4 1.3 Converting Units
It is sometimes important to convert between different types of units. Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km).
The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers.
Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer.
Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:
Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit.
Here’s another way to think about it:
Consider a simple example: how many cm are there in 4 meters? You may simply think there are 400cm in 4 meters. How did you make this determination? Well, if there are 100 cm in 1 m and there are 4 meters, then there are 4 × 100 = 400cm in 4 meters.
This is correct, of course, but it is informal. Let us formalize it in a way that can be applied more generally. We know that 1 m equals 100 cm:
1 m = 100 cm
In math, this expression is called an equality. The rules of algebra say that you can change (i.e., multiply or divide or add or subtract) the equality (as long as you don’t divide by zero) and the new expression will still be an equality. For example, if we divide both sides by 2, we get
1/2 m = 100/2 cm
We see that one-half of a meter equals 100/2, or fifty cm—something we also know to be true, so the above equation is still an equality. Going back to the original equality, suppose we divide both sides of the equation by 1 meter (number and unit):
1/1m = 100cm/1m
The expression is still an equality, by the rules of algebra. The left fraction equals 1. It has the same quantity in the numerator and the denominator, so it must equal 1. The quantities in the numerator and denominator cancel, both the number and the unit. When everything cancels in a fraction, the fraction reduces to 1:
1 = 100cm/1m
We have an expression, 100 cm / 1m, that equals 1. This is a strange way to write 1, but it makes sense: 100 cm equal 1 m, so the quantities in the numerator and denominator are the same quantity, just expressed with different units. The expression 100 cm / 1m is called a conversion factor, and it is used to formally change the unit of a quantity into another unit. (The process of converting units in such a formal fashion is sometimes called dimensional analysis or the factor label method.)
To see how this happens, let us start with the original quantity:
4 m
Now let us multiply this quantity by 1. When you multiply anyt