7 Unit Lists: Conversion to Sums of Units

Outside of the SI, it is sometimes desirable to convert a single unit to a sum of units—for example, feet to feet plus inches. The conversion from sums of units was described in Sums and Differences of Units, and is a simple matter of adding the units with the ‘+’ sign:

You have: 12 ft + 3 in + 3|8 in
You want: ft
        * 12.28125
        / 0.081424936

Although you can similarly write a sum of units to convert to, the result will not be the conversion to the units in the sum, but rather the conversion to the particular sum that you have entered:

You have: 12.28125 ft
You want: ft + in + 1|8 in
        * 11.228571
        / 0.089058524

The unit expression given at the ‘You want:’ prompt is equivalent to asking for conversion to multiples of ‘1 ft + 1 in + 1|8 in’, which is 1.09375 ft, so the conversion in the previous example is equivalent to

You have: 12.28125 ft
You want: 1.09375 ft
        * 11.228571
        / 0.089058524

In converting to a sum of units like miles, feet and inches, you typically want the largest integral value for the first unit, followed by the largest integral value for the next, and the remainder converted to the last unit. You can do this conversion easily with units using a special syntax for lists of units. You must list the desired units in order from largest to smallest, separated by the semicolon (‘;’) character:

You have: 12.28125 ft
You want: ft;in;1|8 in
        12 ft + 3 in + 3|8 in

The conversion always gives integer coefficients on the units in the list, except possibly the last unit when the conversion is not exact:

You have: 12.28126 ft
You want: ft;in;1|8 in
        12 ft + 3 in + 3.00096 * 1|8 in

The order in which you list the units is important:

You have: 3 kg
You want: oz;lb
        105 oz + 0.051367866 lb

You have: 3 kg
You want: lb;oz
        6 lb + 9.8218858 oz

Listing ounces before pounds produces a technically correct result, but not a very useful one. You must list the units in descending order of size in order to get the most useful result.

Ending a unit list with the separator ‘;’ has the same effect as repeating the last unit on the list, so ‘ft;in;1|8 in;’ is equivalent to ‘ft;in;1|8 in;1|8 in’. With the example above, this gives

You have: 12.28126 ft
You want: ft;in;1|8 in;
        12 ft + 3 in + 3|8 in + 0.00096 * 1|8 in

in effect separating the integer and fractional parts of the coefficient for the last unit. If you instead prefer to round the last coefficient to an integer you can do this with the --round (-r) option. With the previous example, the result is

You have: 12.28126 ft
You want: ft;in;1|8 in
        12 ft + 3 in + 3|8 in (rounded down to nearest 1|8 in)

When you use the -r option, repeating the last unit on the list has no effect (e.g., ‘ft;in;1|8 in;1|8 in’ is equivalent to ‘ft;in;1|8 in’), and hence neither does ending a list with a ‘;’. With a single unit and the -r option, a terminal ‘;’ does have an effect: it causes units to treat the single unit as a list and produce a rounded value for the single unit. Without the extra ‘;’, the -r option has no effect on single unit conversions. This example shows the output using the -r option:

You have: 12.28126 ft
You want: in
        * 147.37512
        / 0.0067854058

You have: 12.28126 ft
You want: in;
        147 in (rounded down to nearest in)

Each unit that appears in the list must be conformable with the first unit on the list, and of course the listed units must also be conformable with the unit that you enter at the ‘You have:’ prompt.

You have: meter
You want: ft;kg
             ^
conformability error
        ft = 0.3048 m
        kg = 1 kg

You have: meter
You want: lb;oz
conformability error
        1 m
        0.45359237 kg

In the first case, units reports the disagreement between units appearing on the list. In the second case, units reports disagreement between the unit you entered and the desired conversion. This conformability error is based on the first unit on the unit list.

Other common candidates for conversion to sums of units are angles and time:

You have: 23.437754 deg
You want: deg;arcmin;arcsec
    23 deg + 26 arcmin + 15.9144 arcsec

You have: 7.2319 hr
You want: hr;min;sec
    7 hr + 13 min + 54.84 sec

Some applications for unit lists may be less obvious. Suppose that you have a postal scale and wish to ensure that it’s accurate at 1 oz, but have only metric calibration weights. You might try

You have: 1 oz
You want: 100 g;50 g; 20 g;10 g;5 g;2 g;1 g;
        20 g + 5 g + 2 g + 1 g + 0.34952312 * 1 g

You might then place one each of the 20 g, 5 g, 2 g, and 1 g weights on the scale and hope that it indicates close to

You have: 20 g + 5 g + 2 g + 1 g
You want: oz;
        0.98767093 oz

Appending ‘;’ to ‘oz’ forces a one-line display that includes the unit; here the integer part of the result is zero, so it is not displayed.

If a non-empty list item differs vastly in scale from the quantity from which the list is to be converted, you may exceed the available precision of floating point (about 15 digits), in which case you will get a warning, e.g.,

You have: lightyear
You want: mile;100 inch;10 inch;mm;micron
        5.8786254e+12 mile + 390 * 100 inch (at 15-digit precision limit)

• Cooking Measure
• Unit List Aliases

You have: (2+1|2) cup / 6
You want: cup;1|2 cup;1|3 cup;1|4 cup;tbsp;tsp;1|2 tsp;1|4 tsp
        1|3 cup + 1 tbsp + 1 tsp

You have: 12.28125 ft
You want: ft;in;1|8 in;
        12 ft + 3 in + 3|8 in

You have: (5+1|4) cup / 3
You want: 1|2 cup;1|3 cup;1|4 cup
        3|2 cup + 1|4 cup

You have: (5+1|4) cup / 3
You want: 1|2 cup;1|3 cup;1|4 cup
        3 * 1|2 cup + 1|4 cup

cup;1|2 cup;1|3 cup;1|4 cup;tbsp;tsp;1|2 tsp;1|4 tsp

hms         time: hours, minutes, seconds
dms         angle: degrees, minutes, seconds
time        time: years, days, hours, minutes and seconds
usvol       US cooking volume: cups and smaller
uswt        US weight: pounds and ounces
ftin        length: feet, inches and 1/8 inches
ftin2       length: feet, inches and 1/2 inches
ftin4       length: feet, inches and 1/4 inches
ftin8       length: feet, inches and 1/8 inches
ftin16      length: feet, inches and 1/16 inches
ftin32      length: feet, inches and 1/32 inches
ftin64      length: feet, inches and 1/64 inches
inchfine    length: inches subdivided to 1/64 inch

You have: anomalisticyear
You want: time
        1 year + 25 min + 3.4653216 sec
You have: 1|6 cup
You want: usvol
        2 tbsp + 2 tsp

You have: screwgauge(10) + 1|64 in
You want: ftin64
        13.16 * 1|64 in
You have: _
You want: ftin32
        6.58 * 1|32 in

You have: dms
        Definition: unit list, deg;arcmin;arcsec

You have: year
You want: day;min;sec
365;348;45.974678

You have: liter
You want: cup;1|2 cup;1|4 cup;tbsp
4;0;0;3.6280454