Mixed Radix Converter
Break a number into places that each have their own base, like days, hours, minutes and seconds, or put the places back together. Time, angles, imperial length and weight are built in, or type your own radices.
How to Use
- Choose split a total or combine the parts.
- Pick the system: time (d h min s), angle (° ′ ″), imperial length (mi yd ft in), weight (st lb oz), or custom radices.
- For custom, type the radices from the second place on, e.g. 24 60 60.
- Type the total in the smallest unit, or the parts separated by spaces or colons, such as 1:3:46:40.
- Read the breakdown and the total; the drawing shows each place with its base.
Worked Example
100,000 seconds as days, hours, minutes and seconds. 100,000 ÷ 60 = 1,666 remainder 40 seconds. 1,666 ÷ 60 = 27 remainder 46 minutes. 27 ÷ 24 = 1 remainder 3 hours, leaving 1 day. So 100,000 s = 1 d 3 h 46 min 40 s. Check: 86,400 + 10,800 + 2,760 + 40 = 100,000.
100,000 inches in miles, yards, feet and inches. ÷ 12 = 8,333 ft 4 in; ÷ 3 = 2,777 yd 2 ft; ÷ 1,760 = 1 mi 1,017 yd. So 1 mi 1,017 yd 2 ft 4 in.
The common mistake: treating time as a decimal. 1.5 hours is 1 h 30 min, not 1 h 50 min, and 3:45 is not 3.45 hours but 3.75. Every place carries at its own base, so you divide by 60, 60 and 24, never by 100.
Show Work
Formulas
Bases That Change from Place to Place
Most human measuring systems are mixed-radix. The 24-hour day and the 60-minute hour descend from Egyptian and Babylonian practice; the 360° circle and its 60 minutes and 60 seconds of arc come from Babylonian astronomy. British money before 1971 ran on pounds, 20 shillings and 12 pence, so 1,000 pence was £4 3s 4d.
The same structure appears in computing. A three-dimensional array of shape 4 × 5 × 6 stores element [i][j][k] at index (i × 5 + j) × 6 + k, a mixed-radix number, and calendars, clocks and date libraries are mixed-radix arithmetic with exceptions such as leap years.
About This Calculator
This converter splits a total in the smallest unit into places with their own bases, or combines the places into a total, for time, angles, imperial length, stones and pounds, or any radices you type. It normalises parts that exceed their base and shows the division or multiplication at each step, with whole numbers of any size.
Everything runs in your browser; nothing is sent anywhere.
Related calculators: Factoradic Converter, Hours Calculator, and Babylonian Numerals.
Frequently Asked Questions
What is a mixed radix system?
A positional system in which each place has its own base. Time is the everyday example: 60 seconds make a minute, 60 minutes an hour and 24 hours a day, so 1:03:46:40 means 1 × 86,400 + 3 × 3,600 + 46 × 60 + 40 = 100,000 seconds. Ordinary decimal is the special case where every base is 10.
How do I convert seconds to days, hours, minutes and seconds?
Divide by 60: the remainder is the seconds. Divide the quotient by 60: the remainder is the minutes. Divide again by 24: the remainder is the hours and the quotient is the days. 100,000 s → 1,666 min 40 s → 27 h 46 min → 1 d 3 h.
How do I add up degrees, minutes and seconds?
Convert each to the smallest unit, add, and convert back. 45° 30′ is 45 × 3,600 + 30 × 60 = 163,800 arcseconds. The same works for feet and inches or stones and pounds, and it avoids carrying mistakes.
What if a place is bigger than its base?
It still has a value: 0:75:00 is 75 minutes. Combining the parts gives the correct total, and the breakdown is normalised, so the 75 minutes appear as 1 hour 15 minutes.
Where else are mixed radices used?
Old British money (pounds, 20 shillings, 12 pence), the Maya Long Count calendar (with a place of 18 × 20 days), array indexing in programming (the index of [i][j][k] in a 3-D array) and the factorial number system, whose bases are 2, 3, 4, ….
How do I use the Mixed Radix Converter?
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
Durations
100,000 s = 1 d 3 h 46 min 40 s; 1,000,000 s = 11 d 13 h 46 min 40 s.
Angles
163,800″ = 45° 30′ 0″.
Imperial length
100,000 in = 1 mi 1,017 yd 2 ft 4 in.
Weight
2,000 oz = 8 st 13 lb 0 oz.
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