Last week, I made the claim that the significant digits of a measurement are not dependent on the units that you use to measure.  Today, I want to look at this from the point of view of units that are linearly related to each other, and I’m going to start with even more specificity and look only at units that are multiplicatively related (for example, this class excludes Fahrenheit to Celsius conversion) using the rules of significant digits.

Suppose that X and Y are units such that:

for:

having infinite precision.

Let x be a measured value in units X with p significant digits.

Then, a x also has p significant digits by the multiplication rule of significant digits (x has fewer significant digits so the product has the same number as x).

Therefore, y has the same number of significant digits as x.

It’s interesting to note that this is not necessarily true if there is an additional term in the unit conversion.  As a counter-example, consider the Celsius to Fahrenheit conversion:

If c = 5, it has one significant digit.  However,

has two significant digits (since we keep everything greater than the ones place).

In fact, the number of significant digits can become arbitrarily large in this scheme (consider, eg, c = .00005 still only has one significant digit, but after the conversion, it would have seven – remember that 32 has infinite precision).

This can go the other way as well, and mimic’s the phenomenon of subtractive cancellation.  If, for example, you take c = -17.775 (five significant figures), then f = .005 (one significant figure).

Thinking about this in terms of the original example, (when and where are we going to lunch?), we don’t need units so long as we agree on the origin time (right now and right here; the start of the UNIX epoch and 0 latitude/0 longitude; the time of the big bang and the center of the universe, etc).  So, I guess it would be more correct to say that we don’t need units, but we do need a point of origin.