OBJECTIVE To compare sizes of objects using logarithms or orders of magnitude.

Our universe presents us with enormous disparities in size—from incredibly tiny subatomic particles to unbelievably huge galaxies. How can we meaningfully compare these sizes? The 1977 short documentary film Powers of Ten, created by Ray and Charles Eames, uses zoom photography to visually compare the relative sizes of different objects in the universe. (This delightful film is available for viewing on several Internet sites.) We also encounter disparities in size in our everyday life—for example, our own income in comparison to trillion dollar national budgets (or deficits). But to bring these disparities in size down to earth, let's just try to fathom the wealth of the billionaire next door.

I. How Rich Is a Billionaire?

A millionaire is rich; he or she must have at least a million dollars in the bank. A billionaire is richer. Let's compare the relative wealth of a millionaire and a billionaire graphically and numerically.

1. Let's use the millionaire measuring tape to measure wealth. Each mark on the tape represents one million dollars. So the mark at $1$ represents one million dollars, the mark at $2$ represents two million dollars, and so on.
   
   
   
   
   1. At what number on the tape is the one billion dollar mark? _______
   2. If the marks on the tape are one inch apart, how many inches to the right of zero is the billion dollar mark? _______
   3. How many feet to the right of zero is the billion dollar mark? _______
2. The billion dollar mark on the millionaire measuring tape in Question 1 is way off the page. So let's use the billionaire ruler, on which the mark at $1$ represents one billion dollars.
   
   
   
   
   1. Where is the $100$ million dollar mark on the ruler?
   2. Place marks on the ruler at $1$ million and $2$ million dollars. Is it easy to visually distinguish the two marks?
   3. Place marks at one thousand and ten thousand dollars on the ruler.
3. The "millionaire measuring tape" and the "billionaire ruler" are not so easy to use.
   • The millionaire measuring tape is just too long to handle easily.
   • The billionaire ruler can't be easily used to measure small amounts.
   So let's try the "logarithm ruler." Each mark on the logarithm ruler is the logarithm (base $10$) of the number it represents. So the mark at $1$ represents $10^1$ dollars, the mark at $2$ represents $10^2$ dollars, and so on. Note that each mark on this ruler represents a number that is ten times the number of the preceding mark.
   
   
   
   
   1. Where is the one thousand dollar mark on the logarithm ruler? Where is the ten thousand dollar mark?
   2. Where are the one million and one billion dollar marks on the ruler?
   3. Is there a one dollar mark on the ruler?
   4. Is the logarithm ruler too long to handle easily? Can the ruler be easily used to measure small numbers of dollars?
4. In Questions 1–3 we used graphical methods to help us understand the difference in size between a million dollars and a billion dollars. Now let's use another method.
   
   

   Money under the mattress
   cbarnesphotography/Shutterstock.com 2009

   
   
   1. Suppose you have one million dollars under your mattress (you don't trust banks). Each day you take out one thousand dollars and spend it. For how many years would you be able to maintain this lifestyle before you become totally broke?
   2. If you have a billion dollars (instead of a million) in part (a), how many years would pass by before you would be broke?

II. How Small Is a Bacterium?

We've talked a lot about bacteria in this book. Let's compare the size of a bacterium with other, more familiar things. Recall that a centimeter (cm) is one hundredth of a meter (m), a millimeter (mm) is a thousandth of a meter, and a micrometer (µm) is a millionth of a meter.

Small child	$1$ m
Housefly	$1$ cm
Mite	$1$ mm
Bacterium	$1$ µm

1. To visually compare the size of the living things in the above list, let's use a meter stick. Mark the size of each one on the meter stick below.
   
   
2. Some of the marks you put on the meter stick in Question 1 are not easily distinguishable. Let's use the logarithmic meter stick shown below.
   
   
   1. Express the size (in meters) of each of the living things on the list on the preceding page in exponential notation.
   2. Label the size of each object on the logarithm meter stick.
   3. Where would a $10$-meter tree and a $100$-meter tree land on the logarithmic meter stick?

III. Orders of Magnitude

Sometimes we want to make rough size comparisons. To do this, we round off numbers to the nearest power of $10$. For example, for a man who is $6$ feet tall, we round his height to $10$ feet. In other words, the height of the man is closer to $10^1$ feet than to $10^0$ feet (or $1$ foot). For a ladybug that is $9.0 × 10^{-3}$ feet long, we round its length to $10^{-2}$ feet. A number rounded to the nearest power of $10$ is called an order of magnitude. The idea is that an object that is $10$ times or so larger than another object is in a different category or "order of magnitude."

To compare the height of a human being ($10^1$ feet) to the height of a ladybug ($10^{-2}$ feet), we find the ratio: $$\frac{\text{human height}}{\text{ladybug length}} = \frac{10^1}{10^{-2}} = 10^{1-(-2)} = 10^3$$

So a human being is roughly $10^3$ or $1000$ times larger than a ladybug. We say that a human being is $3$ orders of magnitude (three powers of $10$) greater than a ladybug.

1. Express the following in terms of order of magnitude.

   Object	Size (m)	Order of magnitude
   Gold atom	$2.8 × 10^{-10}$
   Bacterium	$3.5 × 10^{-6}$
   Mite	$2.0 × 10^{-3}$	$10^{-3}$
   Ladybug	$7.0 × 10^{-3}$	$10^{-2}$
   Human	$1.8 × 10^0$
   Earth	$1.3 × 10^7$	$10^7$
   Sun	$1.4 × 10^9$	$10^9$
   Betelgeuse	$8.3 × 10^{11}$

2. Perform the following order of magnitude comparisons using the results of Question 1.
   1. The sun is 2 orders of magnitude larger than the earth.
   2. A bacterium is _______ orders of magnitude larger than a gold atom.
   3. A lady bug is _______ orders of magnitude smaller than the earth.
   4. Betelgeuse is _______ orders of magnitude larger than a bacterium.