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One of the most useful skills that you
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can pick up, in scientific
quantitative analysis,
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Is the ability to look at and manipulate
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the units that are attached to the
numbers.
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This is called the factor-label method.
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And once you get it, it seems very
intuitive, but it seems to me
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that nobody is born knowing how to do
this, so you
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might have to work at it a little bit to
get it.
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Say we have a number which is a velocity
in units of kilometers per hour.
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And then we want to convert that into
different units of meters per second.
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Instead of trying to remember a formula to
do that, we
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can actually use the kilometers and the
hours, meters, and seconds.
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To construct the formula that we need to
do the conversion such
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that it will make the units work.
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So we'll start with a velocity in
kilometers
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per hour, say we're driving along and it's
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100 kilometers per hour, and you want to
know how many meters per second that is.
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So, we need to find factors that can
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convert these units into the ones that we
want.
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And so let's start with the time
units.
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We want to convert from hours to seconds.
And so we'll start by writing a factor
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which can cancel out the units of hours
here, and give us something shorter.
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And I don't remember in my head how many
seconds
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there are in an hour, so I'll write it
out.
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Do this in two stages, that there are 60
minutes
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in an hour, and so now we can cancel the
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units of hours here and that's the
first step in our
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our factor label method.
Now because an hour equals 60 minutes.
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If you take a quantity which is the same
on the top and the
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bottom of a fraction, then that quotient is equal to one.
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So by multiplying our
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original information, which is 100
kilometers
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per hour, by this conversion factor, we're
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just multiplying by one, and so we're not
actually changing the value at all.
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We're changing the units and so that
changes
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the numbers, but the actual quantity stays
the same.
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So the next factor will be from minutes
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to seconds, where 60 seconds equals one
minute.
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Now the minute units are gone away and we
have kilometers per second,
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so that's getting closer, but now we have to
deal with the kilometers to meters.
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And so the conversion factor here is a
kilometer is equal to 100 meters.
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And I have to put the kilometer on the
bottom
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I can see, because I have to cancel
kilometer here.
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And end up with meters on the top.
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So after we have constructed this whole
thing to cancel out the units.
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It's meters.
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So we're left with what we want, meters
per second.
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all we have to do is multiply the numbers.
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100 times 1000 divided by 60 divided by 60
gives us this answer.
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And so the units have told us how to do
the math.