# NumberExpand

NumberExpand[x]

gives a list of the decimal digits of x multiplied by their corresponding powers of 10.

NumberExpand[x,b]

expands x in base b.

NumberExpand[x,b,len]

gives a list of length len.

# Details

• For any number x, Total[NumberExpand[x,]]==x.
• For an integer x, NumberExpand[x] returns a list of integers.
• For a rational x, the fractional part of x is added to the last element of .
• For a non-exact number x, all elements of NumberExpand[x] but the last are exact.
• For an exact number x, the length of NumberExpand[x] equals the number of digits in the integer part of x.
• For a non-exact number x, NumberExpand[x] normally returns a list of length Round[Precision[x]].
• For a non-exact number x and an exact base b, NumberExpand[x,b] normally returns a list of length Round[Precision[x] Log[b,10]].
• If len is larger than Precision[x] Log[b,10], the remaining parts of the expansion are filled in as Indeterminate.
• The base b in NumberExpand[x,b] can be a real number greater than 1.
• For any number x of absolute value less than 1, the first element of NumberExpand[x,] is 0 or 0..
• NumberExpand[0.] returns a list of length Floor[Accuracy[0.]]+2.

# Examples

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## Basic Examples(3)

Expand a number into a list of multiples of powers of 10:

Expand a number in base 2:

Specify the length of the output:

## Scope(5)

Expand an integer into a list of multiples of powers of 10:

Expand a rational number in base 2, obtaining a rational remaining part:

Expand a machine-precision real number, obtaining a machine-precision remaining part:

Expand an exact complex number in base 7:

Expand an inexact complex number in base 10:

## Generalizations & Extensions(5)

Expand a negative integer:

Expand a real number in a rational base:

Expand a number using a machine-precision base:

Expand a rational number in a real base:

Expand a real number in a real base:

## Properties & Relations(9)

For an integer, when the length of the output is required to be larger than needed, NumberExpand pads with 0s on the right:

For a rational number with a finite-length decimal part, when the length of the output is required to be larger than needed, NumberExpand pads with 0s on the right:

For a rational number with an infinite-length decimal part, the last element of the output list is always nonzero:

For any number n, Total[NumberExpand[n,]] equals n:

The total of the expansion of an exact number in an integer base is the number itself:

If the base is non-exact, the total will have a different precision:

For an exact number expanded into inexact parts, the difference with the total is smaller than the last part of the expansion:

Then Rationalize may be able to recover the original exact number:

When a non-exact number is expanded in an exact base, all the elements of the output list but the last are exact:

The last element is not necessarily zero:

For non-exact numbers, NumberExpand returns a list of parts corresponding to the digits of RealDigits:

Small variations of the input may result in representations containing multiple 9s:

The precision of Total[NumberExpand[]] is effectively determined by the minimum precision of the input arguments:

## Possible Issues(1)

Parts of the expansion unknown at the available precision are filled in as Indeterminate:

In this situation, the original number cannot be reconstructed:

Wolfram Research (2016), NumberExpand, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberExpand.html.

#### Text

Wolfram Research (2016), NumberExpand, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberExpand.html.

#### CMS

Wolfram Language. 2016. "NumberExpand." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberExpand.html.

#### APA

Wolfram Language. (2016). NumberExpand. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberExpand.html

#### BibTeX

@misc{reference.wolfram_2023_numberexpand, author="Wolfram Research", title="{NumberExpand}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/NumberExpand.html}", note=[Accessed: 22-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_numberexpand, organization={Wolfram Research}, title={NumberExpand}, year={2016}, url={https://reference.wolfram.com/language/ref/NumberExpand.html}, note=[Accessed: 22-April-2024 ]}