**Unit Conversions**

can also be called:

Dimensional analysis Unit analysis

Factor label method

# How do I change units on a number?

Unit conversions in the geosciences

Unit conversions in the geosciences

## Introduction to unit conversions

In the geosciences, we think about how the Earth works on a variety of scales. For example, the San Andreas Fault that runs nearly the length of California's coastline is over 1200 km long.## How do I do a unit conversion?

You can do any unit conversion if you follow a few simple steps. Although there is no single "right" way to do unit conversions, these steps provide one way to learn to do unit conversions. DO NOT SKIP ANY STEPS! Although it may seem tedious, working through unit conversions requires that each of these steps be followed so that you can be sure that you end up with what you want, especially when you are just starting out with learning to do unit conversions.

Below, you can download and print some tables for your use when doing unit conversions:

- If you are converting from one metric unit to another, this list of metric prefixes (Acrobat (PDF) 7kB Aug31 11) will be useful.
- When converting from metric to imperial (sometimes called English) units (or
*vice versa*), this conversion chart (Acrobat (PDF) 40kB Sep3 09) might be helpful.

When you do any unit conversion, you should always know what units you started with and what units you want to end up with. This is key to success at unit conversions.

## The Steps

The steps to successfully completing a unit conversion are outlined below. To illustrate the steps, lets use a geologic example:**In Southern California, slip on the San Andreas Fault is on the order of 25 km/Myr. How many cm does the San Andreas Fault move in one year?**

- Write (copy) out the units that you are given as a fraction.

- Write out the units that you want at the end of the conversion as a fraction:

- Determine appropriate conversion factors. Use tables in your textbook or download one of the tables listed above (for this particular example, you probably only need to know the prefixes for metric system (Acrobat (PDF) 7kB Aug31 11)).

For our example of movement on the San Andreas Fault, we need to convert 1 km to cm and Myr (Mega-years or Million years) to years, so with the handy guide to metric system prefixes, we need three conversion factors:- 1 km = 1,000 m
- 1 m = 100 cm
- 1 Myr = 1,000,000 yr

- Evaluate the appropriate arrangement for conversion factors. That is, copy the conversion factor(s) from step 3 into fraction form so that the units end up canceling. Remember that when you multiply fractions (as you will in step 6 below), you can cancel units
*ONLY*when they appear in the numerator of one fraction and the denominator of another.

We're going to have to do several steps in this conversion and we have to start somewhere. Let's start with converting km/year to cm/year. Because km is on top, we're going to want to arrange the conversion fraction so that km is in the bottom position (denominator) like so:Now we have what we want in the numerator but we still have Myr in the denominator. So, we need a conversion factor that has Myr in the numerator: - Set up the conversion by writing the fractions in a row with multiplication signs in between.
- Evaluate. Do the original units cancel so that you end up with what the question is asking for? If not, repeat steps 2 and 3 until they do!
- Now, lets do some arithmetic: To complete the conversion, we multiply all the numerators and then all the denominators (or multiply across the top and bottom).

If you are unsure about how I got the numbers here, you can review what is meant by "multiply across top and bottom" below:When fractions are multiplied, you need to keep track of numbers on the top (also called the numerator) and bottom (also called the denominator).- It's easier to keep track of these numbers if you write the fractions like this: . In this case, the numerator is 25 km and the denominator is 1 Myr.
- To multiply fractions, treat top and bottom numbers separately (multiply top numbers and then multiply bottom numbers).

- Write out your new numbers as a fraction (the top product as the numerator; the bottom product as the denominator)

- Reduce the fraction by dividing numerator by denominator once you have the top and bottom multiplied.
- EVALUATE (again). Is this a reasonable number?
Considering that plate motions are in the range of 1-10 cm/yr, 2.5 cm/yr is a reasonable answer!

### You can download and print a sheet with the unit conversion steps (Acrobat (PDF) 44kB Apr11 08)

to use while completing the practice problems.**Just a quick note about the metric system:**Although the US is one of the last hold outs on the imperial or English system of measurement (inches, miles, °F, etc.), the metric system is often much more intuitive because it is a system of base 10 numbers. The nice thing about dividing numbers in the metric system is that when you get to the end, you can actually cancel out the zeros in the fraction (remember, you have to cancel the same number on the top and bottom - there may be a few left over in either the numerator or the denominator). For example in the question above:

so that you only have to divide 25 by 10. If you are rusty about what the prefixes mean for the metric system, you can review the order of magnitude for a "femtogram" or "gigabyte" or any other SI unit using the metric prefixes (Acrobat (PDF) 7kB Aug31 11) document available earlier on the page.

## Where are unit conversions used in the geosciences?

**Plate tectonics**- converting rates of plate motion, etc.**Topographic maps**- converting scales**Rivers and Streams**- converting rates of flow, slope, etc.**Groundwater**- converting rates of flow**Glaciers**- converting rates of flow (or retreat), etc.**Geologic time**- converting time, rates of deposition, etc.

**And many other topics...**

## Next Steps

I'm ready to practice! (These problems have worked answers.)(See the links below for more help with unit conversions).

I still need more help!

## More Help with Unit Conversions

**Some tables of conversion factors**

**NIST (National Institute of Standards and Technology)**has an extensive list of weights and measures. There are specific tables for metric units of measure, US units of measure and US and metric equivalents.**SOSMath**has several tables of conversion factors for linear measure (length/distance), square (area) measure and http://www.sosmath.com/algebra/unitconv/unit4/unit4.html 'cubic (volume) measure'.**The geology department at Rutgers**has a list of important conversion factors in geology.- The engineering department at
**Auburn University**has a long list of conversion factors.

**Help with unit conversions (dimensional analysis)**

**Alabama Learning Exchange**has a handout about basic mathematical skills needed for the workplace.**Alysion.org**has a page with step by step instructions for dimensional analysis and some fun with dimensional analysis.**Oak Road Systems**(Stan Brown) has a discussion of converting units and how to go about it.

**Help with multiplying fractions**

**SOS Math**has an entire unit on fractions. These pages include a list of rules for the mathematics of fractions, reducing fractions, multiplying fractions.

^{This page was written and compiled by Dr. Jennifer M. Wenner, Geology Department, University of Wisconsin Oshkosh, and Dr. Eric M. Baer, Geology Program, Highline Community College}