Mixed-Radix Converter — Time, Angle & Custom Radix Vectors
Break a whole number into a mixed-radix (multi-base) breakdown where each place has its own base. Built-in Time (days:hours:minutes:seconds), Angle (degrees:arcminutes:arcseconds) and Imperial length presets, plus a custom radix vector you type. Reverse it too: enter the components, get the total. Live, BigInt-accurate, in your browser.
How to Use
- Pick a preset — Time, Angle or Imperial length — or choose Custom and type a radix vector like “24,60,60”.
- Type a whole number in the “Total” box: it is decomposed into each place instantly.
- Or type the individual place values (days, hours, minutes …) and read the recombined total back.
- Click any value to copy it. The place-value row shows the cumulative product each column is worth.
[∞, 24, 60, 60] · place values When one base per column isn’t enough
Every everyday number system you know is single-radix: decimal uses base 10 in every column, binary base 2, and so on. But plenty of real quantities don’t roll over at a single, regular base. A clock counts seconds up to 60, minutes up to 60, then hours up to 24 before a new day begins; an angle uses 60 arcseconds to an arcminute and 60 arcminutes to a degree; an imperial length packs 12 inches into a foot, 3 feet into a yard, and 1760 yards into a mile. Each of these is a mixed-radix (multi-base) system, described by a radix vector — the list of bases for each place. This tool decomposes any whole number into those places and, just as usefully, rebuilds the total from the parts.
How the decomposition works
Write the radix vector with the smallest place on the right, e.g. Time is [∞, 24, 60, 60] for days : hours : minutes : seconds. To split a total, start at the rightmost place and work left: the digit there is total mod radix, then you replace the total with floor(total ÷ radix) and step to the next place. The leftmost place has no cap (its radix is shown as ∞) and simply keeps whatever remains. So 3661 seconds gives 3661 mod 60 = 1 second, carrying 61; 61 mod 60 = 1 minute, carrying 1; 1 mod 24 = 1 hour, carrying 0; and 0 days — written 0 d 1:01:01. The reverse simply multiplies each place by its place value (the running product of all the radices to its right) and sums them. Because those place values grow quickly, every calculation here uses BigInt for exact results.
The factorial number system
One mixed-radix system deserves a special mention: the factorial base, with radix vector […, 5, 4, 3, 2, 1]. Here place i (counting from the right, starting at 1) is worth i! and its digit ranges from 0 to i. Its digits map one-to-one onto the permutations of a set in lexicographic order, which makes it the standard tool for indexing and enumerating permutations. Type 5,4,3,2,1 as a custom radix vector to see it in action — note that the rightmost factorial digit is always 0.
Quick reference
About the Mixed-Radix Converter — Time, Angle & Custom Radix Vectors
The Mixed-Radix Converter — Time, Angle & Custom Radix Vectors is a simple, free helper for everyday tasks that runs entirely on your own device. Break a whole number into a mixed-radix (multi-base) breakdown where each place has its own base. Built-in Time (days:hours:minutes:seconds), Angle (degrees:arcminutes:arcseconds) and Imperial length presets, plus a custom radix vector you type. Reverse it too: enter the components, get the total. Live, BigInt-accurate, in your browser.
How it works
Type a value, then pick what you want to change it into. The answer appears straight away. It all happens on your own device, so it is fast and nothing you type is sent away. Just check that you picked the right “from” and “to” so you get the answer you wanted.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
What is a mixed-radix number system?
An ordinary positional system uses one base for every column — base 10 means each place is worth ten times the one to its right. In a mixed-radix system each place can have a different base. Clock time is the everyday example: seconds roll over at 60, minutes at 60, hours at 24, but days have no upper limit. The radix vector [∞, 24, 60, 60] captures exactly that.
How is a number decomposed?
Working from the rightmost (smallest) place leftward: take the total mod the radix of that place to get the digit, then integer-divide the total by that radix and move left. The leftmost place is unbounded and keeps whatever is left. So 3661 seconds with [∞,24,60,60] gives 3661 mod 60 = 1 second, 61 mod 60 = 1 minute, 1 mod 24 = 1 hour, 0 days → 0 d 1:01:01.
What is the factorial number system?
It is the special mixed radix where the places are …, 4, 3, 2, 1 (the rightmost place is always 0). Each place i is worth i! and its digit ranges 0…i. It is used to enumerate permutations: the factorial digits of n directly index the n-th permutation in lexicographic order. Type “5,4,3,2,1” as a custom vector to experiment with it.
Why use BigInt?
Place values are cumulative products of the radices and grow fast, so totals can exceed what a normal JavaScript number represents exactly (2⁵³). Every step here uses BigInt, so even huge mixed-radix values recombine to the exact integer with no rounding.
Is anything uploaded?
No. All the arithmetic runs in your browser with JavaScript — nothing is sent to a server.
How do I use the Mixed-Radix Converter — Time, Angle & Custom Radix Vectors?
Just type or paste your value. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Time arithmetic
Split a duration in seconds into days, hours, minutes and seconds — or build a total from the parts.
Angles & coordinates
Convert between decimal arcseconds and degrees-minutes-seconds (DMS) for astronomy, surveying and GPS.
Imperial measurement
Turn a length in inches into miles, yards, feet and inches with the irregular 1760 / 3 / 12 factors.
Permutations & combinatorics
Explore the factorial number system to index permutations or teach mixed-radix counting.
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