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2 Exercises: Quantities

Exercise 2. 1 Give three different named units for each of J.E. Thompson’s eight “common measurements”

Length
Surface, or area
Volume, or capacity
Weight, or force of gravity
Time
Angles, latitude and longitude
Temperature, or intensity of heat
Money, or value

Exercise 2. 2 Name five units for energy. Remember that “power” has a different dimension than energy.

Exercise 2. 4 Perhaps unsurprisingly, the power of an automobile engine has the dimension of power.

Imagine a car with a 100 horsepower engine. What is the equivalent value in units of watts.

light-bulb-e3

Exercise 2. 5 Chapter 2 quoted a list from Arithmetic for the practical man by J.E. Thompson (1931). He wrote:

Common measurements are of several kinds …. The chief of the common measurements are those of

Length

Surface, or area

Volume, or capacity

Weight, or force of gravity

Time

Angles, latitude and longitude

Temperature, or intensity of heat

Money, or value

Annotate each item of Thompson’s list with the dimension. Indicate whether it is a basic dimension or a derived dimension.

Exercise 2. 6 The Economist is considered one of the world’s leading news magazines. Here is part of an article entitled “Fear the deficit-populism doom loop” from Aug 24, 2025.

You are a finance minister after a decade of meagre economic growth, shocks from a financial crisis, a pandemic and sky-high energy prices. Public debt is worth more than your country’s gross domestic product, interest rates are at their highest in years and merely servicing outstanding debt is taking up an ever-greater share of tax revenue. Inflation is stubborn. America’s profligacy is satisfying much of the world’s appetite for government bonds, meaning your debt must pay more to attract investors. You lie awake worrying about how to make the numbers add up.

What are the dimensions of public debt?

What are the dimensions of gross domestic product? S T^-1

It’s quantitatively incorrect to compare public debt with gross domestic product.
1. What dimension must the quantity be that multiplies gross domestic product in order to convert it to a quantity that is directly comparable with public debt?
2. Let’s give the name B to the quantity in (i). You know the dimension of B, but to perform the multiplication you need to choose a value for B. Is there a value for B that is obviously correct? If you say “no,” explain why not. If you say “yes,” explain the reasoning behind its being obvious.

The Economist article continues:

This is the bind facing governments in much of the rich world. The average fiscal deficit in the OECD, a club of mostly rich countries, hit 4.6% of GDP last year, up from an average of 2.9% in the four years before the covid-19 pandemic; interest payments on outstanding debt came to 3.3% of GDP, only just below the amount Nato members hope to spend on defence by 2035. The political-science literature offers some comfort—austerity is not usually a barrier to re-election—but also a warning. Research shows a link between spending cuts and populist success. Indeed, in Britain, France and Germany such parties are already ascendant. Call it the deficit-populism doom loop: ministers face both big deficits and voter revolts, and there is little way of satisfying both the bond markets and the barbarians at the gate.

If fiscal deficit can be stated as a percentage of GDP, what does that say about the dimension of fiscal deficit?

Explain what’s different between fiscal deficit and debt.

Fiscal deficit can be reckoned as the sum of 1) interest payments on the debt (3.3%) and 2) the amount of other spending (on education, pensions, military, etc.) in excess of government income (taxes, etc.) By what percentage would spending need to decrease or taxes need to increase in order to eliminate the fiscal deficit?

Ready for testing

Assigned to DD

Exercise 2. 18 Everyone is in favor of reducing deaths due to pollution, illness and disease. It makes sense then to measure the reduction in deaths. In everyday speech and in the media, reduction in deaths is operationalized by lives saved.

BEFORE MOVING ON, answer question 1 without looking at the remainder of this exercise.

Do you see any flaws in measuring lives saved? What are they? Or, feel free to anticipate a claimed flaw and argue that it isn’t a flaw at all.

Blank space to keep you from looking ahead.

Suppose—a nasty hypothetical ethics problem—there are two people, A and B, whose lives you could save. But you can only save one of them. Whether this is a realistic scenario is not at issue here; we’ll see in later chapters that it often arises, with “you” replaced by “society.”

Further suppose that person A is twenty years old and person B is eighty years old.

Which of these actions would you take? a. Save the younger person. b. Save the older person. c. Flip a coin to determine whom to save.

We’re not suggesting that there is a right answer but we want you to think about whether some choices are better than others.

Explain (briefly) whether you think the coin flip option is better than either (a) or (b). Justify your answer.

For those of you who selected option (a), we want to provide a means to operationalize avoiding deaths. But, since everyone dies, let’s turn things around: operationalizing life. In particular, we will measure the impact of your intervention in terms of ☞ life-years ☜.

A catch is that its hard to know when the saved person will eventually die, so its hard to know at the decision time how many life-years will be gained. As an expedient, we’ll do the calculation based on life expectancy, a statistical measure of the average number of years that a cohort of people are expected to survive. For instance, for the 20-year old, the life-years will be “life expectancy at age 20.” Table 1 shows life expectancy as a function of age in the US.

Table E2. 1: Life expectancy in the US as it depends on age. Tabulated in 2021. Source: US CDC

Age	Overall	Males	Females
0	76.4	73.5	79.3
1	75.8	73.0	78.7
5	71.9	69.1	74.8
10	66.9	64.1	69.8
15	62.0	59.2	64.9
20	57.1	54.4	60.0
25	52.5	49.8	55.2
30	47.8	45.4	50.4
35	43.3	41.0	45.7
40	38.8	36.6	41.1
45	34.4	32.3	36.5
50	30.1	28.2	32.1
55	26.0	24.2	27.8
60	22.1	20.5	23.7
65	18.4	17.0	19.7
70	14.9	13.7	16.0
75	11.6	10.6	12.5
80	8.7	7.9	9.4
85	6.3	5.6	6.7
90	4.4	3.9	4.6
95	3.0	2.7	3.2
100	2.2	1.9	2.2

Let’s assume that our hypothetical 20-year old and 80-year old are both female.

How many life years would saving the 20-year old produce?

How many life years would saving the 80-year old produce?

prb-80-elxl

Suppose you could save multiple 80-year olds, but only one 20-year old. How many 80-year olds would correspond in life years to the 20-year old?

How comfortable or uncomfortable would you be making decisions on such a basis?

A “refined” operationalization of lives saved takes into account the “quality” of life. This is called “Quality Adjusted Life Years,” or QALYs for short.

Exercise 2. 9 Figure 2.1 displays child mortality versus gross domestic product (GDP) on a country-by-country basis. Child mortality is operationalized as the number of children who die between birth and their fifth birthday. It is a “rate” because the number given is a ratio of the number of deaths in this group to the total number of children in this group, but is given in units of “deaths per 1000 children.” That is, the ratio is dimensionless with dimension C C^-1 = [1], (Strictly speaking, the rate should be called “Under-five mortality.”)

GDP is often presented as the value of goods and services produced by a country in one year. The quantity shown here is also a rate, described in the label of the graph as “GDP per capita.” The dimension of the GDP is W T^-1 C^-1. The operationalization of GDP is defined in a way that makes it relatively easy to measure from government and commercial records. All goods and services for which money is exchanged are included. No goods and services for which money is not exchanged are included. Thus, GDP ignores the value provided by, for instance, home child care and home meal preparation. A country which switched from home-cooked meals to fast food would see a sharp increase in GDP but, perhaps, a decrease in nutrition. Paid work need not be productive to be included in GDP. No adjustment is made for losses due to accidents, disasters, or depreciation of buildings and other capital.

Another factor not included in GDP is the price level in each country. Price levels vary from country to country and also from year to year. In Figure 2.1, not only is a comparison made between countries, but also between years. GDP numbers used for international or inter-year comparison are often, as here, adjusted for the “price level” in each country in each year. The price level is determined by what it costs to buy a “market basket” of goods. And that price level varies over the years, a process we call “inflation.”

The GDP per capita presented in Figure 2.1 is adjusted for price level. The adjustment is often referred to as PPP, short for purchasing power parity. The units are 2011 “International Dollars,” a hypothetical currency for comparison between countries.

Figure E2. 1: Child mortality (deaths per 1000) versus GDP for 1960 and 2020. Source: Bolt and van Zanden – Maddison Project Database 2023 – with minor processing by [Our World in Data].

Find two countries (in the same year) that have almost the same GDP but very different mortality rates.

Find two countries (in the same year) that have almost the same mortality, but very different GDPs.

Describe the overall relationship between GDP and child mortality, as best you can discern from these data.

2.1 Drafty or too technical

Either promote or delete …

the following.

Exercise 2. 10

Next status step: Refine the draft

… to make it ready for testing.

Assigned to DTK

Operationalize the notion of the position of the sun in the sky. How do you turn it into a quantity? Is more than one quantity needed? Imagine that your instrumentation consists of

a compass
a stick
a ruler

You can start with this result from an AI search on “how to measure the position of the sun”:

You can measure the sun’s position using either physical methods or mathematical calculations, which determine its altitude (height above the horizon) and azimuth (compass direction). For a simple physical measurement, use a stick and measure the length of its shadow to find the altitude, as the tangent of the sun’s altitude angle is the ratio of the stick’s height to the shadow’s length.

File ID: croc-pull-table Luminous intensity

2.2 Drafty

Exercise 2. 11 Computing with units and dimensions

[I’m playing around with this. Can I get it simple enough that students can use it?]

v <- set_units(60, mi/hr)
g <- set_units(9.8, m/s2)
t <- set_units(1, s)
units(g*t^2)

$numerator
[1] "m"

$denominator
character(0)

attr(,"class")
[1] "symbolic_units"

v*t

0.01666667 [mi]

A routine task in quantitative analysis is to check that computations make dimensional sense and to avoid the pitfalls that often arise from named units whose dimensions are easy to confuse. There are many unfamiliar units used in the sciences. Even more are added by journalists who like to express areas as “football fields” and volumes as “olympic swimming pools.” In geophysics, river and ocean flows may be quantified as “Sverdrup” (Sv). Typical flow in the Amazon River is 0.2 Sv, compared to the Gulf Stream’s roughly 100 Sv. A journalist seeking to make the Amazon flow accessible in everyday terms could use “80 olympic swimming pools in the blink of an eye.” Calculations involving such quantities can easily go awry.

In this exercise, you are going to work with a computing notation for units and dimensions.

File ID: bird-run-roof

Exercise 2. 12 EXERCISE: Windy. How to operationalize windiness. Sea state, Beauford scale, Richter Scale, Tornado and hurricane scales.

File ID: avoid-put-socks

Exercise 2. 13 EXERCISE: GDP as an operationalization of the size of the economy, unemployment (which has a specific definition and can do odd things like go down when people give up on looking for work), labor-force participation, consumer confidence, market capitalization of a company

File ID: seaweed-bend-laundry

Exercise 2. 14 Turn this into an exercise about the physics of things like seconds-cubed.

Velocity, L T^-1 where the unit names—e.g. miles per hour, meters per second—serve as reminders of the fundamental dimensions involved.
Acceleration, L T^-2. You may be aware that “acceleration due to gravity” on the Earth’s surface is approximately 9.8 meters per second per second.
Force, M L T^-2. Unfortunately, the unit name “pound” gives no clue about the fundamental dimensions involved. And the unit name “kilogram” is not actually for force; it’s simply M
Energy, M L² T^-2.
Power, M L² T^-3, which is simply energy per unit time.
Electrical current is itself a fundamental dimensions, denoted I
Electrical voltage amounts to energy per unit electrical charge, that is M L² T^-3 I^-1.

In some cases, derived dimensions seem absurd to the untrained mind. After all, what kind of thing could possibly be the result of multiplying volume times surface area, or dividing dollars by time, or multiplying weight times itself? A sensible person might reasonably object: “I know what time is, but there is no such thing as ‘inverse time’ or ‘time squared’ or ‘time cubed.’ But to say”no such thing” is really to mean, “I haven’t encountered any such thing.” Eventually, however, you might. For instance, you will admit that the ages of a random group of people will vary one from another. In statistics, there is need to measure the amount of variation. For this purpose, statisticians have invented a quantity called the “variance.” The variance of ages has dimension T².

File ID: seal-tug-kitchen physics quantities

Exercise 2. 15 DO WE WANT TO BRING IN mass vs weight as different operationalizations?

Mass as operationalized on a scale by the force on a spring, or the force produced by a leveraged counterweight on a balance.

Similarly, you may recall that weight (M L T^-2) is not the same kind of thing as mass ({{ var dim.mass >}}). The idea of weight goes back to antiquity, but “mass” is much more recent, emerging from Newton’s work with gravity in the late 1600s.

File ID: kitten-bend-door

Exercise 2. 16 DEMONSTRATION to rewrite as an exercise. To illustrate an important style of calculation that we will not use in this book, we have an example: the link between spatial and arithmetic reasoning is provided by the simple problem of finding the mean of two values. Arithmetically, of course, the mean is the sum of the quantities divided by two. More generally, the mean of n quantities is the sum of them divided by n.

Spatially, the two quantities are represented by a single point in a two-dimensional coordinate space. To calculate the mean, we consider another geometrical object: a line. The line needed is the one that runs diagonally between the coordinate axes, that is, the set of points where the coordinates are equal. The mean is the position on that line closest to the single point defined by the two quantities.

If presented with a point in coordinate space and a line running through the space, most people can do a pretty good job finding the place on the line that is closest to the point. This operation is called “projection.” In Section 1 we will meet a technique for constructing functions as “linear combinations.” The technique is called “linear regression” and amounts, mathematically, to the projection of a point onto a …. Well, here we find ourselves having to work with objects in high-dimensional space. So it could be projection of a point onto a line, or onto a plane, or onto an entire three dimensional space, or onto a higher-dimensional space.

File ID: arm-bring-clock

DO WE REALLY NEED THIS? Revisions to SI system

Exercise 2. 17

SI system https://en.wikipedia.org/wiki/2019_revision_of_the_SI

File ID: doe-feel-closet

DO WE REALLY NEED SOMETHING about lumens, a dimension that we won’t use.

include _exercises/girl-lose-hamper.qmd