Units

Metric Units

Physical quantities can be expressed in terms of these metric base units:

distance: meter (m)
time: second (s)
mass: kilogram (kg)

Metric Prefixes

We use metric prefixes to indicate multiplication or division by powers of ten. Converting from 10km to meters means moving the decimal 3 spaces larger.

$$10 \, \mathrm{{\color{#f00}k}m} = 10\left({\color{#f00}1000}\right) \mathrm{m} = 10\,000 \, \mathrm{m}$$

Converting from 10 ms to meters means moving the decimal 3 spaces smaller.

$$10 \, \mathrm{{\color{#00f}m}s} = 10\left({\color{#00f}0.001 }\right) \mathrm{s} = 0.01\, \mathrm{s}$$

In practice I just move the decimal to the left for negative exponents and to the right for positive.

$$10 \, \mathrm{km} = 10.\overrightarrow{\undergroup{0}\undergroup{0}\undergroup{0}} \, \mathrm{m} = 10\,000 \, \mathrm{m}$$ $$10 \, \mathrm{ms} = 0\overleftarrow{\undergroup{0}\undergroup{1}\undergroup{0}}. \, \mathrm{s}= 0.01 \, \mathrm{s}$$
Name Symbol Factor Power
giga G 1 000 000 000 10 9
mega M 1 000 000 10 6
kilo k 1 000 10 3
centi c 0.01 10 −2
milli m 0.001 10 −3
micro μ 0.000 001 10 −6
nano n 0.000 000 001 10 −9
Example: 120 km is how many meters?
solution $$ \mathrm{k} = 1000 $$ $$120\, \mathrm{km} = 120(1000)\, \mathrm{m} = 120\,000\, \mathrm{m}$$
Example: 450 nm is how many meters?
solution $$ \mathrm{n} = 10^{-9} $$ $$450\,\mathrm{nm} = 450\left( 10^{-9}\right) \mathrm{m} = 0.000\,000\,45 \, \mathrm{m}$$

Nonmetric Units

Converting outside the metric system is more complex. You can do a google search to find out how two units are related. You then build a conversion fraction with the new unit on top and the old unit on the bottom to be canceled out. The top and the bottom of the fraction must be equal to each other.

Lets convert 50 minutes into seconds.

$$ 50 \,\mathrm{min} \to \mathrm{s}$$ In order to cancel minutes we want to build a fraction with minutes on top and the seconds on the bottom. $$ 50 \,\mathrm{min} \left( \mathrm{\frac{s}{min}} \right)$$ The fraction must be equal to one to not change the value we are converting. This means the top and bottom must equal each other. One minute equals 60 seconds. $$ 50 \, \mathrm{min} \left( \frac{60\, \mathrm{s}}{1\,\mathrm{min}} \right)$$ We can cancel units. $$ 50 \, \textcolor{red}{ \mathrm{min}} \left( \frac{60\, \mathrm{s}}{1 \,\textcolor{red}{\mathrm{min}} } \right)$$ $$ 50 \left( \frac{60\, \mathrm{s}}{1} \right)$$ Multiply and divide to simplify $$3000\, \mathrm{s}$$
Example: A marathon is 26.2 miles. How far is a marathon in kilometers?
(1 mile = 1.6 kilometers)
solution $$26.2 \, \textcolor{red}{ \mathrm{mile}} \left( \frac{1.6 \, \mathrm{km}}{1 \,\textcolor{red}{ \mathrm{mile}}} \right) = 41.92\, \mathrm{km}$$
Example: I'm 6 feet and 1 inch tall. How many meters tall am I?
solution $$\mathrm{ft \to inches}$$ $$6\, \mathrm{ft}\left( \frac{12\,\mathrm{in}}{1\, \mathrm{ft}}\right) + 1\, \mathrm{in}$$ $$72\, \mathrm{in} + 1\, \mathrm{in}$$ $$73\, \mathrm{in}$$
$$\text {a google search returns: } 1 \, \mathrm{m} = 39.37 \, \mathrm{in}$$ $$73\,\textcolor{red}{ \mathrm{in}} \left( \frac{1\, \mathrm{m}}{39.37 \, \textcolor{red}{ \mathrm{in}}}\right) $$ $$73 \left( \frac{1\, \mathrm{m}}{39.37}\right) $$ $$1.85\, \mathrm{m}$$
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