RealDigits

The digits of a real number and the position of its decimal point, with the repeating block of a rational marked as a sublist.

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RealDigits(x)the exact digits of a rational (integers included), any repeating block nested

RealDigits(x, base)the same, in the given base

RealDigits(x, base, len)len digits — the exact expansion for a rational, or len digits of a numeric approximation (base 10 only) for anything else

Typed signatures

Type	Package	Applies
(value, integer?, integer?) -> list	numerals	always

Domain: Numeral systems

Details

• The result is {{d₁, d₂, ...}, n}: the value is 0.d₁d₂d₃... × baseⁿ, so n is the count of digits before the point
• A rational's expansion is EXACT: a terminating one stops, a repeating one nests its period as the list's last element — RealDigits(1/7) is {{{1,4,2,8,5,7}}, 0}
• When the integer part is 0, leading fractional zeros are dropped from the digit list and folded into a more negative exponent instead of spelled out — RealDigits(1/8, 2) is {{1}, -2}, not {{0,0,1}, 0}
• For anything without an exact rational value — Pi, a Sqrt, ExponentialE — len is required, and the digits are TRUNCATED, not rounded, from a numeric approximation computed a little beyond len digits
• Sign is dropped, like every other digit head here

Examples

Basic #

the repeating block of $1/7 = 0.\overline{142857}$

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In$\mathrm{RealDigits}(\frac{1}{7})$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack1, 4, 2, 8, 5, 7\bigr\rbrack\bigr\rbrack, 0\bigr\rbrack$

the first 10 digits of $\pi$ (truncated, not rounded)

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In$\mathrm{RealDigits}(π, 10, 10)$Out$\bigl\lbrack\bigl\lbrack3, 1, 4, 1, 5, 9, 2, 6, 5, 3\bigr\rbrack, 1\bigr\rbrack$

The first 25 digits of $\pi$ with exponent 1

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In$\mathrm{RealDigits}(π, 10, 25)$Out$\bigl\lbrack\bigl\lbrack3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3\bigr\rbrack, 1\bigr\rbrack$

A rational repeats: $19/7=2.\overline{714285}$

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In$\mathrm{RealDigits}(\frac{19}{7}, 10, 25)$Out$\bigl\lbrack\bigl\lbrack2, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5\bigr\rbrack, 1\bigr\rbrack$

Scope #

$19/7 = 2.\overline{714285}$, one digit before the point

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In$\mathrm{RealDigits}(\frac{19}{7})$Out$\bigl\lbrack\bigl\lbrack2, \bigl\lbrack7, 1, 4, 2, 8, 5\bigr\rbrack\bigr\rbrack, 1\bigr\rbrack$

a terminating rational

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In$\mathrm{RealDigits}(\frac{5}{4})$Out$\bigl\lbrack\bigl\lbrack1, 2, 5\bigr\rbrack, 1\bigr\rbrack$

$1/8 = 0.001_2$

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In$\mathrm{RealDigits}(\frac{1}{8}, 2)$Out$\bigl\lbrack\bigl\lbrack1\bigr\rbrack, -2\bigr\rbrack$

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In$\mathrm{RealDigits}(\sqrt{2}, 10, 5)$Out$\bigl\lbrack\bigl\lbrack1, 4, 1, 4, 2\bigr\rbrack, 1\bigr\rbrack$

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In$\mathrm{RealDigits}(e, 10, 5)$Out$\bigl\lbrack\bigl\lbrack2, 7, 1, 8, 2\bigr\rbrack, 1\bigr\rbrack$

an integer: six digits before the point

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In$\mathrm{RealDigits}(123\,456)$Out$\bigl\lbrack\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, 6\bigr\rbrack$

A shorter digit count for $\pi$

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In$\mathrm{RealDigits}(π, 10, 20)$Out$\bigl\lbrack\bigl\lbrack3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4\bigr\rbrack, 1\bigr\rbrack$

About

Examples	11
Checked against	Wolfram Language 11 agree
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See also: IntegerDigits, FromDigits