FromDigits

The integer a digit string denotes, in any numeral system. The inverse of IntegerDigits — and the round trip is the whole specification of a system.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFromDigits!↗

FromDigits([d1, d2, …])integer formed from a digit list in base 10.

FromDigits(string, base?)integer formed from a digit string, 0-9 then a-z.

SageMathsum(d * (base?)^k for k, d in enumerate(reversed(string)))Rustfrom_digits(string, base?)

FromDigits([d1, d2, …], base)integer formed from a digit list in the given base.

FromDigits(digits, system)read the digits in that system

SageMathsum(d * (system)^k for k, d in enumerate(reversed(digits)))Rustfrom_digits(digits, system)

FromDigits(roman, "Roman")the integer a Roman numeral spells, the inverse of RomanNumeral.

SageMathsum(d * ("Roman")^k for k, d in enumerate(reversed(roman)))Rustfrom_digits(roman, "Roman")

FromDigits(digits, x)a symbolic base gives the polynomial the digits spell in x.

SageMathsum(d * (x)^k for k, d in enumerate(reversed(digits)))Rustfrom_digits(digits, x)

FromDigits(digits, negativeBase)a negative base reads with the same Horner reduction as a positive one.

SageMathsum(d * (negativeBase)^k for k, d in enumerate(reversed(digits)))Rustfrom_digits(digits, negativeBase)

FromDigits({digits, exponent})reads the single {digits, exponent} pair RealDigits itself returns, base 10.

SageMathsum(d * (exponent})^k for k, d in enumerate(reversed({digits)))Rustfrom_digits({digits, exponent})

Typed signatures

Type	Package	Applies
(collection<any>, integer?) -> integer	compute-engine	always
(collection<any>, any?) -> integer	numeralsreplaces compute-engine	always

Domain: Numeral systems

Details

• A digit string that denotes NO integer leaves the call standing: two adjacent Zeckendorf ones, an out-of-range mixed-radix digit, residues that no integer satisfies
• In a residue system with moduli that are not pairwise coprime the map is not a bijection, and an inconsistent string has no value
• Zeckendorf is the clearest case of a system whose digits are constrained by a forbidden PATTERN rather than a per-place bound
• Reading a ResidueNumerals numeral IS the Chinese remainder theorem: the digits are the ResidueClass classes of $n$, and ChineseRemainder of those classes gives it back
• In a fixed base, the inverse of IntegerDigits: $\mathrm{FromDigits}(\mathrm{IntegerDigits}(n))=n$ for $n\ge0$.
• Digits need not be restricted to $0..\mathrm{base}-1$ -- a digit $\ge$ base simply carries into higher place values.
• An empty digit list has no natural value; compute-engine leaves it unevaluated rather than returning 0.
• compute-engine's FromDigits takes a list of digits, not a digit string.

Examples

14 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack5, 1, 2, 8\bigr\rbrack)$Out$5\,128$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 0, 1, 1, 0, 1, 1\bigr\rbrack, 2)$Out$91$

1d 2h 3m 4s back to seconds

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$93\,784$

CRT reconstruction

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack2, 3, 2\bigr\rbrack, \mathrm{ResidueNumerals}(\bigl\lbrack3, 5, 7\bigr\rbrack))$Out$23$

1 day 4 h 25 min 41 s converted to seconds

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 4, 25, 41\bigr\rbrack, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$102\,341$

Scope #

The digits can be given as a string, 0-9 then a-z

live ↗

In$\mathrm{FromDigits}(\text{1923})$Out$1\,923$

a digit string in a given base

live ↗

In$\mathrm{FromDigits}(\text{1011011}, 2)$Out$91$

Roman numerals, with the base "Roman"

live ↗

In$\mathrm{FromDigits}(\text{XVII}, \text{Roman})$Out$17$

a mixed radix, spelled MixedRadix: 1d 4h 25m 41s back to seconds

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 4, 25, 41\bigr\rbrack, \mathrm{MixedRadix}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$102\,341$

the inverse of IntegerDigits(137, MixedRadix([3, 12]))

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack3, 2, 5\bigr\rbrack, \mathrm{MixedRadix}(\bigl\lbrack3, 12\bigr\rbrack))$Out$137$

digits past 9 are plain integers: $1 \cdot 16 + 15$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 15\bigr\rbrack, 16)$Out$31$

a symbolic base gives the polynomial $x^2 + 2x + 3$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 2, 3\bigr\rbrack, x)$Out$x^2+2x+3$

a negative base: $1 \cdot 4 - 1 \cdot 2 + 0 = 2$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 1, 0\bigr\rbrack, -2)$Out$2$

a {digits, exponent} pair — the shape RealDigits gives — reads as $1.415$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack\bigl\lbrack1, 4, 1, 5\bigr\rbrack, 1\bigr\rbrack)$Out$\frac{283}{200}$

ordinary base 2 as a system value

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1\bigr\rbrack, \mathrm{PositionalNumerals}(2))$Out$2\,147$

Place weights $36,12,1$ from radices $[3,12]$: $3\cdot36+2\cdot12+5$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack3, 2, 5\bigr\rbrack, \mathrm{MixedRadixNumerals}(\bigl\lbrack3, 12\bigr\rbrack))$Out$137$

Applications #

a Lehmer code back to its rank

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack0, 2, 1, 0\bigr\rbrack, \mathrm{FactorialNumerals})$Out$5$

Properties #

FromDigits inverts IntegerDigits

live ↗

In$\mathrm{FromDigits}(\mathrm{IntegerDigits}(58\,127))=58\,127$Out$\mathrm{True}$

the round trip holds in any base

live ↗

In$\mathrm{FromDigits}(\mathrm{IntegerDigits}(58\,127, 2), 2)$Out$58\,127$

Days-hours-minutes-seconds again, inverse of IntegerDigits

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 3, 22, 14\bigr\rbrack, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$98\,534$

Possible issues #

An empty digit list is left unevaluated rather than treated as 0

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack\bigr\rbrack)$Out$\mathrm{FromDigits}(\bigl\lbrack\bigr\rbrack)$

two adjacent ones is not a numeral, so it denotes nothing

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack1, 1\bigr\rbrack, \mathrm{ZeckendorfNumerals})$Out$\mathrm{FromDigits}(\bigl\lbrack1, 1\bigr\rbrack, \mathrm{ZeckendorfNumerals})$

live ↗

In$\mathrm{Map}(\mathrm{FromDigits}, \bigl\lbrack\text{7}, \text{8}, \text{9}, "A", \text{B}, "C", \text{D}\bigr\rbrack)$Out$\bigl\lbrack7, 8, 9, 10, 11, 12, 13\bigr\rbrack$

Neat examples #

Digits ≥ the base carry into higher place values: $7\times10^5+11\times10^4+122=810122$

live ↗

In$\mathrm{FromDigits}(\bigl\lbrack7, 11, 0, 0, 0, 122\bigr\rbrack)$Out$810\,122$

About

Examples	38
Checked against	Wolfram Language 25 agree2 differSageMath 11 agree1 unevaluated7 errormathlib4 10 agree1 unevaluated8 errorRust 4 agree6 error
Crosswalk	Wolfram LanguageSageMath ×2Rust

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

A list of digits and an integer base from 2 to 256.

mappedmathlib4packages/oracle/src/mappings.ts

Nat.ofDigits is Horner's rule over a little-endian list, so the list is reversed; digits past the base carry as ours do. Natural digits only, and an empty list is 0 where ours stays unevaluated.

mappedmathlib4packages/oracle/src/mappings.ts

As the one-argument row, in an integer base read over ℤ so a negative base is the same Horner reduction; a digit string or symbolic base is a type error.

See also: IntegerDigits, NumeralSystemShape, ChineseRemainder