Other Nonlinear Units (Units: A Unit Conversion Program and Scientific Calculator)

### 6.3 Other Nonlinear Units ¶

Some other examples of nonlinear units are numerous different ring sizes and wire gauges, screw gauges, pipe and tubing sizes, the grit sizes used for abrasives, the decibel scale, shoe size, scales for the density of sugar (e.g., baume). The standard data file also supplies units for computing the area of a circle and the volume of a sphere. See the standard units data file for more details.

Diameters of American wire sizes can be found using the ‘`wiregauge()`’ function or its alias ‘`awg()`’:

    You have: wiregauge(11)
    You want: inches
            * 0.090742002
            / 11.020255

    You have: 1 mm
    You want: wiregauge
            18.201919

Wire and screw gauges with multiple zeroes are signified using negative numbers, where two zeroes (“00”; “2/0”) is ‘`-1`’, three zeros (“000”; “3/0”) is ‘`-2`’, and so on. Alternatively, you can use the synonyms ‘`g00`’, ‘`g000`’, or ‘`g2_0`’, ‘`g3_0`’, and so on that are defined in the standard units data file.

    You have: brwiregauge(g00)
    You want: inches
            * 0.348
            / 2.8735632

In North America, wire sizes larger than 0000 (“4/0”) are usually given in terms of area, either in kcmil or the older initialism MCM (thousand circular mils). Outside of North America, all wire sizes are usually given in terms of area in mm2. Wire area can be obtained using ‘`wiregaugeA()`’ or its alias ‘`awgA()`’:

    You have: awgA(g6_0)
    You want: kcmil
            * 336.45718
            / 0.0029721464
    You have: awgA(12)
    You want: mm^2
            * 3.3087729
            / 0.30222685

The closest standard metric sizes are 2.5 mm2 and 4 mm2; in general, there isn’t an exact correlation between American and metric wire sizes.

Though based on the long-established iron pipe size (IPS) given in inches, nominal pipe size (NPS) is a dimensionless quantity that corresponds to the inch size. Pipe size can be equivalently specified using metric diamètre nominal (DN), which roughly corresponds to the diameter in mm. For a given pipe size, outside diameter is constant while inside diameter varies with schedule. For example, for NPS 2½ pipe,

    You have: npsOD(2+1|2)
    You want: in
            * 2.875
            / 0.34782609
    You have: nps40(2+1|2)
    You want: in
            * 2.469
            / 0.40502228
    You have: nps80(2+1|2)
    You want: in
            * 2.323
            / 0.43047783

Pipe size can be given equivalently in terms of the metric DN by using the ‘`DN()`’ function, which converts nominal metric size to nominal inch size:

    You have: npsOD(DN(65))
    You want: mm
            * 73.025
            / 0.01369394
    You have: _
    You want: in
            * 2.875
            / 0.34782609

Unlike with wire sizes, actual NPS and metric DN pipe dimensions are the same.

    You have: grit_P(600)
    You want: grit_ansicoated
            342.76923

The last example shows the conversion from P graded sand paper, which is the European standard and may be marked “P600” on the back, to the USA standard.

You can compute the area of a circle using the nonlinear unit, ‘`circlearea`’. You can also do this using the circularinch or circleinch. The next example shows two ways to compute the area of a circle with a five inch radius and one way to compute the volume of a sphere with a radius of one meter.

    You have: circlearea(5 in)
    You want: in2
            * 78.539816
            / 0.012732395

    You have: 10^2 circleinch
    You want: in2
            * 78.539816
            / 0.012732395

    You have: spherevol(meter)
    You want: ft3
            * 147.92573
            / 0.0067601492

The inverse of a nonlinear conversion is indicated by prefixing a tilde (‘`~`’) to the nonlinear unit name:

    You have: ~wiregauge(0.090742002 inches)
    You want:
            Definition: 11

You can give a nonlinear unit definition without an argument or parentheses, and press `Enter` at the ‘`You want:`’ prompt to get the definition of a nonlinear unit; if the definition is not valid for all real numbers, the range of validity is also given. If the definition requires specific units this information is also displayed:

    You have: tempC
            Definition: tempC(x) = x K + stdtemp
                        defined for x >= -273.15
    You have: ~tempC
            Definition: ~tempC(tempC) = (tempC +(-stdtemp))/K
                        defined for tempC >= 0 K
    You have: circlearea
            Definition: circlearea(r) = pi r^2
                        r has units m

To see the definition of the inverse use the ‘`~`’ notation. In this case the parameter in the functional definition will usually be the name of the unit. Note that the inverse for ‘`tempC`’ shows that it requires units of ‘`K`’ in the specification of the allowed range of values. Nonlinear unit conversions are described in more detail in Defining Nonlinear Units.