# Units

Units let us know what property is being measured, and they tell us the magnitude scale.

Metric units follow powers of ten. There are 100 cm in a meter, and 1000 mm in a meter. English units have less predictable relationships. There are 3 feet in a yard, and 12 inches in a foot, and 36 inches in a yard.

# Metric Units

The metric system was designed to be a unified and rational system of measures. The metric system improves calculation, communication, and conversion. Amazingly, it has been adopted by almost every country on Earth.

Annoyingly, United States have chosen to not fully adopt the metric system, but the American sciences do use metric. We use metric units on this site.

The metric system defines all units from seven base units. The base units come from measuring physical constants. For example, the meter is defined as the distance light travels in (1 / 299 792 458) seconds.

Quantity Name Symbol
time second s
length meter m
mass kilogram kg
current ampere A
temperature kelvin K
amount mole mol
light intensity candela cd

Derived units are combinations of these base units. For example, velocity is combination of length divided by time (m/s).

Quantity Name Symbols
area meters squared
volume meters cubed
velocity meters per seconds m/s
acceleration meters per seconds squared m/s²
momentum kilogram meters per seconds kg m/s

Some derived units get a special abbreviation normally written as a capital letter. For example, the unit of force is N, but it stands for kg m/s².

Quantity Name Abbreviation Symbols
force newton N kg m/s²
energy joule J kg m²/s²
power watt W kg m²/s³
frequency hertz Hz 1/s
volume liter L 10⁻³ m³

# Metric Prefixes

We use metric prefixes to indicate multiplication or division by powers of ten. For example we can replace 1000 with the letter "k".

$$1000 \, \mathrm{m} = 1 \, \mathrm{km}$$ $$22\,500 \, \mathrm{m} = 22.5 \, \mathrm{km}$$
Name Symbol Power
giga G 109
mega M 106
kilo k 103
centi c 10-2
milli m 10-3
micro μ 10-6
nano n 10-9
Name Symbol Factor Power
tera T 1 000 000 000 000 1012
giga G 1 000 000 000 109
mega M 1 000 000 106
kilo k 1 000 103
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
pico p 0.000 000 000 001 10-12

Converting 10 km into meters means multiplying by 1000.

$$10 \, \mathrm{km} = 10\left({\color{DeepPink}1000}\right) \mathrm{m} = 10\,000 \, \mathrm{m}$$ $$10\,000 \, \mathrm{m} = 10\,({\color{DeepPink}1000}) \, \mathrm{m} = 10 \, \mathrm{km}$$

Converting 10 ms into meters means multiplying by 0.001.

$$10 \, \mathrm{ms} = 10\left({\color{DodgerBlue}0.001 }\right) \mathrm{s} = 0.01\, \mathrm{s}$$ $$0.01 \, \mathrm{s} = 10\left({\color{DodgerBlue} 0.001 }\right) \mathrm{s} = 10\, \mathrm{ms}$$
Example: Convert 120 kilometers into meters.
solution $$\mathrm{k} = 1000$$ $$120\, \mathrm{km} = 120(1000)\, \mathrm{m} = 120\,000\, \mathrm{m}$$
Example: How many meters is 450 nanometers?
solution $$\mathrm{n} = 10^{-9}$$ $$450\,\mathrm{nm} = 450\left( 10^{-9}\right) \mathrm{m} = 0.000\,000\,45 \, \mathrm{m}$$

In practice, conversions are just moving the decimal to the left for negative exponents and to the right for positive.

$$10 \, \mathrm{km} = 10.\overrightarrow{\undergroup{0}\undergroup{0}\undergroup{0}}{\color{DeepPink}.} \, \mathrm{m} = 10\,000 \, \mathrm{m}$$ $$10\,000 \, \mathrm{m} = 10{\color{DeepPink}.} \overleftarrow{\undergroup{0}\undergroup{0}\undergroup{0}}. \, \mathrm{m} = 10 \, \mathrm{km}$$
$$10 \, \mathrm{ms} = 0{\color{DodgerBlue}.}\overleftarrow{\undergroup{0}\undergroup{1}\undergroup{0}}.0 \, \mathrm{s}= 0.01 \, \mathrm{s}$$ $$0.01 \, \mathrm{s} = 0.\overrightarrow{\undergroup{0}\undergroup{1}\undergroup{0}}{\color{DodgerBlue}.}0 \, \mathrm{s}= 10 \, \mathrm{ms}$$
Example: How many meters is 10 Mm?
solution $$10 \, \mathrm{Mm} = 10.\overrightarrow{\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}}{\color{DeepPink}.} \, \mathrm{m} = 10\,000\,000 \, \mathrm{m}$$
Example: How many seconds is 10 μs?
solution $$10 \, \mathrm{\mu s} = 0{\color{DodgerBlue}.}\overleftarrow{\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{1}\undergroup{0}}.0\, \mathrm{s}= 0.000\,01 \, \mathrm{s}$$

# Nonmetric Units

Converting outside the metric system is more complex. Other unit systems don't always use powers of ten, so we can't simply move the decimal left or right. To convert we need to multiply by a conversion fraction.

1. Build a conversion fraction.
2. Put the old unit on the bottom of the fraction.
3. Put the new unit on top of the fraction.
4. Find out how two numbers are equal. Add numbers to the top and bottom of the fraction so that they are equal.
5. Multiply the number you are converting by the conversion fraction. The old unit should cancel to leave just the new unit.

Let's 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.

$$50 \,\mathrm{min} \left( \mathrm{\frac{s}{min}} \right)$$

Our fraction can't change the actual value, so it must be equal to one. The top and bottom of the fraction must equal each other.

$$\text{{\color{DeepPink}1 minute} = \color{DodgerBlue}60 seconds}$$ $$50 \, \mathrm{min} \left( \frac{{\color{DodgerBlue}60\, \mathrm{s}} }{{\color{DeepPink}1\,\mathrm{ min}}} \right)$$ $$50 \, \cancel{\mathrm{min}} \left( \frac{60\, \mathrm{s}}{1 \,\cancel{\mathrm{min}}} \right)$$ $$50 \left( \frac{60\, \mathrm{s}}{1} \right)$$ $$3000\, \mathrm{s}$$
Example: Convert 50 minutes into hours.
solution $$50 \,\mathrm{min} \to \mathrm{hour}$$ $$50\,\mathrm{\color{DeepPink}min} \left( \frac{1\,\mathrm{hour}}{60\,\mathrm{\color{DeepPink}min}} \right)$$ $$\frac{50}{60}\,\mathrm{hour}$$ $$0.8\overline{33}\,\mathrm{hour}$$
Example: A marathon is 26.2 miles. How far is a marathon in kilometers?
(1 mile = 1.6 kilometers)
solution $$26.2 \, \textcolor{DeepPink}{ \mathrm{mile}} \left( \frac{1.6 \, \mathrm{km}}{1 \,\textcolor{DeepPink}{ \mathrm{mile}}} \right) = 41.92\, \mathrm{km}$$
Example: I'm 6 feet and 1 inch tall. How many meters tall am I?
solution $$\mathrm{ft \to inches}$$ $$6\, \mathrm{ft} + 1\, \mathrm{in}$$ $$6\, \textcolor{DeepPink}{\mathrm{ft}}\left( \frac{12\,\mathrm{in}}{1\, \textcolor{DeepPink}{\mathrm{ft}}}\right) + 1\, \mathrm{in}$$ $$72\, \mathrm{in} + 1\, \mathrm{in}$$ $$73\, \mathrm{in}$$
$$\mathrm{inches \to meters}$$ $$\text {a google search returns: } 1 \, \mathrm{m} = 39.37 \, \mathrm{in}$$ $$73\,\textcolor{DeepPink}{ \mathrm{in}} \left( \frac{1\, \mathrm{m}}{39.37 \, \textcolor{DeepPink}{ \mathrm{in}}}\right)$$ $$73 \left( \frac{1\, \mathrm{m}}{39.37}\right)$$ $$1.85\, \mathrm{m}$$