Becimal Calculator

[See Repeating Decimal Expansions in Different Bases published in EJMT for more background.]

The application below will perform basic arithmetic operations on rational numbers. Numbers can be entered and displayed is various bases. This app also handles repeating becimal and $p$-adic expansions. Detailed instructions are displayed below the app.

Note: The "dec" in "decimal number" refers to our native base 10, so we will refer to our arbitrary base expansions as becimal numbers. Likewise, instead of "digit" we will say "bigit".

$p$-Adics: This app allows for $p$-adic number inputs and outputs. While the fields of $p$-adic and real numbers are distinct, they do share the rational numbers in common (which is the domain for this app). A $p$-adic number is rational if and only if its $p$-adic expansion repeats. We enter such expansions as "[XXX]YYY.ZZZ" which represents the $p$-adic expansion "...XXX XXX XXX YYY.ZZZ". Note that while we allow $p$-adic expansions for arbitrary bases, $p$-adic expansion outputs require the output basis to be a prime number.

Mixed Bases: Inputs are by default interpreted according to the specified "Input Base". However, if one wishes to enter an input using several other bases, an underscore can be used to override our default input base. For example, no matter what "Input Base" is specified, "ABC.DEF[123]_16" will be interpreted as the hexidecimal number whose whole part is "ABC", delay (i.e., non-repeating part) is "DEF", and cycle (i.e., repeating part) is "123".

Order of Operations: The calculator uses standard order of operations (moving left to right): exponentiations first, then multiplications and divisions, then additions and subtractions. Do keep in mind that an underscore to override the default input base is bound before all else. For example, if the "Input Base" was 2, then "10^10_3" would be interpreted as the number "2^3" (i.e., 8).

Outputs: The bases themselves are displayed in our usual base ten. Numbers are displayed as a list of alphanumeric digits (up to base 36) and as boxed digits for larger bases. There are check boxes at the bottom of the app to switch on and off various kinds of outputs.

• Checking "expansions?" makes the app display the becimal or $p$-adic expansions as a sum of bigits times powers of our base. Components of such expansions are displayed in our native base 10.
• Checking "becimals in text?" makes the app display text boxes with non-abbreviated becimal expansions of our computed result in both the input and output bases.
• Checking "suggested base?" makes the app display the computed result in the smallest base such that our number has a terminating becimal expansion. This number is the product of the distinct prime factors in the denominator of our computed result written as a reduced fraction. If the number is an integer (i.e., the denominator is 1), we do not suggest a base.

Truncations: Some numbers have absurdly long repeating becimal or $p$-adic expansions. For becimal expansions exceeding 10000 bigits or $p$-adic expansions exceeding 1000 bigits, the app's code terminates early and truncated outputs are displayed. If such a truncation occurs, a note of warning will be displayed.

The computation above is powered by SageMath. The Sage code is embedded in this webpage's html file. To view the code instruct your browser to show you this page's source. For example, in Chrome, right-click and choose "View page source".

• Wikipedia entry for our place number system.
• Wikipedia entry for $p$-adic numbers.
• A lovely blurb by Keith Conrad about $p$-adic expanions of rational numbers.