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Compound[10] unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches, gallons and pints, pounds shillings and pence, and so on. Prior to the use of decimal-based systems of money and units of measure, the use of compound unit arithmetic formed a significant part of commerce and industry. Basic arithmetic operations[edit] The techniques used for compound unit arithmetic were developed over many centuries and are well-documented in many textbooks in many different languages.[11][12][13][14] In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions: Reduction where a compound quantity is reduced to a single quantity, for example conversion of a distance expressed in yards, feet and inches to one expressed in inches.[15] Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to 1 lb, 8 oz. Normalization is the conversion of a set of compound units to a standard form – for example rewriting "1 ft 13 in" as "2 ft 1 in". Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic. Principles of compound unit arithmetic[edit] There are two basic approaches to compound unit arithmetic: Reduction–expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day. On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the on-going normalization method as applied to addition is shown below. UK pre-decimal currency 4 farthings (f) = 1 penny 12 pennies (d) = 1 shilling 20 shillings (s) = 1 pound (£) MixedUnitAddition.svg The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results. The total in the pence column is 25. Since there are 12 pennies in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column. It should be noted that for the sake of simplicity, the example chosen did not have farthings. Operations in practice[edit] A scale calibrated in imperial units with an associated cost display. During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings and "Ready Reckoners" – books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet[16] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound". The cumbersome nature of compound unit arithmetic has been recognized for many years – in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth")[17] in which he declared that the universal introduction of decimal coinage, measures, and weights to be merely a question of time while in the modern era, many conversion programs, such as that embedded in the calculator supplied as a standard part of the Microsoft Windows 7 operating system display compound units in a reduced decimal format rather than using an expanded format (i.e. "2.5 ft" is displayed rather than "2 ft 6 in"). |
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