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A P P E N D I X A
Units and Scientific Notation
A.1 UNITS AND DIMENSIONS
When we refer to a physical quantity, it must always have associated with it a set of dimensions,
and also in many circumstances, a set of units.
1
In this context the word “dimension” refers
not to spatial dimensions, but rather to the type of physical quantity. For example, length is a
fundamentally different type of quantity than time. One cannot add a length to a time, nor can
one subtract one from the other, because that would equal nonsense. Note that this is not the
same thing as apples and oranges. Unlike length and time, one can add apples and oranges (it
equals fruit salad).
But on the other hand, it is just fine to multiply or divide a length by a time. is produces
something with different dimensions, that are a combination of the two. For example, if one
divides a length by a time, the result is something that has dimensions of length/time (“length
per time”). Often these combined dimensions have special names. is example of length/time
has the special name of velocity or speed. And so any time one divides a length by a time,
something with dimensions of length/time results.
But what about the actual numbers one plugs into the calculator in a specific case? What if
one has a specific length, and a specific time, and wants to calculate a specific speed? Whenever
actual numbers are involved, there must also be units.
A length of 12.0345 is ambiguous. Is it 12.0345 meters or 12.0345 furlongs? e meter
and the furlong are examples of units, which are agreed-upon standards for attaching a numerical
value to a particular physical quantity. And so the meter is a unit of the dimension of length,
and so is a furlong. One can convert between units of the same dimension, by establishing an
equivalence between them. And so 1 meter D 3.280 feet D 39.37 inches D 0.00497 furlongs,
etc.
In the physical sciences we mostly use a particular international system of units, called SI,
which stands for “International System” (in French). e SI unit of length is the meter, while
the SI unit of time is the second. Every SI unit has an official abbreviation. e abbreviation for
the meter is m, and for the second it is s (it matters that they are lower-case). Table
A.1 lists
some common SI units, with their dimensions and official abbreviations.
Just as we can derive new dimensions by multiplying or dividing dimensions by each other
(length/time, for example), we can do the same for units. And so we can divide meters by seconds
to get a new derived unit, which we write m=s (called “meters per second”). What if we want
1
Parts of this chapter appeared, in a somewhat different form, in Beaver [2018].
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