All systems have something in common: they measure the same physical quantities, but the standard
“amount” may vary from system to system.
Because different units of distance still measure
distance
,
you can easily convert from one system to another if you know the
conversion factor
(an “exchange
rate”) between systems.
Consider a conversion of 1 ft to cm knowing 1 in = 2.54 cm
1 ft
12 in
2.54 cm
=
30.48 cm
ft
in
In the conversion above, begin with the amount needing conversion.
Because we don’t know the
conversion factor from cm to ft, we will need to use a
second
conversion to go from ft to in, then inches
to cm.
Multiply the original value by the conversion factor in a way that makes sense (this means it is
not 1 ft × 1 ft, instead we use 1 ft × 12 inches / ft).
Use the second conversion factor next.
This
“chart”
method is a convenient way to keep track of conversions without using a large number of ( ) × ( ) ×
( )’s.
Once all the necessary conversion factors are charted up, cancel units that match on top and bottom.
Multiply across the top and multiply across the bottom.
Simplify the result if it remains a fraction.
More often than not, your conversions will be within the same unit system (from one prefix to another).
Because the decimal system is based in tens, it is very easy to work with because we can simply shift the
decimal point left or right.
For example, convert 34.6 mm to m.
Our intuition tells us to move the
decimal point
three places to the left
because it’s divided by 1000, for a result of 0.0346 m.
Check it.
34.6 mm
1 m
=
0.0346 m
1000 mm
A common mistake students make is to
use this logic on areas, volumes, or anything that’s squared.
For
example, convert 36 cm
2
to m
2
.
An easy trap to fall into is to move the decimal two places and call it
0.36m
2
.
Let’s
explore that.
36 cm
2
(1 m)
2
=
36 cm
2
1 m
2
=
36 m
2
=
0.0036 m
2
(100 cm)
2
10000 cm
2
10000
When the conversion happens for a square, it happens twice: once for the “length” and once for the
“width.”
This logic applies to any squared quantity, even if “squared

time”
as a unit is hard for us to
understand.
Because the math is much more difficult to remember to do mentally, it is a good idea to
do any unit conversions
before
plugging them into squares.
The same mentality applies to volumes and
anything
raised to a power.
23
Developing a Sense for Units
Unless you come from
anywhere
north, south, east or west of the United States, you probably grew up
with the
English System
of units (inches, feet, pounds) and have no
instinct
about the metric system (or
SI).
Science is a global community and it is always conducted in SI, so American students can find
themselves at a disadvantage because we don’t have a good sense of what the units
mean
.
The point of doing unit analysis is as a safeguard for errors.
Bad units will tell you a calculation was done
incorrectly (the
hr/mi
example above), but they also mean to give us sense of scale for our results.
Work
on trying to visually translate a value into units you understand, and compare that to your life
experience to let your “gut feeling” inform you about an answer.
For
the example above,
36 cm
2
looks like a 6 cm by 6 cm square.