Metric system

### The measurement system of science(and nearly everything else)

As of 2015, there are only three countries on Earth which do not use the metric system as their official system of weights and measurements: Myanmar (Burma), Liberia and the United States.

Back in the 1970's, President Jimmy Carter tried to convert the US to the metric system, but there was a lot of resistance to change. We were making progress when a gasoline shortage hit the country just as gas stations were converting from gallons to liters, the standard metric unit of volume. Because the difference between liters and gallons is significant (there are a little less than 4 liters in a gallon), enough customers were convinced that the change was an attempt to charge more for gas that it was pretty much the end of the US conversion to metric units.

The metric system is, and has been for a long time, the universal system of measurement in science. Because it's a base-10 system, it is very easy to use and completely compatible with scientific notation.

Unlike feet, inches, miles, ounces, pounds and other measurements used in the US, the metric system is a base-ten system. For example, in our so-called imperial units

Photo: Associated Press

system (inherited from the English), there are 5,280 feet in a mile, three feet in a yard, and12 inches in a foot. There's not much consistency there, no real pattern. In the metric system, the analogous units would be the Kilometer, with 1000 meters in a Kilometer, 10 decimeters in a meter, 10 centimeters in a decimeter and 10 millimeters in a centimeter.

The metric system is a base-10 system, and all conversions within it involve powers of 10, and that arithmetic involving powers of 10 is nothing more than moving a decimal point left or right.

#### Powers of ten

The metric system is a base-ten system. All conversions between units in the metric system involve factors and powers of 10.

### Base units in the metric system

The common base units of measurement of the metric system are shown in this table. The base unit of time is the second, which is not a base 10 unit, but it has become so entrenched in every nation that we are unlikely to convert to a base-10 time scheme in your lifetime. Below we'll discuss some of the advantages of a base 12 or base 60 system, like our time system.

Measurement Base unit Abbrev.
length meter (m)
mass gram (g)
volume liter (L)
time second (s)

### Larger and smaller units are built from the base units

In the metric system, we build larger and smaller units from the base units. Let's focus first on length. The metric base unit of length is the meter. The meter is defined (actually it was redefined in 1983 to be this) as the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. That doesn't really have to matter to you, suffice to say that the length of the standard meter is known and won't likely change.

One consequence of this definition is that we can state the exact speed of light in vacuum, c = 2.99792458 x 108 m/s.

For scale, your height is probably between 1 and 2 meters. In order to build larger and smaller units from the meter, we multiply or divide the meter by powers of ten. To some of those we give common names and abbreviations. Here are many of the multiples and fractions of the meter and some of the common names.

You can see from the table above that appending prefixes to the name of the base unit (meters in this case) is how we identify units by relative size. 1000 meters is called a Kilometer (Kilo is from the Latin for 1000), but we don't have a special word for 10,000 m or 100,000 m. With some exceptions, we name units about every 1000, or three powers of 10: A kilometer is 1000 m, a millimeter is 1/1000th of a meter.

When we divide meters into smaller units, we make some exceptions to that rule of thumb: 1 centimeter (cm, "centi" means 1/100th) is 1/100th of a meter, which means that there are 100 cm in every 1 m. A decimeter is a tenth of a meter and there are 10 dm in 1 m.

You should memorize all of the metric prefixes in that table and understand what they mean. Some day you'll get into a conversation where they're being thrown around and you won't want to be lost.

#### Definition of the meter

1 meter (m) is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. A typical adult human is between 1 and 2 m tall.

### Converting from one unit to another

#### ... but first an important mathematical reminder

It's very important in a great many fields to be able to convert between units. It's not only important in science and engineering, but think about business and finance: How many Euros or Yuan in a Dollar? You don't want to make a mistake there.

This section has many examples of how to convert from one unit to another, but first, we ought to take a mathematical diversion. I find that a lot of students forget about the important properties of multiplication and division, namely that they are commutative and associative. These are absolutely crucial for doing unit conversions – and really for doing almost any algebraic operation.

Here's an example. Simplify this expression:

$$\frac{a^2 \cdot b \cdot d \cdot e}{d^2 \cdot e \cdot b \cdot a}$$

It's possible to start "canceling" some terms – that really means dividing elements into one-another, but let's see how it works first. Because multiplication is commutative (a·b = b·a), we can write

$$\frac{a^2 \cdot b \cdot d \cdot e}{d^2 \cdot e \cdot b \cdot a} = \frac{a^2 \cdot b \cdot d \cdot e}{a \cdot b \cdot d^2 \cdot e}$$

And because multiplication is associative, we can regroup, putting like terms together like this:

$$= \left( \frac{a^2}{a} \right) \left( \frac{b}{b} \right) \left( \frac{d}{d^2} \right) \left( \frac{e}{e} \right)$$

Now it's obvious what to do: a2 divided by a is a; b divided by b is 1, and so on.

$$\require{ cancel } \ \left( \frac{a^{\cancel{2}}}{\cancel{a}} \right) \left( \frac{\cancel{b}}{\cancel{b}} \right) \left( \frac{\cancel{d}}{d^{\cancel{2}}} \right) \left( \frac{\cancel{e}}{\cancel{e}} \right) = \frac{a}{b}$$

But we don't have to go through all of those steps once we realize that we could. We can just start dividing terms, even clear across the expression as it was originally written. Follow this diagram as I divide (cancel) terms one at a time, a through e:

\require{ cancel } \begin{align} \frac{a^{\cancel{2}} b d e}{d^2 e b \cancel{a}} \color{#E90F89}{\longrightarrow} \frac{a \cancel{b} d e}{d^2 e \cancel{b}} \\[5pt] \color{#E90F89}{\rightarrow} \frac{a d \cancel{e}}{d^2 \cancel{e}} \color{#E90F89}{\rightarrow} \frac{a \cancel{d}}{d^{\cancel{2}}} \color{#E90F89}{\rightarrow} \frac{a}{d} \end{align}

You'll see why this is important as we work through the examples below.

One other important thing: We'll always treat units just like algebraic constants, multiplying and dividing them just like x's and y's.

#### Units

An object doesn't care what units you use to describe it. Converting the length of a steel bar from 1.2 m to 47-¼ inches doesn't change the length of the bar, only the way we describe it. It's important to realize that there are all kinds of units we use to describe matter or its condition, and changing from one to the other is something we do frequently, but it doesn't actually change the matter or its situation.

### Example 1

Convert 128 meters (128 m) to centimeters (cm)

The first thing I like to do when doing unit conversions like this is to write my starting point, 128 m, and right next to it, a big set of empty parenthesis with an empty fraction bar. I know I want the unit of meters (m) to cancel, so I'll put that in the denominator of my empty fraction with the unit I'm going for in the numerator:

$$\require{ cancel } 128 \; \cancel{m} \left( \frac{\phantom{000} cm}{\phantom{0000} \cancel{m}} \right)$$

Now I need some sort of relationship between meters and centimeters (cm). I get that from my memory, or from the table above: There are 100 cm in every 1 m, so I'll plug that into my fraction:

$$\require{ cancel } 128 \; \cancel{m} \left( \frac{100 \; \color{#E90F89}{cm}}{1 \; \cancel{m}} \right)$$

The m's cancel; remember I'm treating those just like numerical constants: m/m = 1. I'm left with units of cm, which is just what I wanted.

Now it's just a matter of following the "instructions" that we've set up for ourselves. Those are "multiply 128 by 100 to get the measurement in cm."

$$12,800 \; \text{cm} = 1.28 \times 10^4 \; \text{cm}$$

You should get into the habit of converting awkward numbers like 12,800 into scientific notation.

### Example 2

Convert 0.1243 m to micrometers (μm)

We'll follow the same pattern here as for the first example. Write down 0.1243 m next to our parenthesis, and set up to cancel the m's:

$$\require{ cancel } 0.001243 \; \cancel{m} \left( \frac{\phantom{000} \mu m}{\phantom{0000}\cancel{m}} \right)$$

The relationship we need is that there are 106 micrometers (μm) in every 1 m (because 1 μm is 10-6 m).

$$\require{ cancel } 0.001243 \; \cancel{m} \left( \frac{10^6 \; \color{#E90F89}{\mu m}}{1 \; \cancel{m}} \right)$$

Now with our units canceled, and noting that μm is the only unit left, as we expected, we just multiply 0.001243 by 106 to get our result:

$$= 1240 \; \mu m = 1.249 \times 10^3 \; \mu m$$

### Example 3

Convert 0.093 nm to picometers (pm)

In this example, we could notice that there are 1000 pm in every 1 m, but let's assume, just for this example, that all we know is that there are 109 nm in every 1 m and 1012 pm in every 1 m. We'll set up our first set of parenthesis as though we're converting from nm to m:

$$\require{ cancel } 0.093 \; \cancel{nm} \left( \frac{\phantom{000} m}{\phantom{0000}\cancel{nm}} \right)$$

If we stopped there we could put 1 in the numerator and 109 in the denominator of our fraction, multiply and convert to meters. But we'll press on and add another set of parenthesis with the unit meters in the denominator and pm in the numerator. We'll often "chain" these conversions together like this.

$$\require{ cancel } 0.093 \cancel{nm} \left( \frac{1 \cancel{m}}{10^9 \cancel{nm}} \right) \left( \frac{\phantom{000} pm}{1 \cancel{m}} \right)$$

Now we'll fill in the number of picometers per meter, 1012

$$\require{ cancel } 0.093 \cancel{nm} \left( \frac{1 \cancel{m}}{10^9 \cancel{nm}} \right) \left( \frac{10^{12} \color{#E90F89}{pm}}{1 \cancel{m}} \right)$$

Here is our final set of instructions for this conversion: Multiply 0.093 by 1012, then divide by 109 to get the measurement in picometers:

$$= \frac{(0.093)(10^{12}) \, pm}{10^9}$$

The arithmetic looks like this. Remember that when dividing powers of the same base (10), we subtract exponents.

\begin{align} &= 0.093 \times 10^{12-9} \, pm \\[5pt] &= 0.093 \times 10^3 \, pm \\[5pt] &= 93 \, pm \end{align}

It's customary not to use scientific notation for numbers less than 100 or 1000, though there's no fixed rule. I prefer to write 93 pm in this case. This also suggests that picometers might be a kind of natural unit for this measurement.

We often look for units that make the numbers we have to use in certain circumstances numbers of the order of 100 or so (i.e. no more than two zeros). It's not always possible, but it's nice.

### Example 4

Mt. Everest is 29,028 ft. tall. Convert that to meters using only these facts: 1 ft. = 12 in., 1 in. = 2.54 cm, and the table of metric units above.

We'll have to do this conversion in many steps, but we can still just chain them all together. We'll start by converting feet to inches:

$$\require{ cancel } 29,028 \cancel{ft.} \left( \frac{12 \, in.}{1 \cancel{ft.}} \right)$$

Now inches to centimeters using the information given: 2.54 cm = 1 in:

\

$$\require{ cancel } 29,028 \cancel{ft.} \left( \frac{12 \, \cancel{in.}}{1 \cancel{ft.}} \right) \left( \frac{2.54 \, cm}{1 \cancel{in.}} \right)$$

Now centimeters to meters using the fact that there are 100 cm in 1 m (isn't the metric system much easier?):

$$\require{ cancel } 29,028 \cancel{ft.} \left( \frac{12 \, \cancel{in.}}{1 \cancel{ft.}} \right) \left( \frac{2.54 \, \cancel{cm}}{1 \cancel{in.}} \right) \left( \frac{1 \, \color{#E90F89}{m}}{100 \cancel{cm}} \right)$$

After canceling all of the repeated units, we're left with units of meters, which is what we wanted. Now it's just a matter of following the instructions with a calculator to get:

\begin{align} &= \frac{29,028)(2)(2.54)}{100} \\[5pt] &= 8,848 \, m \end{align}

By the way, you can enter that calculation into your calculator in one line, with just multiplication and division, and without parenthesis: 29028 [ × ] 2 [ × ] 2.54 [ / ] 100 [ Enter ]. Remember that division is just multiplication by the reciprocal of the divisor, and that multiplication is commutative (a·b = b·a) Here's a picture:

### Practice problems

Perform these unit conversions. Always make sure to write the result with the proper units:

 1 0.00243 μm to nanometers (nm) 2 101,235 m to Kilometers (Km) 3 1,288 cm to Km 4 0.0034 liters to mL 5 2.92 × 10-1 m3 to liters 6 2343 pL to nanoliters (nL) 7 432.2 mg to grams 8 5.221 × 107 mg to Kg 9 122,309 g to Kg 10 27 μL to nanoliters (nL)

### Other base units: Mass & volume

The base unit of mass in the metric system is the gram. For scale, a U.S. penny has a mass of about 2.5 grams (2.5 g). We use multiples and divisors of powers of ten, and the same prefixes and abbreviations to name multiples and divisions of the gram.

For example, 1000 grams is 1 Kilogram (1000 g = 1 Kg) and 1/1000 of a gram is 1 milligram (1 mg).

Here are the commonly-used mass units:

##### Commonly-used metric units of mass
Unit abbr. in g in 1 g
1 picogram pg 10-12 g 1012
1 nanogram ng 10-9 g 109
1 microgram μg 10-6 g 106
1 milligram mg 10-3 g 103
1 gram g 1 g
1 Kilogram Kg 103 g 10-3
1 metric ton t 106 g 10-6

The base unit of volume in the metric system is the liter (lee·ter) – the British spell it "litre." The same multiples, prefixes and abbreviations apply, but we most often divide liters (about a fourth of a U.S. gallon) into smaller units. For example, a milliliter (1 ml) is 1/1000 of a liter, and 1 microliter is 1-millionth of a liter (1 μl = 1 x 10-6 liter).

Here is a table of commonly-used multiples and divisions of liters.

##### Commonly-used metric units of volume
Unit abbr. in L in 1 L
1 picoliter pL 10-12 L 1012
1 nanoliter nL 10-9 L 109
1 microliter μL 10-6 L 106
1 milliliter mL 10-3 L 103
1 deciliter dL 10-2 L 100
1 Liter L 1 L
1 m3   1 L

### Time and base 12 units

Did you ever wonder why our clocks have 60 seconds per minute and 60 minutes per hour, why we use base 12 in our length measurements here in the U.S.?*

These fractions come to us from a time when cultures, such as the Mesopotamians, were developing mathematics, and when much of that math revolved around supporting commerce – the exchange of money and goods.

Now in our base 10 system, consider the even fractions of 10. They are 1, 1/2, and multiples of 1/5 and 1/10. But consider all of the integer fractions we can get from a base of 60 divisions:

If you're interested in the history of mathematics and more interesting things like this, you might want to check out the books of Princeton Prof. Victor J. Katz.

### Video examples

Here are four worked examples in conversion between metric and other units.

#### Example 1

Here are two examples of unit conversions using metric units (and some non-metric ones, too).