Scale Work Page

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But in beginning to understand scale, perhaps the best place to start is with length and its close relation, area. There are many websites[1] where you can explore and begin to grasp the incredible scale of lengths that exist around us. 

Before you visit one of these sites, you need to understand a scale concept called “order of magnitude.”   Formally defined,[2] an order of magnitude is “the class of scale of any amount, where each class contains values of a fixed ratio to the class preceding it. The ratio most commonly used is 10.”  For example, think of money.  A penny is one tenth (1/10) of a dime.  So a dime is one order of magnitude more valuable than a penny.  A dollar is worth 100 times as much as a penny.  So the dollar is two orders of magnitude (10x10 = 100) more valuable than a penny.  Conversely, a penny is one order of magnitude (1/10 or 0.1) less valuable than a dime and two orders of magnitude (1/100 or 0.01) less valuable than a dollar. 

In the examples above, the relative values of a penny to a dime and dollar were expressed as fractions or decimals.  But scientists most commonly express numbers using scientific notation.  Scientific notation is a way of writing numbers that allow you to express values too large or small to be conveniently written in fractional or decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers.[3]

For example, let's say the average height of a human is about 1.7 meters (about 5' 7").  For the sake of simplicity, let's round off 1.7 meters to the nearest power of 10, which is 1 meter.  We are not saying that the average height of a person is 1 meter, but rather the average height is closer to 1 meter than it is to 10 meters.  Similarly, rounding the height of an ant (about 8 x 10-4 meters) to the nearest power of ten results in 10-3 meters.  Another way of saying this is that the order of magnitude of the height of an ant is 10-3 meters. Now, if we compare the height of a human being (1 meters) with the height of an ant (10-3 meters), we come up with the ratio human height/ant height = 1/10-3.  So a human being is roughly 1,000 times (or 103 times) taller than an ant.  In other words, a human being is 3 orders of magnitude (3 powers of 10) taller than an ant.

That height comparison example could have been easily done with either decimals or scientific notation.  But consider these much more complex examples:

·         An electron's mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 26 kilograms. In scientific notation, this is written 9.1093826x10−31 kilograms (or kg).

·         A proton's mass is 0.000 000 000 000 000 000 000 000 001 672 6 kg or, in scientific notation,  1.6726×10−27 kg

How do the masses of an electron and proton compare?  Instead of (mis)counting all those zeros, the masses can be compared by comparing the exponents.  In this case, the proton’s 10−27 is four orders of magnitude larger than the electron’s 10−31 and so the proton is roughly four orders of magnitude (about 10,000 times) more massive than the electron.[4] 

Some other mass- and length-based orders of magnitude are shown in the table on the next page along with some practice problems to help you better understand orders of magnitude.

Orders of Magnitude Worksheet

1.      Round off the following numbers to the nearest power of ten

a.       The order of magnitude for 2.3 is ___________

b.      The order of magnitude for 0.04 is ___________

c.       The order of magnitude for 501 is ___________

d.      The order of magnitude for 499 is ___________

e.       The order of magnitude for 4.57 x 1012  is ___________

f.       The order of magnitude for 4.57 x 10-5  is ___________

Using values from the table, answer the following two questions:

2.      An amoeba is ____________ orders of magnitude (heavier/lighter) than a virus.

3.      The earth’s radius is about ___________  times (smaller/larger) than the sun’s radius.

4.      Light is a form of energy which can be described by either wavelength or frequency.  The frequency of an infra-red lamp averages about 4 x 1013 hertz.  The frequency of a UV-tanning lamp is about  8 x 1014 hertz.  Round each number to its nearest order of magnitude and then compare their orders of magnitudes.

5.      The density of solid carbon dioxide is 1.6 kg/L (kilograms per liter) while the density of carbon dioxide gas is 1.98 g/L (grams per liter) at room temperature.  Convert the densities for the solid and gas forms of CO2 to the same units and compare them using orders of magnitudes.

6.      The speed of sound at room temperature in dry air is 1,235 m/hr.  The speed of light is 299,792,458 meters per second. 

a.       Compare the two speeds using orders of magnitudes (careful to convert them to the same units).

b.      For this case, explain in your own words, why an orders of magnitude comparison might be more appropriate than comparing the actual numbers.

c.       You might have learned that to find the distance in miles to a lightning strike, you should count the seconds between a lightning flash and its thunder and divide by five.  If the lightning flash is so many orders of magnitude faster than its sound, why do you only divide by five?

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