irrational Numbers
Irrational numbers cannot be represented by floats, rationals, or integers.
You'd have to make a special type to represent irrationals, and of course
it would only represent as many of them as you choose to assign values to
(for example, your type might represent a floating point number times the
square root of two -- all of the values of that type would be irrational as
well except for its NaN, zeros, and infinities.
On Wed, Jun 13, 2018 at 11:17 AM Roger Hui
wrote:
> Mathematically, all finite precision floating point numbers (e.g. 64-bit
> floats) are rational since they are all ratios of integers. You have to
> specify something (size of repeating pattern? size of denominator? relative
> size of numerator/denominator? ??) before the question can be answered.
>
> For example, with your second vector a,
>
>]a=.(20?.20){(100*%1+i.10),(10?.20)*o.1
> 15.708 18.8496 37.6991 9.42478 16.6667 50 25 100 14.2857 50.2655 10 11.
> 3.14159 0 33. 43.9823 12.5 20 40.8407 56.5487
>
>x: a
> 5646741500662r359482728877 4414041858589r234172193603
> 376716722090r9992721411 4414041858589r468344387206 50r3 50 25 100 100r7
> 1681140150209r33445220617 10 100r9 1285290289249r409120605684 0 100r3
> 10299431003375r234172193603 25r2 20 15619071123724r382438827...
>
>x: 4 5$a
> 5646741500662r359482728877 4414041858589r234172193603
> 376716722090r9992721411 4414041858589r468344387206
> 50r3
> 50 25
> 100 100r7 1681140150209r33445220617
> 10 100r9
> 1285290289249r409120605684 0
> 100r3
> 10299431003375r234172193603 25r2
>20 15619071123724r382438827053 188358361045r3330907137
>
>x:!.0 ]4 5$a
> 4421398595017775r281474976710656 2652839157010665r140737488355328
> 2652839157010665r70368744177664 2652839157010665r281474976710656
> 2345624805922133r140737488355328
> 50 25
>100100r7
> 884279719003555r17592186044416
> 10100r9
> 884279719003555r2814749767106560
> 2345624805922133r70368744177664
> 3094979016512443r70368744177664 25r2
> 20 1436954543380777r35184372088832
> 1989629367757999r35184372088832
>
> Some are "obviously" rational and some aren't but all are mathematically
> rational.
>
>
>
> On Wed, Jun 13, 2018 at 11:06 AM, Skip Cave
> wrote:
>
> > Roger,
> > You're right. The array i generated was all rational numbers. I'll try
> > again:
> >
> > ]a=.(20?20){(100*%1+i.10),(10?20)*o.1
> >
> > 6.28319 10 11. 16.6667 25.1327 100 0 21.9911 33. 56.5487 12.5
> > 31.4159 53.4071 14.2857 25 43.9823 50 12.5664 59.6903 20
> >
> >
> > Is that a better combination of rational and irrational numbers?
> >
> >
> > I think I was originally thinking that floating-point numbers that have
> > infinitely repeating patterns after the decimal are irrational. but that
> is
> > not the correct definition of irrational. However, a verb that could find
> > those kinds of numbers (infinitely repeating pattern) in a vector of
> > floating point numbers could be useful.
> >
> >
> > Skip
> >
> >
> >
> >
> > On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
> > wrote:
> >
> > > What's an irrational number in this context? Your list a are
> reciprocals
> > > of integers and so are all rational. On the other hand, going just by
> > the
> > > display, 0.5 is a rational number (1%2), but since the display is to 6
> > > significant digits, for all you know 0.5 could be
> > > 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
> > >
> > >
> > > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave
> > > wrote:
> > >
> > > > Here's another problem similar to my previous one about finding
> > integers
> > > in
> > > > a floating point array:
> > > >
> > > > Find the irrational numbers in a floating-point array:
> > > >
> > > > Given the vector a:
> > > >
> > > > ]a =. % 1+i.20
> > > >
> > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1
> 0.0909091
> > > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > > > 0.0526316 0.05
> > > >
> > > >
> > > > Create a function that will generate a boolean array indicating the
> > > > locations of the irrational numbers in a.
> > > >
> > > >
> > > > Skip
> > > >
> --
> > > > For information about J forums see
> http://www.jsoftware.com/forums.htm
> > > --
> > > For information about J forums see http://www.jsoftware.com/forums.htm
> > --
> > For information about J forums see http://www.jsoftware.com/forums.htm
>
Re: [Jprogramming] Finding Irrational Numbers
Ok. I see.
We know pi is an irrational number. However, in J:
x: o.1
1285290289249r409120605684
J converts pi to a rational fraction approximation of pi. So I'm not sure
how to generate a vector of truly irrational numbers in J. Can it be done,
or is there no way to define/create true irrational numbers in J? I expest
it would be hard to support irrational numbers on a finite word-size
machine, but then I also thought extended integers would be impossible. I'm
guessing you might have to create a new noun type - irrational?
On a similar note, is there a way to detect which numbers in a vector of
rational fractions will result in infinitely repeating floating-point
numbers?
Skip
On Wed, Jun 13, 2018 at 1:17 PM Roger Hui wrote:
> Mathematically, all finite precision floating point numbers (e.g. 64-bit
> floats) are rational since they are all ratios of integers. You have to
> specify something (size of repeating pattern? size of denominator? relative
> size of numerator/denominator? ??) before the question can be answered.
>
> For example, with your second vector a,
>
>]a=.(20?.20){(100*%1+i.10),(10?.20)*o.1
> 15.708 18.8496 37.6991 9.42478 16.6667 50 25 100 14.2857 50.2655 10 11.
> 3.14159 0 33. 43.9823 12.5 20 40.8407 56.5487
>
>x: a
> 5646741500662r359482728877 4414041858589r234172193603
> 376716722090r9992721411 4414041858589r468344387206 50r3 50 25 100 100r7
> 1681140150209r33445220617 10 100r9 1285290289249r409120605684 0 100r3
> 10299431003375r234172193603 25r2 20 15619071123724r382438827...
>
>x: 4 5$a
> 5646741500662r359482728877 4414041858589r234172193603
> 376716722090r9992721411 4414041858589r468344387206
> 50r3
> 50 25
> 100 100r7 1681140150209r33445220617
> 10 100r9
> 1285290289249r409120605684 0
> 100r3
> 10299431003375r234172193603 25r2
>20 15619071123724r382438827053 188358361045r3330907137
>
>x:!.0 ]4 5$a
> 4421398595017775r281474976710656 2652839157010665r140737488355328
> 2652839157010665r70368744177664 2652839157010665r281474976710656
> 2345624805922133r140737488355328
> 50 25
>100100r7
> 884279719003555r17592186044416
> 10100r9
> 884279719003555r2814749767106560
> 2345624805922133r70368744177664
> 3094979016512443r70368744177664 25r2
> 20 1436954543380777r35184372088832
> 1989629367757999r35184372088832
>
> Some are "obviously" rational and some aren't but all are mathematically
> rational.
>
>
>
> On Wed, Jun 13, 2018 at 11:06 AM, Skip Cave
> wrote:
>
> > Roger,
> > You're right. The array i generated was all rational numbers. I'll try
> > again:
> >
> > ]a=.(20?20){(100*%1+i.10),(10?20)*o.1
> >
> > 6.28319 10 11. 16.6667 25.1327 100 0 21.9911 33. 56.5487 12.5
> > 31.4159 53.4071 14.2857 25 43.9823 50 12.5664 59.6903 20
> >
> >
> > Is that a better combination of rational and irrational numbers?
> >
> >
> > I think I was originally thinking that floating-point numbers that have
> > infinitely repeating patterns after the decimal are irrational. but that
> is
> > not the correct definition of irrational. However, a verb that could find
> > those kinds of numbers (infinitely repeating pattern) in a vector of
> > floating point numbers could be useful.
> >
> >
> > Skip
> >
> >
> >
> >
> > On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
> > wrote:
> >
> > > What's an irrational number in this context? Your list a are
> reciprocals
> > > of integers and so are all rational. On the other hand, going just by
> > the
> > > display, 0.5 is a rational number (1%2), but since the display is to 6
> > > significant digits, for all you know 0.5 could be
> > > 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
> > >
> > >
> > > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave
> > > wrote:
> > >
> > > > Here's another problem similar to my previous one about finding
> > integers
> > > in
> > > > a floating point array:
> > > >
> > > > Find the irrational numbers in a floating-point array:
> > > >
> > > > Given the vector a:
> > > >
> > > > ]a =. % 1+i.20
> > > >
> > > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1
> 0.0909091
> > > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > > > 0.0526316 0.05
> > > >
> > > >
> > > > Create a function that will generate a boolean array indicating the
> > > > locations of the irrational numbers in a.
> > > >
> > > >
> > > > Skip
> > > >
> --
> > > > For information about J forums see
> http://www.jsoftware.com/forums.htm
> > >
Re: [Jprogramming] Finding Irrational Numbers
Mathematically, all finite precision floating point numbers (e.g. 64-bit
floats) are rational since they are all ratios of integers. You have to
specify something (size of repeating pattern? size of denominator? relative
size of numerator/denominator? ??) before the question can be answered.
For example, with your second vector a,
]a=.(20?.20){(100*%1+i.10),(10?.20)*o.1
15.708 18.8496 37.6991 9.42478 16.6667 50 25 100 14.2857 50.2655 10 11.
3.14159 0 33. 43.9823 12.5 20 40.8407 56.5487
x: a
5646741500662r359482728877 4414041858589r234172193603
376716722090r9992721411 4414041858589r468344387206 50r3 50 25 100 100r7
1681140150209r33445220617 10 100r9 1285290289249r409120605684 0 100r3
10299431003375r234172193603 25r2 20 15619071123724r382438827...
x: 4 5$a
5646741500662r359482728877 4414041858589r234172193603
376716722090r9992721411 4414041858589r468344387206
50r3
50 25
100 100r7 1681140150209r33445220617
10 100r9
1285290289249r409120605684 0
100r3
10299431003375r234172193603 25r2
20 15619071123724r382438827053 188358361045r3330907137
x:!.0 ]4 5$a
4421398595017775r281474976710656 2652839157010665r140737488355328
2652839157010665r70368744177664 2652839157010665r281474976710656
2345624805922133r140737488355328
50 25
100100r7
884279719003555r17592186044416
10100r9
884279719003555r2814749767106560
2345624805922133r70368744177664
3094979016512443r70368744177664 25r2
20 1436954543380777r35184372088832
1989629367757999r35184372088832
Some are "obviously" rational and some aren't but all are mathematically
rational.
On Wed, Jun 13, 2018 at 11:06 AM, Skip Cave wrote:
> Roger,
> You're right. The array i generated was all rational numbers. I'll try
> again:
>
> ]a=.(20?20){(100*%1+i.10),(10?20)*o.1
>
> 6.28319 10 11. 16.6667 25.1327 100 0 21.9911 33. 56.5487 12.5
> 31.4159 53.4071 14.2857 25 43.9823 50 12.5664 59.6903 20
>
>
> Is that a better combination of rational and irrational numbers?
>
>
> I think I was originally thinking that floating-point numbers that have
> infinitely repeating patterns after the decimal are irrational. but that is
> not the correct definition of irrational. However, a verb that could find
> those kinds of numbers (infinitely repeating pattern) in a vector of
> floating point numbers could be useful.
>
>
> Skip
>
>
>
>
> On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
> wrote:
>
> > What's an irrational number in this context? Your list a are reciprocals
> > of integers and so are all rational. On the other hand, going just by
> the
> > display, 0.5 is a rational number (1%2), but since the display is to 6
> > significant digits, for all you know 0.5 could be
> > 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
> >
> >
> > On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave
> > wrote:
> >
> > > Here's another problem similar to my previous one about finding
> integers
> > in
> > > a floating point array:
> > >
> > > Find the irrational numbers in a floating-point array:
> > >
> > > Given the vector a:
> > >
> > > ]a =. % 1+i.20
> > >
> > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091
> > > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > > 0.0526316 0.05
> > >
> > >
> > > Create a function that will generate a boolean array indicating the
> > > locations of the irrational numbers in a.
> > >
> > >
> > > Skip
> > > --
> > > For information about J forums see http://www.jsoftware.com/forums.htm
> > --
> > For information about J forums see http://www.jsoftware.com/forums.htm
> --
> For information about J forums see http://www.jsoftware.com/forums.htm
--
For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
Roger,
You're right. The array i generated was all rational numbers. I'll try
again:
]a=.(20?20){(100*%1+i.10),(10?20)*o.1
6.28319 10 11. 16.6667 25.1327 100 0 21.9911 33. 56.5487 12.5
31.4159 53.4071 14.2857 25 43.9823 50 12.5664 59.6903 20
Is that a better combination of rational and irrational numbers?
I think I was originally thinking that floating-point numbers that have
infinitely repeating patterns after the decimal are irrational. but that is
not the correct definition of irrational. However, a verb that could find
those kinds of numbers (infinitely repeating pattern) in a vector of
floating point numbers could be useful.
Skip
On Wed, Jun 13, 2018 at 12:44 PM Roger Hui
wrote:
> What's an irrational number in this context? Your list a are reciprocals
> of integers and so are all rational. On the other hand, going just by the
> display, 0.5 is a rational number (1%2), but since the display is to 6
> significant digits, for all you know 0.5 could be
> 0.50314159265358979... (0.5+ pi*1e_7) and irrational.
>
>
> On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave
> wrote:
>
> > Here's another problem similar to my previous one about finding integers
> in
> > a floating point array:
> >
> > Find the irrational numbers in a floating-point array:
> >
> > Given the vector a:
> >
> > ]a =. % 1+i.20
> >
> > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091
> > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056
> > 0.0526316 0.05
> >
> >
> > Create a function that will generate a boolean array indicating the
> > locations of the irrational numbers in a.
> >
> >
> > Skip
> > --
> > For information about J forums see http://www.jsoftware.com/forums.htm
> --
> For information about J forums see http://www.jsoftware.com/forums.htm
--
For information about J forums see http://www.jsoftware.com/forums.htm
Re: [Jprogramming] Finding Irrational Numbers
What's an irrational number in this context? Your list a are reciprocals of integers and so are all rational. On the other hand, going just by the display, 0.5 is a rational number (1%2), but since the display is to 6 significant digits, for all you know 0.5 could be 0.50314159265358979... (0.5+ pi*1e_7) and irrational. On Wed, Jun 13, 2018 at 10:29 AM, Skip Cave wrote: > Here's another problem similar to my previous one about finding integers in > a floating point array: > > Find the irrational numbers in a floating-point array: > > Given the vector a: > > ]a =. % 1+i.20 > > 1 0.5 0.33 0.25 0.2 0.17 0.142857 0.125 0.11 0.1 0.0909091 > 0.083 0.0769231 0.0714286 0.067 0.0625 0.0588235 0.056 > 0.0526316 0.05 > > > Create a function that will generate a boolean array indicating the > locations of the irrational numbers in a. > > > Skip > -- > For information about J forums see http://www.jsoftware.com/forums.htm -- For information about J forums see http://www.jsoftware.com/forums.htm
