Fifty Shades of J/Chapter 15
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Principal Topics
- x: (extended precision) , numerical data types, conversion rules, rational approximations.
Rational Numbers in J
There are various forms of number representation in J, as well as rules and conditions for converting between them. To start with, in complex arithmetic +. transforms a complex number in to a list of its components
+.2j3 2 3
An analogous problem is that of transforming a rational number into a list {numerator,denominator} following any possible cancellation. This is provided by the primitive verb x: with left argument 2
2 x: 3r5 6r10 14r7 3 5 3 5 2 1
but notice
%/2 x:8r6 4r3
does not convert 4%3 into decimal format as you might expect. Instead say either of the following
_1 x:8r6 1.33333 x:^:_1 (8r6) 1.33333
There are conversion rules which apply in calculations in which the six different data types (Boolean, Integer, Extended Integer, Rational, Floating Point and Complex) are mixed. It is not usually necessary to be explicitly aware of the thirty possible conversions - the main ones are
(1) the presence of just one decimal notated number in a calculation is enough to force decimal notation for the result
1r1 2r1 3r1 + 3r5 3r5 14r7 8r5 13r5 5 1r1 2r1 3r1 + 3r5 0.6 14r7 1.6 2.6 5
(2) mixing j and r results in conversion to floating point
1r9j2r3 NB. (1/9) + (2/3)i 0.111111j0.666667
Also rational notation cannot be used to perform rational complex division
1j9r2j3 NB. not (1+9i) / (2+3i) |ill-formed number
(3) where present, floating point and complex are dominant. Rational is also “strong” except as under (1), Boolean and integer are “weak”.
2+3j2+7r2 8.5j2
The full conversion table is:
| Boolean | Integer | Extended | Rational | Float | Complex | |
|---|---|---|---|---|---|---|
| Boolean | Boolean | Integer | Extended | Rational | Float | Complex |
| Integer | Integer | Integer | Extended | Rational | Float | Complex |
| Extended | Extended | Extended | Extended | Rational | Float | Complex |
| Rational | Rational | Rational | Rational | Rational | Float | Complex |
| Float | Float | Float | Float | Float | Float | Complex |
| Complex | Complex | Complex | Complex | Complex | Complex | Complex |
Pure rational calculations perform cancellation automatically
]t=.+/\1r3 1r4 1r5 1r3 7r12 47r60 t*/t 1r9 7r36 47r180 7r36 49r144 329r720 47r180 329r720 2209r3600 t +/ .*t 213r200 1r9 + 49r144 + 2209r3600 213r200
Interestingly cancellation can also be done using +. in its dyadic role of greatest common denominator
dac=.%@(1&+.)
gives the denominator after cancelling, and multiplying this by the number itself gives the numerator, so ((*dac), dac) does the job!
((*dac), dac)8r6 4 3
Again subsequent calculations do not convert to decimal format unless such a number appears
%/((*dac), dac)8r6 4r3 1.6* %/((*dac), dac)8r6 2.13333
To extend to objects with rank, say
((*dac),"0 dac)8r6 12r2 34r51 4 3 6 1 2 3
Some ingenious alternatives have been discussed of which (,&1)%(+.&1) proposed by Roger Hui avoids the double calculation of dac and has a certain appealing symmetry, extending to ,"0&1 % +.&1 to deal with general rank
2 x:2 3$8r6 12r2 34r51 4 3 6 1 2 3 4 3 6 1 2 3
To obtain close rational approximations to irrational numbers such as π and e, say
x: o.1 1285290289249r409120605684 NB. 1,285,290,289,249 / 409,120,605,684 _1 x: x: o.1 3.14159 _1 x: x: ^1 2.71828
Code Summary
dac=.%@(1&+.)