### It's not the thing that changes, just how we describe it.

These titanium bars each have a fixed length. To within a few layers of atoms, we could measure that length, and we'd expect it never to change.*

Whether we express the length of one of those bars in feet, inches, centimeters, Angstroms, microns, fathoms or parsecs doesn't matter at all to the bar. Those are just descriptions of its length that are convenient (or not so convenient) for us. We try to choose units that "feel about right" for the situation. It's important that we understand how units are chosen, and how to convert from one unit to the other.

This section will help you get a better feel for units, how and why they're helpful, and how to manipulate them like a pro.

* OK, sure, the length of the bar would fluctuate a little with temperature and over time it might corrode (though titanium is pretty inert), but you get the point.

#### Why are units important?

1. They help to organize a calculation. If we pay attention to units throughout a calculation, no matter how complicated, it's hard to get the wrong answer.

2. Even when you forget an equation or formula, knowing the units of the answer can usually help you reconstruct it.

3. Units are a key component of expressing and sharing results and information. If I tell you I'm 1.83 tall, that doesn't make any sense. But if I say 1.83 meters (which is about 6 feet), it makes perfect sense. In science, all numbers should have units.

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.

### 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:

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

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

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

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:

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:

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:

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."

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:

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

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:

### 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:

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.

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

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:

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

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:

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

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

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:

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
1. 2343 pl to nanoliters (nl)

2. 432.2 mg to grams

3. 5.221 × 107 mg to Kg

4. 122,309 g to Kg

5. 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:

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.

### 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 problem solving

#### Example 1

Two introductory examples of unit conversion, done two ways.