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.

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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.

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

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

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

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

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

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

Typed signatures

Type	Package	Applies
`(collection<any>, integer?) -> integer`	`compute-engine`	always
`(collection<any>, any?) -> integer`	`numerals`replaces `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 #

In

\mathrm{FromDigits}(\bigl\lbrack5, 1, 2, 8\bigr\rbrack)

Out

5\,128

In

\mathrm{FromDigits}(\bigl\lbrack1, 0, 1, 1, 0, 1, 1\bigr\rbrack, 2)

Out

91

1d 2h 3m 4s back to seconds

In

\mathrm{FromDigits}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))

Out

93\,784

CRT reconstruction

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

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

In

\mathrm{FromDigits}(\text{1923})

Out

1\,923

a digit string in a given base

In

\mathrm{FromDigits}(\text{1011011}, 2)

Out

91

Roman numerals, with the base "Roman"

In

\mathrm{FromDigits}(\text{XVII}, \text{Roman})

Out

17

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

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]))

In

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

Out

137

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

In

\mathrm{FromDigits}(\bigl\lbrack1, 15\bigr\rbrack, 16)

Out

31

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

In

\mathrm{FromDigits}(\bigl\lbrack1, 2, 3\bigr\rbrack, x)

Out

x^2+2x+3

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

In

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

Out

2

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

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

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 $[3,12]$ : $3\cdot36+2\cdot12+5$

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

In

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

Out

5

Properties #

FromDigits inverts IntegerDigits

In

\mathrm{FromDigits}(\mathrm{IntegerDigits}(58\,127))=58\,127

Out

\mathrm{True}

the round trip holds in any base

In

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

Out

58\,127

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

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

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

In

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

Out

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

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: $7\times10^5+11\times10^4+122=810122$

In

\mathrm{FromDigits}(\bigl\lbrack7, 11, 0, 0, 0, 122\bigr\rbrack)

Out

810\,122

About

Examples	38
Checked against	Wolfram Language 25 agree2 differ
Crosswalk	Wolfram Language

See also: IntegerDigits, NumeralSystemShape, ChineseRemainder