"Strong typing vs. strong testing"

[Rob Warnock]

EXECUTIVE SUMMARY:

    1 inch + 1 second = ~4.03e38 grams.

GORY DETAILS:

Tim Bradshaw  <tfb at tfeb.org> wrote:
+---------------
| Malcolm McLean said:
| > he problem is that if you allow expressions rather than terms then
| > the experssions can get arbitrarily complex. sqrt(1 inch + 1 Second),
| > for instance.
| 
| I can't imagine a context where 1 inch + 1 second would not be an 
| error, so this is a slightly odd example.  Indeed I think that in 
| dimensional analysis summing (or comparing) things with different 
| dimensions is always an error.
+---------------

Unless you convert them to equivalent units first. For example, in
relativistic or cosmological physics, one often uses a units basis
wherein (almost) everything is scaled to "1":

    http://en.wikipedia.org/wiki/Natural_units

When you set c = 1, then:

    Einstein's equation E = mc2 can be rewritten in Planck units as E = m.
    This equation means "The rest-energy of a particle, measured in Planck
    units of energy, equals the rest-mass of a particle, measured in
    Planck units of mass."

See also:

    http://en.wikipedia.org/wiki/Planck_units
    ...
    The constants that Planck units, by definition, normalize to 1 are the:
    * Gravitational constant, G;
    * Reduced Planck constant, h-bar;  [h/(2*pi)]
    * Speed of light in a vacuum, c;
    * Coulomb constant, 1/(4*pi*epsilon_0) (sometimes k_e or k);
    * Boltzmann's constant, k_B (sometimes k).

This sometimes leads people to do things that would appear sloppy
or even flat-out wrong in MKS or CGS units, such as expressing mass
in terms of length:

    Consider the equation A=1e10 in Planck units. If A represents a
    length, then the equation means A=1.6e-25 meters. If A represents
    a mass, then the equation means A=220 kilograms. ...
    In fact, natural units are especially useful when this ambiguity
    is *deliberate*: For example, in special relativity space and time
    are so closely related that it can be useful to not specify whether
    a variable represents a distance or a time.

So it is that we find that the mass of the Sun is 1.48 km or 4.93 us, see:

    http://en.wikipedia.org/wiki/Solar_mass#Related_units

In this limited sense, then, one could convert both 1 inch and 1 second
to masses[1], and *then* add them, hence:

    1 inch + 1 second = ~4.03e38 grams.

;-}  ;-}


-Rob

[1] 1 inch is "only" ~3.41e28 g, whereas 1 second is ~4.03e38 g,
    so the latter completely dominates in the sum.

-----
Rob Warnock			<rpw3 at rpw3.org>
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