README.md

binaries

Arithmetic system of 2-adic numbers in scheme language

2-adic numbers are represented by infinite streams in binaries. All arithmetic operations provided are accurate.

Run with Chez Scheme

Construction and display

binaries-zero:

2-adic zero.
binaries-one:

2-adic one.
binaries-minus-one:

2-adic minus-one.
(r-binaries r):

Construct a 2-adic number from a rational number r.
(h-binaries h):

Construct a 2-adic number from an infinite stream of hexes h.
(display-binaries b n):

Display a 2-adic number b by hex digits.

If n is a positive integer or #t, scientific notation is enabled, and n hex digits or infinite digits will be displayed.

If n is a non-positive integer or #f, the fractional part will be displayed in full and the integer part will show n or infinite digits.

If n is omitted, it will be regarded as #f.

Example

Construct and display a rational number ⁷/₁₃.

> (display-binaries (r-binaries 7/13) -50)
.36726726726726726726726726726726726726726726726726...

Construct an infinite series

2 + 3 × 16 + 5 × 16² + 7 × 16³ + 11 × 16⁴ + ⋯ + P(n+1) × 16ⁿ + ⋯

where P(k) is the k-th prime. Display in scientific notation.

> (define primes
    (letrec ([integers (stream-cons 3
                         (stream+ (ns 1) integers))])
      (stream-cons 2
        (stream-filter
          (lambda (n)
            (let ([m (isqrt n)])
              (let loop ([ps primes])
                (cond [(> (stream-car ps) m) #t]
                      [(zero? (remainder n (stream-car ps))) #f]
                      [else (loop (stream-cdr ps))]))))
          integers))))
> (define binaries-primes (h-binaries ($stream-carry primes)))
> (display-binaries binaries-primes 50)
[1]99ABDE02478BDE0478BDE147BDE024BD0278BE0478DE028E02...

Basic arithmetic operations and potential zero

(binaries+ b1 b2 ... bn):

Construct a 2-adic number b1 + b2 + ⋯ + bn.
(binaries- b1 b2):

Construct a 2-adic number b1 - b2.

If b1 is omitted, it will be regarded as zero.
(binaries* b1 b2):

Construct a 2-adic number b1 × b2.
(binaries*r b r):

Construct a 2-adic number b × r, where r is a rational number.
(binaries-shift b n):

Construct a 2-adic number b × 2ⁿ, where n is an integer.
(binaries/ b1 b2):

Construct a 2-adic number b1 / b2.

If b1 is omitted, it will be regarded as one.

Since all the results are represented by infinite streams, a 2-adic number that is theoretically zero cannot be calculated to be zero. In binaries, if a certain number of zero digits are detected, it will be judged as a potential zero. Functions below are provided for dealing with potential zeros.

(binaries-zero? b):

Test if a 2-adic number b is zero, non-zero or potential zero.

If b is zero, it will return #t; if b is non-zero, it will return #f.

If b is a potential zero, it will return an integer n indicating that b is divisible by 2ⁿ.
(binaries-eval b n):

Evaluate a potential zero to the given binary bits n until it be determined to be non-zero.

It will return #t if b is evaluated to be non-zero, #f else.

If n is omitted, then there is no limit to the number of bits in the evaluation before determination. Any theoretical zero number will result in endless computations.

Potential zeros cannot be used as a divisor.

Example

Ordinary operations may yield potential zeros:

> (define pot-zero-0 (binaries- one-binaries one-binaries))
> (binaries-zero? pot-zero-0)
200

A construction from infinite stream may also yield potential zeros:

> (define pot-zero-1 (h-binaries (hexes-shift one-hexes 200)))
> (binaries-zero? pot-zero-1)
200

A shift operation will not change the state of zero:

> (define non-zero-small (binaries-shift one-binaries 800))
> (binaries-zero? non-zero-small)
#f
> (binaries-zero? (binaries-shift pot-zero-1 -800))
-600
> (binaries-zero? (binaries-shift zero-binaries -800))
#t

Addition of multiple parameters is recommended, as this will reduce the possibility of potential zeros appearing:

> (define one-plus-small (binaries+ one-binaries non-zero-small))
> (define pot-zero-2 (binaries- one-plus-small one-binaries))
> (binaries-zero? pot-zero-2)
200
> (binaries-zero? (binaries+ one-binaries one-plus-small minus-one-binaries))
#f

We can see that three numbers pot-zero-1, non-zero-small and pot-zero-2 are theoretically equal. Do some more calculations involving potential zeros:

> (define pot-zero-3 (binaries+ pot-zero-1 pot-zero-2))
> (binaries-zero? pot-zero-3)
200
> (binaries-zero? (binaries+ non-zero-small pot-zero-2))
#f
> (define pot-zero-4 (binaries- non-zero-small pot-zero-2))
> (binaries-zero? pot-zero-4)
1000
> (define pot-zero-5 (binaries* pot-zero-3 pot-zero-3))
> (binaries-zero? pot-zero-5)
400
> (binaries-eval pot-zero-5 1000)
#f
> (binaries-eval pot-zero-5)
#t
> (display-binaries pot-zero-5 50)
[1602]10000000000000000000000000000000000000000000000000...
> (display-binaries (binaries/ pot-zero-3 non-zero-small) -50)
.20000000000000000000000000000000000000000000000000...
> (display-binaries (binaries/ non-zero-small pot-zero-3) -50)
Exception in binaries/: Division by potential zero.
> (binaries-eval pot-zero-3)
#t
> (display-binaries (binaries/ non-zero-small pot-zero-3) -50)
8.00000000000000000000000000000000000000000000000000...

Advanced arithmetic operations

(binaries-square b):

Construct a 2-adic number b².
(binaries-expt b n):

Construct a 2-adic number bⁿ, where n is an integer.
(binaries-sqrt? b):

It will return #t if b is zero or non-zero that has square roots, #f else.
(binaries-sqrt b index):

Construct a 2-adic number √b. If integer index is even, then the odd part in the scientific notation of result ≡ 1 mod 4 , else ≡ 3 mod 4.

If index is omitted, it will be regarded as 0.
(binaries-root? b n):

It will return #t if b is zero or non-zero that has n-th roots, #f else.
(binaries-root b n index):

Construct a 2-adic number ⁿ√b, where n is a non-zero integer. In the case that n is even, if integer index is even, then the odd part in the scientific notation of result ≡ 1 mod 4 , else ≡ 3 mod 4.

If index is omitted, it will be regarded as 0.
(binaries-exp b):

Construct a 2-adic number exp(b). It should hold that b ≡ 0 mod 4.
(binaries-log b):

Construct a 2-adic number log(b). It should hold that b ≡ 1 mod 2.
(binaries-pow b1 b2):

Construct a 2-adic number pow(b1, b2). It is defined to be exp(b2 × log(b1)).
(binaries-sin b):

Construct a 2-adic number sin(b). It should hold that b ≡ 0 mod 4.
(binaries-cos b):

Construct a 2-adic number cos(b). It should hold that b ≡ 0 mod 4.
(binaries-tan b):

Construct a 2-adic number tan(b). It should hold that b ≡ 0 mod 2.
(binaries-asin b):

Construct a 2-adic number arcsin(b). It should hold that b ≡ 0 mod 4.
(binaries-acos b index):

Construct a 2-adic number arccos(b). It should hold that b ≡ 1 mod 8 and that 1 - b² has square roots. If integer index is even, then the odd part in the scientific notation of result ≡ 1 mod 4 , else ≡ 3 mod 4.

If index is omitted, it will be regarded as 0.
(binaries-atan b):

Construct a 2-adic number arctan(b). It should hold that b ≡ 0 mod 2.
(binaries-sinh b):

Construct a 2-adic number sinh(b). It should hold that b ≡ 0 mod 4.
(binaries-cosh b):

Construct a 2-adic number cosh(b). It should hold that b ≡ 0 mod 4.
(binaries-tanh b):

Construct a 2-adic number tanh(b). It should hold that b ≡ 0 mod 2.
(binaries-asinh b):

Construct a 2-adic number arcsinh(b). It should hold that b ≡ 0 mod 4.
(binaries-acosh b index):

Construct a 2-adic number arccosh(b). It should hold that b ≡ 1 mod 8 and that b² - 1 has square roots. If integer index is even, then the odd part in the scientific notation of result ≡ 1 mod 4 , else ≡ 3 mod 4.

If index is omitted, it will be regarded as 0.
(binaries-atanh b):

Construct a 2-adic number arctanh(b). It should hold that b ≡ 0 mod 2.

Example

The following code constructs a number α satisfying α⁻²⁸ + 15 = 0 and verifies that α²³ - α⁹ / √-15 = 0.

> (define r-15 (r-binaries -15))
> (define sqrt-15 (binaries-sqrt r-15))
> (display-binaries sqrt-15 -50)
.9D44EED7085922F66E2026FB6C24DF22C0523BCA43D8A7B7E5...
> (define alpha (binaries-root r-15 -28))
> (display-binaries alpha -50)
.540CE8D70746CCB36DD9D74161E23D98177655CEA52763E19A...
> (display-binaries
    (binaries-
      (binaries-expt alpha 23)
      (binaries/
        (binaries-expt alpha 9)
        sqrt-15))
    -50)
.00000000000000000000000000000000000000000000000000...

The following code calculates ((43 √-15 + 26 √41)² + 2⁴ + 1 + 1 + 1)⁶.

> (define r+41 (r-binaries 41))
> (define sqrt+41 (binaries-sqrt r+41))
> (display-binaries sqrt+41 -50)
.DC66CC00EFB622B945B363D9C09AC784C104795047307E7C73...
> (define beta (binaries+
                 (binaries*r sqrt-15 43)
                 (binaries*r sqrt+41 26)))
> (display-binaries beta -50)
.54108CB31641E4C6C4388FC1AA2633D39EBB37391A11C0B098...
> (define x (binaries-expt
              (binaries+
                (binaries-square beta)
                (binaries-shift one-binaries 4)
                one-binaries one-binaries one-binaries)
              6))
> (display-binaries x -50)
.000966C0B552B5CC09E3112AFFFFFFFFFFFFFFFFFFFFFFFFFF...

From the last output we can guess that the result is an integer. So if we don't limit the number of digits for display, the result will be

> (display-binaries x)
.000966C0B552B5CC09E3112AF(3920)^C
break>

The display function will detect consecutive occurrences of digit 0 or F in unlimited display mode. In that case, the number of 0 or F currently evaluated will be placed in the parenthesis and changes over time. We can press Ctrl-C to break the procedure.

The output indicates that the result of our expression is an hex integer -0x5DEEC16F33A4DAA4F3997000, that is -29070743726494498752382464000.

Recall that we have constructed a 2-adic number binaries-primes by infinite series before. Now we will calculate some trigonometric functions involving it and verify some identities.

> (define p (binaries+ binaries-primes (r-binaries 4)))
> (define 2p (binaries-shift p 1))
> (define sin+2p (binaries-sin 2p))
> (display-binaries sin+2p -50)
.CCCFB54B2A5E34A98B527FB5480B811129524A1EFFED9057DC...
> (define cos+2p (binaries-cos 2p))
> (display-binaries cos+2p -50)
.999550FF477BC9A04080262801F89A6ECE380027B2AFEB21FD...
> (define tan+2p (binaries-tan 2p))
> (display-binaries tan+2p -50)
.CAA54E78CB2101842651A3A8AC142FEE07749383649730BDBD...
> (define tan+p (binaries-tan p))
> (display-binaries tan+p -50)
.EF7B00122CF11254C3CF1E40CDB58E6706C5B53A7CEF7A31CF...

sin(2p)² + cos(2p)² = 1:

> (display-binaries (binaries+
                      (binaries-square sin+2p)
                      (binaries-square cos+2p))
                    -50)
.10000000000000000000000000000000000000000000000000...

sin(2p) = cos(2p) × tan(2p):

> (display-binaries (binaries-
                      sin+2p
                      (binaries* cos+2p tan+2p))
                    -50)
.00000000000000000000000000000000000000000000000000...

tan(2p) = 2 tan(p) / (1 - tan(p)²):

> (display-binaries (binaries-
                      tan+2p
                      (binaries-shift
                        (binaries/
                          tan+p
                          (binaries-
                            one-binaries
                            (binaries-square tan+p)))
                        1))
                    -50)
.00000000000000000000000000000000000000000000000000...

Then we will try inverse trigonometric functions:

> (display-binaries (binaries-asin sin+2p) -50)
.C6AE6B380D1E6B30D1E6B70DD6B380D638C1EA30D16B380A38...
> (display-binaries (binaries-acos cos+2p) -50)
.495194C7F2E194CF2E1948F2294C7F29C73E15CF2E94C7F5C7...
> (display-binaries (binaries-atan tan+2p) -50)
.C6AE6B380D1E6B30D1E6B70DD6B380D638C1EA30D16B380A38...
> (display-binaries (binaries-atan tan+p) -50)
.6357BD148E07BD18E07BD38E6BD1486B14E07D18E0BD140D14...
> (display-binaries (binaries-cosh (binaries-acosh r+41)) -50)
.92000000000000000000000000000000000000000000000000...

Notice that the result arccos(cos(2p)) is -2p, because p's odd part in the scientific notation ≡ 3 mod 4 . We can add the index parameter to get another result:

> (display-binaries (binaries-acos cos+2p 1) -50)
.C6AE6B380D1E6B30D1E6B70DD6B380D638C1EA30D16B380A38...

Exponential function exp always return an odd that ≡ 1 mod 4, so we have exp(log(p / 2)) = - p / 2:

> (define p/2 (binaries-shift p -1))
> (display-binaries p/2 -50)
.B9ABDE02478BDE0478BDE147BDE024BD0278BE0478DE028E02...
> (define log+p/2 (binaries-log p/2))
> (display-binaries log+p/2 -50)
.C527A125DC5C044112B1DB2F15539006E0C6AE9946F856F586...
> (define exp+log+p/2 (binaries-exp log+p/2))
> (display-binaries exp+log+p/2 -50)
.565421FDB87421FB87421EB8421FDB42FD8741FB8721FD71FD...
> (display-binaries (binaries+ exp+log+p/2 p/2) -50)
.00000000000000000000000000000000000000000000000000...

So pow(b, 1) is not necessarily equal to b, it may be equal to -b. We can think of it as some kind of absolute value function.

Recall that we have constructed some potential zeros, some of them are theoretically zero (pot-zero-0, pot-zero-4) and others are actually non-zero (pot-zero-1, pot-zero-2, pot-zero-3, pot-zero-5). We can apply analytic functions on them:

> (define exp+pot-zero-1 (binaries-exp pot-zero-1))
> (display-binaries exp+pot-zero-1 -50)
.10000000000000000000000000000000000000000000000000...
> (display-binaries exp+pot-zero-1 -2000)
.1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000085555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555BAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D28D2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2BC2B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B21B2116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB0A448D766B4FE282116F99D2DBB09E...
> (define sin+pot-zero-2 (binaries-sin pot-zero-2))
> (binaries-zero? sin+pot-zero-2)
800
> (binaries-eval sin+pot-zero-2)
#t
> (binaries-zero? sin+pot-zero-2)
#f
> (display-binaries sin+pot-zero-2 2000)
[800]1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598598BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C95A298BC6405D33671FAF7ED02C986...
> (define atan+pot-zero-4 (binaries-atan pot-zero-4))
> (binaries-zero? atan+pot-zero-4)
1000
> (display-binaries atan+pot-zero-4)
.0(284980)^C
break>

We can see from here that all the calculations are certified.

License

WTFPL

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

README.md

README.md

binaries

Construction and display

Example

Basic arithmetic operations and potential zero

Example

Advanced arithmetic operations

Example

License

Files

README.md

Latest commit

History

README.md

File metadata and controls

binaries

Construction and display

Example

Basic arithmetic operations and potential zero

Example

Advanced arithmetic operations

Example

License