Time Calculator

Clock time is a mixed-radix quantity: seconds and minutes run in base 60 while hours run in base 24 or base 12, so adding two durations column by column requires carrying at 60 rather than at 10. The reliable method, and the one used here, is to normalize both operands into a single unit. Each value becomes hours times 3600 plus minutes times 60 plus seconds, the addition or subtraction is done on plain integers, and the total is decomposed back with integer division by 3600 and 60. This avoids the classic error of writing 2h 45m plus 3h 30m as 5h 75m. The decimal-hours output exists because payroll and billing systems do not work in base 60. A duration of 2h 45m is 2.75 hours, and 2h 45m 30s is 2.7583 hours; multiplying decimal hours by a rate gives pay directly, whereas multiplying 2.45 by a rate is simply wrong. In US payroll practice, recorded time is often rounded to the nearest quarter hour under the rounding allowance in 29 CFR 785.48, which tolerates rounding to 5, 6, or 15 minute increments provided it does not systematically favor the employer. Note that this tool computes pure durations: it has no calendar, so it does not account for daylight saving transitions, date boundaries, or leap seconds, and the minute and second fields are expected to stay within 0 to 59.

The base-60 structure being manipulated here is Babylonian. Sumerian and Old Babylonian scribes used a sexagesimal place-value system from roughly 2000 BC, chosen in part because 60 divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. Greek astronomers inherited it: Ptolemy's Almagest, written about 150 AD, expresses angles and time in sexagesimal fractions, and the Latin terms for those fractions, pars minuta prima and pars minuta secunda, are the direct ancestors of the words minute and second. For centuries the subdivisions were computational conveniences rather than things a clock displayed. Mechanical clocks acquired minute hands only in the sixteenth century, and seconds became practically observable after Christiaan Huygens built the first pendulum clock in 1656. The second was eventually detached from astronomy entirely: in 1967 the 13th General Conference on Weights and Measures redefined the SI second as 9,192,631,770 periods of a specific hyperfine transition of caesium-133. Decimal time was tried and abandoned; revolutionary France legislated a 10-hour day with 100-minute hours in 1793 and gave it up within a few years. Decimal hours survived only in the narrow accounting sense used today, spreading with industrial timekeeping after Willard Bundy patented a workman's time recorder in 1888.