In elementary work, it is very doubtful if units should be omitted, especially if units are to be changed.
– Hassler Whitney
You carry out the work on an assigned exercise and finally arrive at the correct answer — correct to you, that is, but unfortunately, not according to your teacher. Your work is marked wrong. Why? Because you answered 2.65 and the required answer was $2.65. Your request for partial credit might even be denied.
Everyone has had that kind of experience in one form or another. At the end of each problem, you are expected to see to it that your answer is correctly labeled. And if you fail to do so, heaven help you.
At the same time, if your training has been anything like mine, you never use labels within arithmetic, algebraic or geometric statements: “2 cows + 3 cows = 5 cows” is only allowed in kindergarten and first grade. And the result of walking for 3 hours at 4 miles per hour may be written as “3 hr. × 4 
= 12 mi.” (with the hours canceling) in science class but never in math class.
Fortunately, although some teachers may continue to reject those labels, Princeton mathematician Hassler Whitney has shown in a pair of 1968 articles that using labels in mathematical statements is perfectly correct: that, in fact, labels can be treated in some of the same ways as the numbers they accompany. For example, Whitney offers as instances of the distributive and associative laws18 the following:
5 cakes + 2 cakes = (5 + 2) cakes = 7 cakes
2 yd = 2 × (3 ft) = (2 × 3) ft = 6 ft
Whitney’s article is replete with mathematical terminology — commutative and divisible semi-groups, idempotence, homomorphisms, Dedekind cuts and birays — but his message is clear: You can incorporate units in equations and still be mathematically correct.
You do not have to, but you can. In this chapter you will meet some examples of how this kind of dimensional analysis can contribute to your understanding of a number of situations.
3.1 Using dimensions in converting units
The most straightforward use of dimensional analysis is in the conversion from one unit system to another.
Such conversions are based on two basic rules of mathematics:
Combining those two rules into one, we have the general form for converting units:
The roles of the b’s and c’s in equations (3.1.2) and (3.1.3) are played by such equivalent pairs as 3 feet and 1 yard, 60 minutes and one hour, one inch and 2.54 centimeters, and (on a given day) 10 U. S. Dollars and 6.87 euros.19
Consider then some examples:
(1) You return home from a trip to France and find that you have a check for e45 that you need to cash. How much should it be worth at the rate of the previous paragraph?
Think about this carefully. First the math: Because the numerator and denominator of that fraction are equivalent, multiplying by that fraction is the same as multiplying by b or 1, since b = c. And multiplication by 1 gives an equivalent result; thus, the two sides of the equation are indeed equal. Next the calculation: You can cancel those two euro labels, just as you cancel factors when you multiply fractions. Thus the remaining label, $, is assigned to the result. (You can carry out 37.45 × 10 ÷ 6.87 by calculator with appropriate rounding.)
Notice that you choose the form of the fraction in such a way that unit labels will cancel. If you were to convert dollars to euros you would use that same equivalence but with euros in the numerator and dollars in the denominator.
(2) How many seconds are there in a week?
Again notice how the fractions are chosen to make them cancel with previous units. And notice too that singular and plural labels are considered the same.
(3) Unfortunately, the United States has retained the British (Imperial) measuring system,20 even after the British abandoned it themselves, so we continue to have to learn to convert between ours and the metric system. One of the conversion equivalents is exact, because it was made that way by international agreement: 2.54 centimeters = 1 inch.21 Using that relationship, derive the number of kilometers in one mile to four digit accuracy. 22
To identify the fractions to use in this conversion, start with 
and work in one direction toward miles and the other way toward kilometers. The result will be:
(4) Change 30 miles per hour to feet per second.
Note how, back in the third paragraph of this chapter we wrote 4 miles per hour as 
. We could also have written it as 
. The words “per” and “for” indicate division in mathematics in such expressions as 2 pounds for $3.59, which can be written as 
, a fraction that has the value 1. (It could also be written as 
.) Thus to make the conversion of this example you have:
so 30 miles per hour is equivalent to 44 feet per second.
Exercises 3.1
(3.1.1) Change the following units:
(a) $150 to euros using the exchange rate used in the text.
(b) €150 to dollars using the exchange rate you used in (a).
(c) €150 to dollars using the current exchange rate posted on the Web.
(3.1.2) The excellent 1997 thriller by John Burdett about the final days of British control of Hong Kong has the title, The Last Six Million Seconds. Convert this into the equivalent number of days, hours, minutes and seconds by paper and pencil calculation.
(3.1.3) Check your answer to (3.1.2) with the program of Panel 3.1.1.
Panel 3.1.1
Change Seconds to Days, Hours, Minutes and Seconds
(3.1.4) Dimensional analysis works well with units that are proportional, which requires that they all equal 0 at the same time. In the formula d = rt, for example, distance, rate and time all begin at zero: t = 0, r = 0, and d = 0. Although this is the case with most units, it is not true of the Fahrenheit, Celsius, Kelvin and Rankine temperature scales. When the Celsius temperature is 0o, for example, the Fahrenheit temperature is 32o, the Kelvin temperature is about 273o and the Rankine temperature is about 492o. These different zero values require adjustments in the conversion formulas. The formula F = 
C + 32 converts Celsius to Fahrenheit temperatures. (That 32 is added because 0oC = 32oF.) To answer exercises (a) and (b), you must search resources to learn the history of the formula:
(a) A German physicist, Daniel Fahrenheit , proposed his temperature scale first in 1724. How did he determine his scale divisions?
(b) Just 17 years later a Swedish astronomer, Anders Celsius, proposed his scale, first called the centigrade scale. Why was the name changed to Celsius in 1948?
(c) Both scales are based on changes in the states of water.
What are the differences between freezing and boiling of water of the two scales? Express them as a proportion: 
.
(d) Cross multiply to get an equation of the form ?F =?C.
(e) Check with the water freezing temperatures to show that this equation is not true. What could you add to make it true?
(f) Does the equation you have constructed in (e) hold for boiling temperatures?
(g) Solve the equation you have developed in (e) for F.
(h) Solve the equation in (e) for C.
(3.1.5) Two additional scales used in science (again use reference sources to answer):
(a) What is the Kelvin (K) scale and how does it relate to the Celsius (C) scale?
(b) Write an equation to convert K to C.
(c) What is absolute zero on the Celsius scale?
(d) What is the Rankine scale and how does it relate to the Fahrenheit scale?
(e) Write an equation relating the Rankine scale to the Fahrenheit scale.
(f) What is absolute zero on the Fahrenheit scale?
Panel 3.1.2
Change Fahrenheit Temperature to the Celsius, Kelvin and Rankin Scales
Panel 3.1.3
Change Celsius Temperature to the Fahrenheit, Kelvin and Rankin Scales
(3.1.6) Use Panel 3.1.2 to find:
(a) the boiling point of water in the Kelvin scale.
(b) the boiling point of water in the Rankine scale.
(3.1.7) Use Panel 3.1.3 to find:
(a) the boiling point of water in the Kelvin scale.
(b) the boiling point of water in the Rankine scale.
(3.1.8) Do your answers to 3.1.6 and 3.1.7 agree? (They should. If they do not, check your work.)
3.2 Using dimensions in calculations
You can calculate this volume in two different ways:
or
In each case you can quickly calculate the volume to be about 49600 cc. Notice that the labels again operate just as do variables: in. in = in.3 and cm . cm . cm = cm3 just as x . x . x = x3. Read in.3 as cubic inch and cm3 as cubic centimeter. This last leads in our final answer to the alternate and more commonly used abbreviation cc.
One thing that confuses many people is handling fractions within fractions, whether working with mathematical equations or equations involving dimensions. The same rules with which we began this chapter apply here as well. Simply choose an appropriate 
with b = c to multiply numerator and denominator to rid them of fractions. Thus we have:
The first of these simplifications demonstrates why you invert and multiply when dividing fractions:
It is important in such simplification to notice which bar governs the overall fraction. Be sure you see the difference between these two:
Now consider fractions with dimensions:
You plan a trip of 500 miles. Driving at an average speed of 50 miles per hour, how long should the trip take? This is a trivial question but it illustrates how dimensions are You know that distance formula rt = d (for rate times time equals distance), which is equivalent to 
= t. Thus:
or alternatively, you can write:
I finish this section with three real world examples of the use – or abuse – of dimensional analysis:
The Drake equation
A 1961 Green Bank, West Virginia conference of astronomers, physicists, biolo- gists, social scientists, and industry leaders came together to discuss the possibility of detecting intelligent life outside of the planet Earth. SETI, which stands for Search for Extraterrestrial Intelligence, was established as a scientific discipline at that con- ference. At the meeting one of participants, Frank Drake, proposed an equation designed to estimate the number of civilizations in the Milky Way galaxy. His equation, which has since come to be called the Drake equation or the Green Bank equation, is:
N = R∗ × fp × ne × fl × fi × fc × L (3.2.1)
where:
N = the number of civilizations in our galaxy with which communication might be possible;
R∗ = the average rate of star formation per year in our galaxy
fp = the fraction of those stars that have planetary systems
ne = the average number of planets that can potentially support life per star that has a planetary system
fl = the fraction of the above that actually go on to develop life at some point
fi = the fraction of the above that actually go on to develop intelligent life
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time such civilizations release detectable signals into space.
The following initial estimates were made (and have since been the subject of intense debate) for each of these factors:
R∗ = 10 stars formed/year
fp = 1 planetary system for every 2 stars formed
ne = 2 planets capable of developing life per planetary system
fl = each planet capable of developing life will develop life
fi = 1 planet with intelligent life per 100 planets
fc = 1 intelligent life form will be able to communicate per 100 intelligent life forms
L = 10,000 years of communication
Thus we have (abbreviating planetary system with ps):
and as a result, Drake’s planetary system equation (together with its questionable assumptions!) predicts that 10 intelligent life forms are out there in our galaxy able to communicate. Should we prepare for alien communication?
Fermi problems
The Nobel Prize-winning Italian physicist Enrico Fermi enjoyed making esti- mates. Such estimates were formerly called back-of-the-envelope problems, because they were often solved with a few pencil markings on (hopefully paper) napkins at the dinner table. In honor of the physicist, however, such estimates are today more often referred to as Fermi problems. But Fermi problems have additional qualities: they not only encourage approximation but they also encourage clearly identifying the assumptions you make. Here, from Wikipedia, is a classic Fermi problem solution that has been attributed to Fermi himself. His problem: “How many piano tuners are there in Chicago?”
A typical solution to this problem would involve multiplying together a series of estimates that would yield the correct answer if the estimates were correct. For example, we might make the following assumptions:
- There are approximately 5,000,000 people living in Chicago.
- Consider 2 people per household.
- Roughly one household in twenty have pianos that are tuned regularly.
- Pianos that are tuned regularly are tuned on average about once per year.
- It takes piano tuners about two hours to tune a piano, including travel time.
- Each piano tuner works eight hours a day, five days a week, and 50 weeks a year.
- We’ll let P represent the number of people, p pianos, h households, T piano tunings and t piano tuners.
From these assumptions we can compute that the number of piano tunings in a single year in Chicago, from assumptions 1-4, is:
We can similarly calculate that the average piano tuner performs
Dividing gives the approximate number of piano tuners necessary to accomplish this:
The Gimli Glider
Just how important the handling of such units is has been demonstrated twice re- cently when conversions between metric and British units caused serious problems. On September 23, 1999, the Mars Climate Orbiter crashed into that planet because of a 100 mile error in its computer program caused by this kind of mistake. Thus a 9-month space mission that cost over $325 million was wasted over a dimensional analysis error.
But that crash did not involve human lives. A truly frightening episode occurred on July 23, 1983, when an Air Canada jet with 61 passengers enroute from Montreal to Edmonton, lost all power. Fortunately, Captain Robert Pearson was an experienced glider pilot, which gave him familiarity with some flying techniques almost never used by commercial pilots. It became clear, however, that Flight 143 would not make it to Winnipeg. First Officer Maurice Quintal proposed his former airforce base at Gimli as a landing site but, unknown to him, the base had become a dragstrip and had decommissioned one of its runways. Furthermore, a “Family Day” was underway at the dragstrip, the area around the decommissioned runway was covered with cars and campers and the runway itself was being used to stage a race. The pilots barely had time to warn the public away from the airstrip.
Without power, the pilots had to lower the aircraft’s main landing gear by a gravity drop, but, due to the airflow, the nose wheel failed to lock into position. As soon as the wheels touched the runway, Pearson “stood on the brakes,” blowing out two of the aircraft’s tires. The unlocked nose wheel collapsed and was forced back into its well, causing the aircraft’s nose to scrape along the ground. The plane slammed into a guardrail which made the plane lose speed and stopped it from careening off the runway. By great good fortune none of the 61 passengers was seriously hurt.
What caused this accident? At the time of the incident, Canada was converting to the metric system, using litres (Canadian spelling) and kilograms instead of gallons and pounds. For the trip to Edmonton, a dipstick check indicated that there were 7,682 litres already in the tanks. A litre of jet fuel weighs 0.803 kg, so the correct calculation was:
Instead, the crew used the fact that a litre of jet fuel weighs 1.77 pounds, so they calculated:
If they had used this analysis, they would have been okay, but in that second calcu- lation they mislabeled the 13597 pounds as 13597 kg, and for that reason assumed that they had a great deal more fuel than they really had. Even after topping this up, they had scarcely enough fuel for half their flight distance.
An expensive airplane was destroyed, but the situation could have been far worse. Two heroic pilots saved those passengers from a tragic episode caused by a simple and avoidable miscalculation.
Exercises 3.2
(3.2.1) Individual bills of our US currency of any denomination ($1, $5, $10, etc.) weigh about 1 gram. There are 28.35 grams in an ounce and 16 ounces in a pound:
(a) How many dollar bills in a pound?
(b) What would be the weight in pounds of $1,000,000 in $20 bills?
(c) Assume for a moment that you’re planning to rob a large bank. You know that, on a busy day, the cashiers have a total of $1,200,000 in their drawers. You have also found out that the average value of the individual bills they are using is $2.83. You have a big sack in which you plan to make off with the Will you be able to carry it? (Note: This is an exercise and is not meant to encourage larceny.)
(3.2.2) Those same individual bills, when new and unwrinkled, are 2.61 inches wide, 6.14 inches long and .0043 inches thick. In the movies, you often see robbers or espionage agents carrying briefcases filled with currency. Estimate the value of $20 bills you could get into a briefcase whose inside dimensions are 20 inches by 13 inches by 3 inches.
(3.2.3) Using the thickness from exercise (3.2.2), determine how high a pile of $1 billion in dollar bills would be.
(3.2.4) A common approximate conversion factor is 1 kg = 2.2 pound. How does the Gimli Glider episode confirm this?
(3.2.5) Railroad tracks are laid with small spaces between track ends to allow for metal expansion. Suppose this is not done and a length of track a mile long fixed at both ends expands one foot over the entire length. If the track bowed evenly over the length to form a triangle as in the figure, what would be the height of that triangle (the dotted line on the figure)?
(a) Before calculating how high it would be off the surface of the solid, answer these questions:
(1) Could you crawl under the wire?
(2) Could you slide a one inch thick book under the wire?
(b) Calculate how high the wire would be above the sphere.
(c) Do the same calculation if the original sphere had been the size of a beach ball with a one foot radius.
(3.2.7) Mathematician Keith Devlin offers the following interesting Fermi problem: “To the best of our knowledge, our species Homo sapiens is 200,000 years old. For a celebration of human evolution, you decide to line up a group of people to represent your entire personal lineage in the species, with you at one end holding hands with your mother next to you, her mother holding hands with her mother, etc.(a) Roughly, how many people will you need, assuming each generation to be about 25 years.
(b) Let each individual occupy a width of 5 feet from clasped hand to clasped hand. How long will the line be?
(c) How long would it take you to go along the line and shake hands with all of your ancestors?”
(3.2.8) The delightful little book Guesstimation by Lawrence Weinstein and John Adam includes many Fermi Problems. As the title indicates, their exercises call for you to make estimates and your answers will differ based on those estimates. Here are some exercises based on a few of them:(a) If all the people on earth were crammed together, how much area would we require? You will need to estimate how much room (in square feet) each person will need to stand on (at least let them breathe!) and you can use 7,000,000,000 as the number of people on earth. Give your answer in square miles.
(b) Extend the problem in (a) to providing each family with a small plot of land on which to build a house. Now how much area will be needed? You’ll have to decide how many people you want to have in an average family and how large a lot to assign. Again give your answer in square miles. How does your answer compare with the area of Alaska (about 660,000 mi.2)?
3.3 Length, area and volume
Figure 3.3.1:
White-breasted
Panel 3.3.2.
red-breasted
White-breasted and red-breasted nuthatches (photos not to scale)24
Two small birds among those that visit feeders across the United States and Canada are the white-breasted nuthatch and the red-breasted nuthatch.25 These birds are quite similar in shape with the white-breasted nuthatch the larger of the two. It averages 5.75 inches in length and weighs just .74 ounces. The smaller red-breasted nuthatch is 4.5 inches long. Dividing 4.5 by 5.75 will show that the red-breast is about 
the length of the white-breast. You might at first think then that it should weigh about as much or about × .74 oz. ≈ .6 oz.
There is something wrong with that calculation because it turns out that the red-breasted nuthatch only weighs about .35 oz. You might know the reason that calculation is so far off if you studied school geometry, but this is important enough for us to rethink this error here.
There are two considerations involved:
- The linear, area and volume dimensions of similar figures are related in special ways.
2. Weight is closely related to the volume of similar figures.
Consideration 1. Similar figures
Similar figures play an important role in geometry. While congruent figures are figures whose size and shape are exactly the same, similar figures retain shape but not necessarily size. Figure 3.1 shows two similar plane figures.
Figure 3.3.3.
Similar Figures
One informal way to think about the similarity of two figures in a plane is by photographing one of them. You can then either enlarge or shrink that photograph to see if it fits exactly on the other. If they can be made to fit, the two figures are similar.26
Figure 3.3.4.
Similar and Non-Similar Figures
An important property of similar figures is that they have proportional dimensions and other measurements: sides for example, are proportional as well. In Figure 3.2, rectangles A and B are similar because 
the sides are in the same ratio and are proportional. B and C are not similar, because 
. Notice that C was constructed so that the sides are each one more than those of B, but that does not make them similar.
It is not so easy to talk about similarity when figures don’t have easy sides to measure. However, you could say that those two nuthatch species are similar by thinking of the two contained in boxes with proportional dimensions. We think of similarity in this way when we say that parent animals (including humans) are often similar in shape to their children.
Now you have enough background to consider a very important point about dimensions. Consider the following diagram for various values of s in Figure 3.3:
Figure 3.3.5.
Length, area and volume
Here is a table giving length, area and volume for a few values of s:
Suppose we choose two s-values from that table, say s1 = 5 and s2 = 7. Then we have for area, A, and volume, V, the following relationships:
or more generally,
Although we have only established this for squares and cubes, the same relationships hold true for any similar figures. Summing up: for similar figures, all linear dimensions are proportional, all areas are proportional to linear dimensions squared, and all volumes are proportional to linear dimensions cubed.
Consideration 2. Weight and volume
Now turn back to our nuthatch problem. Since weight is proportional to volume, we can rewrite the last proportion as:
This proportion is just what we need. We have the ratios for white-breasted nuthatch to red-breasted nuthatch in that last proportion:
and solving that proportion yields the more accurate weight of .35 oz. for the red-breasted nuthatch. Although it was 
as long as the white-breasted nuthatch, it weighs less than half as much. The exercises will show some other often unexpected consequences of these relationships.
Exercises 3.3
(3.3.1) Think carefully about the concept of similar figures and then answer the following questions:
(a) Are congruent figures similar?
(b) Are all line segments similar?
(c) Are all circles similar?
(d) Are all spheres similar?
(3.3.2) In William Wordsworth’s poem, My Heart Leaps Up When I Behold, appears the rather startling line, ”The Child is Father of the Man.” Interpret this line as it may relate to (approximate) physical similarity.
(3.3.3) I said that a property of similar figures is proportional dimensions and line lengths.
(a) Draw two non-similar parallelograms, one with sides twice as long as the other (thus with sides proportional) to show that proportional sides is not enough to establish similar
(b) Draw two pentagons that have all their sides equal but are neither similar nor (You may have to look up the technical meaning of regular.)
(3.3.4) The whale shark is about twice as long as the great white shark, and great white sharks weigh approximately 6000 pounds.
Figure 3.3.6.
Great white shark and Whale shark
(a) What would you predict would be the weight of the whale shark?
(b) The average weight of a whale shark is about 45,000 pounds. What are some possible reasons for the difference between this weight and your answer to (a)?
(3.3.5) A famous example of the length-area-volume relationships in literature occurs in Jonathan Swift’s Gulliver’s Travels. In the first of his four voyages Gulliver is ship- wrecked on the island of Lilliput, where he finds himself 12 times as tall as the Lilliputians.27 Overcoming some initial problems, Gulliver befriends the tiny people and they set out to nurture him. Immediately they run into problems.
(a) To clothe Gulliver, the area of cloth for the giant’s suit must be how many times that of the Lilliputians’ own suits?
(b) As if that isn’t bad enough, how many times the size of Lilliputian meals would feed the giant?
(c) To provide Gulliver’s sleeping arrangements, Swift writes, “The emperor gave orders to have a bed prepared for me. Six hundred beds of the common measure were brought in carriages; a hundred and fifty of their beds, sewn together, made up the breadth and length; and these were four double: which, however, kept me but very indifferently from the hardness of the floor.” Was the area of the bed satisfactory?
(d) How many mattresses high would have made Gulliver’s bed as comfortable as the Lilliputians’ beds were to them? (Think carefully before you answer.)
(3.3.6) When a model is offered, its scale (or scale factor) is often indicated. This is the ratio of linear dimensions of the model to linear dimensions of the concept being modeled. For example, one child’s model dump truck is scaled 1:32. That means that every inch on the model represents 32 inches on the real dump truck.28 (That’s almost an inch to a yard.) Here are some questions related to models:
(a) What are the corresponding area and volume ratios for that dump truck model?
(b) O-gauge model railroads have a scale of 1:48. One cubic inch of a model freight car’s interior would correspond to how many cubic feet of the real car?
(c) There is a simple way to solve (b) that you may not have used in that exercise. The ratio 1 in.:48 in. can also be written 1 in.:4 ft. Use this ratio to answer (b).
(d) A road map scale is 1 in.:11 mi. A city on this map has an area of .3 sq. What is that city’s true area?
(3.3.7) Another way to describe similar figures is to indicate the percent change going from one to the other. This must be interpreted carefully because, although scale and percent change are related, they are not the same. For example, the scale 1:3 is represented as a 200% increase.
Figure 3.3.7.
Percent change for linear, area and volume dimensions
translates linear percent change into the corresponding percent change for area and volume. Use this program to find the following:
(a) the area and volume percent change for a linear increase of 50%.
(b) the area and volume percent change for a linear decrease of 50%.
(c) the area and volume percent change for a scale factor of 1:2.
(d) by experimenting, the percent change in length that increases area by 100%. Round your answer to percent units.
(e) by experimenting, the percent change in length that doubles volume. Round your answer to tenths of a percent.
(f) Use your answer to part (d) to find the approximate change in volume when area is doubled.
(3.3.8) A prospective skydiver builds a model glider that carries a ten pound weight and sails it from the top of a building. It glides gently to the ground. “Aha,” he reasons, “At 150 pounds I am 15 times as heavy as that weight so I will build a glider to carry me with dimensions 15 times those of the model.” His reasoning is faulty and could lead to a serious accident. Why? (To answer, think about what holds a glider up and what forces it down.)
3.4 Measurement precision and scientific notation
In a famous Bible passage (I Kings: 23-26) a circular “molten sea” is constructed in the Temple of Solomon that is 10 cubits in diameter and 30 cubits in circumference. (A cubit is usually assumed to be about 18 inches.) This passage has often been cited as suggesting that, since C = πd for a circle, the Biblical value of π is 3.
At the opposite extreme, consider what might have happened if your calculator had been available to the author of this passage. The writer might then have used it to calculate the circumference as 31.41592654 cubits, this with a measurement unit that could hardly be considered accurate.
In an attempt to make sense of such situations, scientists have come up with some quite complicated rules for dealing with measurements. Even those rules do not respond to some situations, however, and common sense often serves at least equally well.
In most measurement situations numbers are rounded. If you are asked, for example, to measure the width of your desk in metric units as accurately as possible, you might try to do so in millimeters. You might find the width to be 623 mm, but then when you measure again it’s 619 mm. How can you express this? One way is to say that the width is 621±2 mm. Since 621 – 2 = 619 and 621 + 2 = 623, that would mean that the number of mm, x, would be in the range 619 ≤ x ≤ 623. That “ ± ” notation is, however, difficult to deal with in calculations. Instead, such variation is usually handled by rounding the measure, in this case to tens of millimeters, 620 mm.
You should see immediately that something is being lost here. Rounding in this way doesn’t tell the person who reads 620 mm, the fact that neither of your measurements was that size. All it does tell us is that the measurement is 620 ± 5 mm or between 615 mm and 625 mm.
To clarify what is going on here we need to talk about significant digits, the digits to which we rounded in this example. To count significant digits is straightforward. For numbers without a decimal point all digits are significant except leading and trailing zeros. For numbers with a decimal point only leading zeros are not significant, but trailing zeros are significant. Thus, each of the following numbers has three significant digits:
583 927,000 .000444 5030 2.09
However, extra digits to the right of the decimal point represent additional accuracy. For example, .10 is accurate to the nearest 100th, whereas .1 is accurate only to the nearest tenth. Notice two things about significant digits: (1) the number of significant digits is often different from the number of decimal digits, and (2) zeros between other significant digits count.
The measure 620 mm of our example has two significant digits. When we round those desk measurements to 620 mm, it is perfectly correct to say that we round to the nearest 10 mm, but it is more general to say that we round to two significant digits.
Significant digits are supposed to represent, as in our example, how precise is our measurement, but another problem arises. Suppose our measures had been 598 mm and 601 mm. Then we would have rounded to 600 mm but, instead of having two significant digits, that number has only one. Ways to express the fact that one of those trailing zeros is really significant have been introduced. The number could be written 600 or 600, but such usage has never been standardized.29
One way to avoid this ambiguity is to use scientific notation. Scientific notation, also called standard or exponential notation, expresses any number in our decimal system as:
a × 10b, with 1 ≤ a < 10 (3.4.1)
Instead of writing expressions in this form, however, they are often written a E b. This scientific notation is especially useful for expressing the values of very large and very small numbers. Thus we have the approximate speed of light, 300,000,000 meters per second as 3 E 8 meters per second. And the hydrogen atom is 37 picometers or .000 000 000 037 meters across; this can be represented as 3.7 E 11 meters. And the googol,30 a number name coined by nine-year old Milton Sirotta in 1938, is 1 followed by 100 zeros:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
This is considerably more simply represented as 10100 or 1 E 100.
But now back to our problem of representing the number 600 as accurate to two digits. Scientific notation allows this. We simply write 6.0 E 2. Be sure you see the value in being able to do this. The number 600 with one digit accuracy (6 E 2) represents all numbers in the range 550 ≤ x < 650; whereas 6.0 E 2 represents numbers in the range 595 ≤ x < 605. Similarly, 6.00 E 2 represents numbers in the range 599.5 ≤ x < 600.5. Although common sense must generally guide you in computations involving inexact or rounded numbers, an often quoted general rule is:
Round answers to the number of significant digits of the least accurate number in the computation.
Recall in this regard how quickly rounding errors caused chaotic problems in Chapter 1.
I close this section with a brief review of metric system names often seen in scientific writing.
Exercises 3.4
(3.4.1) That Bible passage about the circumference of the circle could be justified by the following:
(a) The 10 cubits diameter has how many significant digits?
(b) Using your calculator with the formula C=πd, what is the circumference of that circle?
(c) One of the rules often given for dealing with measurements says that your answer should have no more significant digits than your given data. You should then round your answer to (b) to the same number of significant digits as your answer to (a). If you obey this rule, what then should be the circumference of the cauldron.
(d) Does this conform to the Biblical passage?
(3.4.2) There is, however, another way of looking at this measurement:
(a) The number 10, considered as having only one significant digit, would mean a measurement between 5 and 15 cubits. What would be the range of measures if 10 were considered with two significant digits?
(c) Based on your answer to 3.4.1 (b), what should be the circumference to two significant digits?
Note: Problems like these make dealing with significant digits often a matter of individual judgment rather than of following set rules.
(3.4.3) Express the following in scientific notation:
(a) 2231.556
(b) 107
(c) .00000256
(d) 4
(3.4.4) Express the following as decimals:
(a) 2.776 E 7
(b) 2.776 E –7
(c) 33.7 × 1010
(d) 5.00 × 10−6
(e) .5000 × 10−6
(3.4.5) The key on your calculator used to represent the E in scientific notation is labeled EE to distinguish it from the alphabet letter E. Use this key to express the numbers in (a) and (b) in scientific notation and to evaluate them:
(a) 2.3 E –2 (b) 2.300 E –2
(c) Notice that your calculator does not help retain the accuracy indication of scientific notation. What do you need to do to your final answer to adjust the calculator’s decimal answer to (b) to retain this indication?
(3.4.6) Ever since the fourth or fifth grade when you began working with decimals, you have written in math classes that 4.0 = 4. But walk down the hall to the physics classroom and the instructor will tell you that those are very different. What is that difference?
(3.4.7) A well-known (but hopefully apocryphal) story tells of a museum guide showing visitors through the hall of ancient civilizations. In describing the ages of various artifacts, there is a kind of similarity to his announcements: “This one is 30,008 years old.” “This one was made just 5,008 years ago.” and “Here’s our oldest exhibit. It is 75,008 years old.” One of the visitors notices this consistency and asks where all those 8’s come from. The guide tells him that he learned the ages of those artifacts when he joined the museum eight years ago and he was keeping them up to date. How does this story relate to significant digits?
(3.4.8) It is worth noting that even scientific notation does not serve us when we talk about some numbers. A good example of this is the googolplex, a number described by Milton Sirotta to his uncle, mathematician Edward Kasner, as “one, followed by writing zeroes until you get tired.” Kasner refined this definition to 10googol.That is the same as
1010100
What happens when you try to express this number in scientific notation?
(3.4.9) Consider how long it would take just to write out the digits in a googolplex. Suppose you have a printer that will print 1000 zeros per second. That means that we must take 10100 ÷ 103 = 1097 seconds to complete the task.
(a) Convert this to years to get your answer.
(b) How long would it take if your machine printed a billion zeros per second? (You shouldn’t have to do the entire calculation again.)
(3.4.10) A function that increases in size rapidly is n!. Recall that for n ≥ 2, n! is the product whose factors are n and all the smaller positive integers down to 1. For example, 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720.
Stirling’s formula estimates the value for large factorials. It is:
Use Panel 3.4.1 to find the Stirling forumla estimate for the 20! you found in (b) and compare your result with your answer to that question. By how much do the two answers differ? What percent is this difference?
Panel 3.4.1
Approximation to n! by Stirling’s formula
(e) Stirling’s formula is less accurate for small values. Test it for values of N from 1 to 12. Which value of N has the largest error?
Panel 3.4.2.
Answers for Chapter 3.
references
20The United States is not quite alone in this rejection. Two other countries continue to use the Imperial System: Myanmar (Burma) and Liberia.
21Unfortunately, there are even problems with this metric equivalent of the inch. The surveyor’s inch in the United States is slightly different, not enough to cause problems with local calculations, however. Many measurement units have interesting histories. Examples: the Roman cubit was the length of the forearm from the elbow to the tip of the middle finger. You can imagine the cheating involved with that definition: to make measured distances small, hire a giant, to make them large, hire a dwarf. Later the English yard was legislated to be the distance from the tip of the nose to the end of the thumb of King Henry I. The inch was similarly the width of someone’s thumb.
22Note that four-digit accuracy, or in fact any degree of accuracy is allowed here because the
equivalents stated are exact and are not measurements. In this case the answer could even be given as 1.609344 km, not a very useful number. In fact most people use 1.6 km = 1 mi.
