Unit conversion

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

These titanium (Ti) 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.

### Two methods

There are two most-used methods of unit conversion, but first, a word on why we use those at all. After all, if you want to change 27 inches into millimeters, you can always go to Google or some other helpful website or app and just ask it to do the calculation for you. Here's a screen shot of a Google™ conversion:

I just typed "27 inches in millimeters" into the search box and we see that 27 inches is 685.8 mm. That's pretty convenient, and very useful from time to time. I use it myself. I admit it.

BUT ... There is every reason to learn to manipulate units by hand. Being good with units can

Help you to rearrange formulas to ensure that you're doing the right arithmetic to get the right result,

Help you to reconstruct formulas you may have forgotten, because you know what the units of the answer have to be.

Any scientist, engineer, economist or any other professional who uses numbers has to be a master of units.

### 1. Big parentheses

I begin with this method because it's the one I use most often. We'll learn it by example, and we'll begin with a very simple one.

We know that there are 100 centimeters (cm) in every one meter (m), so let's ask the question: How many meters is 123 cm? Now this is a pretty easy question to do just by looking at it, knowing your metric prefixes and moving a decimal place, but let's do the unit conversion anyway.

We begin by writing the number we know with its units: 123 cm.

$$123 \; cm$$

Next to it, on the right, write a big open parenthesis with an empty fraction inside.

$$123 \, cm \left( \frac{\phantom{0000}}{\phantom{0000}} \right)$$

Now we know that our final answer should have units of meters (m), so we want to "get rid" of the units cm. By "get rid of" we mean to treat the unit cm as an algebraic variable, and divide it by itself to get 1, a trivial multiplier that changes nothing. So our units (neglecting numbers for now) will look like this:

$$123 \, cm \left( \frac{\phantom{0000}m}{\phantom{000}cm} \right)$$

Finally, we need some known fact that relates those two units. We have already said that 1 m = 100 cm, so we can use that relationship and plug it in ...

$$123 \, cm \left( \frac{1 \, m}{100 \, cm} \right)$$

Next we cancel the units that divide to one, leaving only one unit, meters, which you might circle just to be sure you've ended with the units you wanted:

$$\require{cancel} 123 \cancel{cm} \left( \frac{1 \, \color{#E90F89}{m}}{100 \cancel{cm}} \right)$$

Now that that's done, the expression is telling you just what arithmetic to do in order to get the correct numerical answer. In this case, it's telling us to divide 123 by 100:

$$= \frac{123}{100} = 1.23 \, m$$

While the conversions can get more complicated, these steps are always the same.

#### Unit conversion steps

1. Write down the starting measurement with its units.
2. Write a parentheses containing an empty fraction.
3. Write in units, arrange so the unit you'd like to discard will cancel the starting unit by division.
4. Write in a known relationship between the two units in the fraction.
5. Cancel the units and make sure that the final unit is the one you want.
6. Do the arithmetic called for by the expression you wrote.

### 2. Grid method or "factor labeling"

The grid method, often called factor labeling, is sometimes preferred to the parentheses method above, though it is really just the same, except for some added structure.

Here's an example. Let's convert 1322 milliliters (ml) to liters (L), noting that there are 1000 ml in every 1 L.

We begin as above (and as always), by writing down the number we have with its units:

Next, construct a grid of lines like this:

Next (we are really just following the steps in the box above), write down the units ml and L so that the unit we want to get rid of (ml) is across the horizontal bar (really a fraction bar) from the starting number:

Now pencil in the known relationship between liters and milliliters, 1000 ml = 1 L:

Cancel units that appear on both sides of the fraction bar, and circle the unit(s) remaining. Check to see if the circled unit(s) are what you want:

Finally, perform the arithmetic called for by your grid. In this case that's dividing 1322 by 1000, and appending the unit L:

That's it! It's very similar to the parentheses method except that we use straight lines to organize the work. Quantities adjacent to one another horizontally are multiplied, and those adjacent vertically are divided.

### Practice problems — one-step conversions

Caclulate the following conversions.

 1 Convert 10.0 meters to yards (1 m = 1.09361 yd) 2 Convert 5 miles to Kilometers (1 mi. = 1.60934 Km) 3 Express 2000 psi to atmospheres (atm.) (1 atm = 14.7 psi; atm and psi are units of pressure)

 4 Express 175˚ in radians (π radians = 180˚; degrees and radians are two units of angle measure.) 5 Convert 7 astronomical units (AU) to light years (ly) (1 AU = 1.581 × 10-5 ly) 6 The British use the unit stones to express weight. Convert 165 lbs. to stones (1 lb. = 0.0714286 stones)

### A length conversion tool

Here is a little length-unit conversion tool. You can use it to play around a bit with length units. For example, type 29028 (the height of Mt. Everest) into the top box, choose units of feet, then choose units of meters in the bottom box. You'll get the height of Everest in meters. I've also included units of fathoms (used to measure deep ocean depths) and furlongs (lengths of horse-racing tracks) just for fun. Enjoy.

### More complicated conversions: chaining

Sometimes we have to do unit conversions that require more than one step. Let's look at one conversion and perform it four different ways.

#### Convert 25 meters per second (m/s) to miles per hour (mi./h).

First we'll do this conversion by performing two separate unit conversions, meters to miles (both units of length), and then seconds to hours (both units of time).

#### Method 1 — parentheses

We begin by writing our number with its units. Write m/s as meters over seconds, not with a diagonal slash. Those can be confusing later.

$$25 \, \frac{m}{s}$$

Add an empty fraction in parentheses:

$$25 \, \frac{m}{s} \left( \frac{\phantom{0000}}{\phantom{0000}} \right)$$

Write in the units. We have a choice here. We can only change one unit at a time, so let's chose miles. that means that meters will go in the denominator so that it divides (to 1) the meters in the numerator:

$$25 \, \frac{m}{s} \left( \frac{\phantom{00} mi.}{\phantom{000} m} \right)$$

Now recognize that $1 \, mi. = 1609 \, m.$ That's something you could look up if you don't know it.

$$\require{ cancel } 25 \, \frac{\cancel{m}}{\color{#E90F89}{s}} \left( \frac{1 \, \color{#E90F89}{mi.}}{1609 \cancel{m}} \right)$$

Canceling units gives us new units of miles/s, and following the arithmetic (remember, that 25 is really 25/1), we get the speed in miles per second:

$$= \frac{25}{1609} = 0.01554 \, \frac{mi.}{s}$$

Now we have to convert seconds to hours with a new beginning point, 0.01554 mi./s (I'm keeping a lot of digits here in the middle of the calculation to avoid round-off errors. In the empty fraction we'll put seconds in the numerator.

$$0.01554 \, \frac{mi.}{s} \left( \frac{\phantom{000}s}{\phantom{000}h} \right)$$

There are 3600 seconds in 1 hour, so our conversion is:

$$0.01554 \, \frac{mi.}{s} \left( \frac{3600 \, s}{1 \, h} \right)$$

Canceling units (seconds) gives us our goal units of mi./h (sometimes written as mph):

$$\require{ cancel } 0.01554 \, \frac{\color{#E90F89}{mi.}}{\cancel{s}} \left( \frac{3600 \cancel{s}}{1 \, \color{#E90F89}{h}} \right)$$

The arithmetic gives us the speed in mi./h.

$$= 0.01554 \cdot 3600 = 55.9 \, \frac{mi.}{h}$$

#### Method 2 – factor labeling

I'll take a couple of shortcuts through the factor-label/grid method. We begin with the starting speed:

Then convert from meters to miles to obtain the speed in miles/s:

The result, of course, is the same:

Now beginning where we left off, convert seconds to hours:

And the final speed is the same.

#### Combining steps: "chaining"

Notice that above we're just taking the first-step result and multiplying it by another unit-conversion fraction. In multiplication (ab)·c = a·b·c (associative property), so we can just join all of those steps together. Here's how it looks with parentheses:

And here it is the in factor-label/grid notation:

Notice now we cancel all of the units and end up with those we desire (circled). The arithmetic is prescribed in both expressions, and the fact that the units work out means that if we do that arithmetic correctly, our answer can't be wrong:

### Example – squared units

An acre is a measure of land area used mainly in the U.S. It is a square with sides of length 208.71 ft. Given that there are 2.54 cm in every inch, calculate the area of an acre of land in square meters (m2)

Solution: Let's convert an area to an area, so we'll first calculate the number of square feet (ft2) in an acre:

$$(208.71 \, ft.)(208.71 \, ft.) = 43,650 \, ft^2$$

Now we're going to convert feet to inches (ignoring the fact that we're working with square feet for now). The way I remember that is that there are 12 inches in every foot and 2.54 cm in every inch. Don't ask me why I remember that; I just do. You can look up the number of feet in a meter if you want.

But we are working with square feet, and therefore we'll need square meters. Do arrive at that, we can take our entire conversion of feet to meters, and square it. Notice that everything will cancel nicely and we'll end up with units of m2.

Canceling units gives us final units of square meters (m2), and the final arithmetic is

$$43,650 \cdot \left[ \frac{(12)(2.54)}{100} \right]^2 = 4055 \, m^2$$

This trick of squaring or cubing our whole unit conversion, or sometimes just a term in it, is common when working with square or cubic units, the units of areas and volumes.

### Example – cubed units

Convert the volume 12,000 gallons to cubic meters (m3)

Solution: To do this conversion, we need one important piece of information, the link between gallons and liters:

In the metric system, the liter is defined as one cubic decimeter, were a decimeter (remember that "deci" means "tenth") is a tenth of a meter. So we have all the links we need to do this unit conversion. Here it is:

$$\require{ cancel } 12,000 \cancel{gal} \left( \frac{3.7854 \cancel{L}}{1 \cancel{gal}} \right) \left( \frac{1 \cancel{dm^3}}{1 \cancel{L}} \right) \left( \frac{1 \, m}{10 \cancel{dm}} \right)^3$$

Notice that the term (1 m/10 dm) is cubed. This must be so to cancel the dm3 in the term before it, and we must cube the entire term in the parentheses. The result of cubing that term shows us that there are 1000 (103) cubic decimeters (or liters) in every cubic meter. The diagram illustrates that.

The result of our calculation is that 12,000 gallons is 45.4 m3.

### Video examples

#### Example 1

Two introductory examples of unit conversion, done two ways.