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# Units and conversions

All can be measured by the standard of the capybara.
Everyone is lesser than or greater than the capybara.
Everything is taller or shorter than the capybara.

--, "Unit of Measure"

## Overview

• Gain familiarity with the units commonly used in Astronomy
• Practice doing some conversions between different units, and determining what units make sense to use

## Introduction

### Part 1: Units

In general, a number all by its self is useless. For example, knowing the weight of an object is 231 doesn't tell you anything. It could be a 231 kiloton ship. It could be a 231 pound man. It could be a 231 gram mouse (note grams is actually a measure of mass, not weight, but we usually use them interchangeably on Earth). Without the unit, you don't know anything meaningful. So, anytime you give a numerical answer, it should include a unit.

Sometimes, the units that go into a problem don't give a sensible answer. For example, it doesn't make much sense to talk about the size of a dorm room in square miles. Or for a more astronomical example, Newton's form of Kepler's 3rd law is very simple if you choose the right units: a3=MP2 will give you a length in astronomical units. This is fine for the distance between a planet and the Sun, but it wouldn't make much sense for the orbit of the space shuttle.

The most common set of units in introductory astronomy is the international system, or SI. It was started in France shortly after the French Revolution, with the invention of the metric system and the deposit of two platinum standards in the Archives de la République. The International Bureau of Weights and Measures "BIPM" still resides in France. More information on all the SI units and their current definition can be found at http://www.bipm.org/en/si/.

The "base units" you will most often deal with are the kilogram "kg", meter "m" and second "s". Other units are generally combinations of these. For example, force and weight are measured in Newtons "N", which is a kilogram meter per second squared: kg-m/s2. Energy is measured in Joules "J", which is a Newton-meter or kilogram meter squared per second squared.

In some cases, the SI units are not the most sensible ones. In that case, you will need to know the conversion factor. For example, the Sun is 149,600,000 km away from Earth. This is a rather large number to have to keep plugging in the equations, so astronomers usually prefer something simpler, like an astronomical unit. An AU is the average distance from the Earth to the Sun, so the Earth-Sun distance is 1 AU. The conversion is 1 AU = 149,600,000 km. Most physics and astronomy books have conversion factors in the appendix. There is also a good resource at the BIPM (link above) or at http://www.physlink.com/Reference/Index.cfm.

#### Conversion Factor

Once you know the conversion, you have to figure out how to use it. Most conversion factors are written as something = something else: 1 foot = 12 inches. You can divide both sides by one of those numbers to get the conversion factor:

This means you can multiply any other number by 1 foot/12inches (the conversion factor) since it equals 1. So, to convert from one unit to another, you simply multiply the number you want to convert by the conversion factor.

Example 1: Convert 24 inches to feet

Note how the inches cancel, leaving only feet as the unit.

Example 2: convert 3.5 feet to inches

Note how the conversion factor was turned upside down compared to the previous example so that the feet would cancel

Example 3: convert 60 miles per hour to miles per minute

1 hour = 60 minutes, so the conversion factor is 1hour/60minutes

again, the choice of which number in the conversion factor went on top was determined by which unit had to be cancelled.

If you need to convert multiple units, you simply multiply by more conversion factors.

Example 4: convert 60 miles per hour to km/s

• 1 km = 0.621 miles
• 1 hour = 3600 seconds

If one of the units is raised to a power, like square meters, you’ll have to raise the entire conversion factor to that power. If you have a square that is 1 foot on a side, the area is 1 foot * 1 foot = 1 foot2, or 12 in*12 in = 144 in2. So, 1 foot2 = 144 in2. It is usually safer to make the conversions before doing any multiplication if possible. That way you can't accidentally forget to square the conversion factor.

Example 5: convert 2 feet2 to inches2.

Example 6: convert 5miles2 to km2

Example 7: convert 3.2miles/hour2 to m/s2

• 1km = 1000m, and
• 1 km = 0.621 miles, so
• 1000 m = 0.621 miles

#### Order of Magnitude Check

Whenever you do a conversion it is important to check and see if the answer makes sense. The quickest way to do this is an order of magnitude check. An order of magnitude is a power of 10. For example, when converting from feet to inches, there are a little more than 10 inches in a foot, so you should expect your answer to be at least 10 times bigger (one order of magnitude), but not as much a 100 times bigger (2 orders of magnitude). If you start with 2.6 feet, your answer should be greater than 26, but less than 260. This might seem like an enormous range, but the most common errors in conversions are things like turning the conversion factor upside down (especially if you don't write down all your steps!)  If you did that for your 2.6 feet, you'd have 2.6/12 = 0.22, which isn't between 26 and 260, so it can't be right.

Example 8: do an order of magnitude check on example3; convert 60 mi/hr to mi/min.

• We didn't convert the miles, so there's nothing to check there. Since there are 60 min to an hour, that's one order of magnitude, or a factor of 10. Since time is on bottom, we expect the number to decrease by at least 10 times. So the answer should be within a order of magnitude of 6, or between 1 and 10. The answer we got was 1, so it checks!

Example 9: do an order of magnitude check on example 4; convert 60mi/hr to km/s.

• Km are only slightly smaller than miles, so you'd expect the conversion from miles to km to be the same order of magnitude.
• Seconds are more than 3 orders of magnitude bigger than hours, but you’re dividing by time, so you'd expect a number the number to become about 1000 times smaller.
• Put together, you have something that stays about the same divided by something 1000 times smaller, so you'd expect an answer around 1000 times smaller than 60, which is 0.06. This tells us we're looking for an answer between 0.01 and 0.1. Our answer was 0.027, so we didn't make any huge conversion errors.

Example 10: check example 7; convert 3.2 mi/hr2 to km/s2.

• meters are about 3 orders of magnitude bigger than miles.
• s2 is 7 orders of magnitude different from hr2.
• So, we have an increase of 3 orders of magnitude and a decrease of 7, giving an overall decrease of 4 orders of magnitude, or about 0.0003. Thus our answer should come out to something between 0.0001 and 0.001, which it did, so again we didn't make any huge errors.

Note an order of magnitude check will not catch close conversions (e.g. if you forget to convert miles to km). It will also not catch small errors like punching 3700 into your calculator instead of 3600. Additionally, you have to think about whether the thing you’re converting to is bigger or smaller than the thing you’re converting from.

#### A final note on working with units.

When adding and subtracting, the units have to be the same. It doesn't make any sense to add 10 seconds to 2 feet!  And expressing numbers as things like 2 feet 9 inches is awkward for calculations.

When multiplying and dividing, the units also multiply and divide. So if you travel 10 miles in 10 minutes, your speed was 10 miles/10minutes = 1 mile/minute.

### Part 2 : Scientific Notation

You may have noticed a lot of these numbers end up being very big or very small. Scientific notation make it much easier to deal with these numbers.

In scientific notation, numbers are expressed as a specific number, usually with only one digit before the decimal place, multiplied by 10 raised to some power. The power also indicates the number of places to move the decimal. For example, 2.63x102 = 2.63x(10*10) = 2.63*100 = 263, which is the same thing as moving the decimal +2 places. 2.63x10-2 = 2.63x(1/102) = 2.63(1/[10*10]) = 2.63/100 = 0.0263, or the same thing as moving the decimal –2 places. If you move the decimal point around, the power changes in the opposite direction: if you move the decimal 2 places to the right (you number gets bigger), the power decreases by 2. For example, 2.63x102 = 26.3x101 = 263x100 = 0.263x103.

The real power of scientific notation is the ability to quickly do calculations:

Move the decimal point so all your numbers have the same power of 10. Then add the numbers normally and keep the same power of 10.

Example 11: add 2.8x106 + 8.2x107

It is usually more obvious to keep all the numbers before the "x10" greater than 1, so start by changing the second number. We need the power to be 6, so the power needs to decrease by 1, which means making the number bigger by a factor of 10. 8.2x107 = 82x106.

2.8x106 + 82x106 = 84.8x106 = 8.48x107.

To avoid having to move the decimal again, you may prefer to make all the powers equal to the greatest power. To do this, you'd change the first number: 2.8x106 = 0.28x107.

0.28x107 = 8.2x107 = 8.48x107.

example 12: subtract 3.58x104 from 4.93x106

4.93x106 – 3.58x104 = 4.93x106 – 0.0358x106 = 4.89x106.

or

4.93x106 – 3.58x104 = 493x104 – 3.58x104 = 489x104 = 4.89x106.

example 13: 2.3x102 +4.72x101 –1.4x10-1

2.3x102 + 0.472x102 – 0.0014x102 = 2.7734x102

or

2300x10-1 + 472x10-1 – 1.4x10-1 = 2773.4x10-1 = 2.7734x102

#### Multiplication and Division

When multiplying and dividing in scientific notation, multiply and divide the numbers before the power normally, then add or subtract the exponent. For example, 2x102 * 3x103 = (2*3)x10(2+3) = 6x105.

Example 14: multiply 2.4x102 by 8.32x104

2.4x102 * 8.32x104 = (2.4*8.32)x10(2+4) = 2.0x106

example 15: divide 2. 4x102 by 8.32x104

2.4x102/8.32x104 = (2.4/8.32)x10(2-4) = 0.29x10-2 = 2.9x10-3.

### Part 3: Significant figures

As a general rule (and probably sufficient for most of your intro astronomy classes) your answer should never have more digits than the smallest number you put into your calculation. So in example 14, 2.4x102 only has two "significant" digits, so the answer should only have 2 significant digits. Any zeros used only to hold the decimal place don't count: 150 = 1.5x102 has 2 significant digits, but 1.50x102 has 3 because the zero is included in the scientific notation.

For multiplication and division, you should keep the smallest number of digits. If you are multiplying 3 numbers and two of them have 4 significant digits but one of them only has two, your answer should have two significant digits.

For addition and subtraction, you keep the figure farthest to the LEFT. In example 13, 2.3x102 has the farthest left decimal place, so the answer in significant digits should have been 2.8x102.

### Activities

Last updated: 2/10/12 by SAM