ExponentialMathematics

The World’s MostImportant Arithmetic

ExponentialMathematics

I

The World’s MostImportant Arithmetic

We will see that exponential number sequences are powerful,self-fueling, selfmisleading, deceptive, counterintuitive, and often dangerous.

Exponential Mathematics"The World's Most Important Arithmetic"

pH scale in chemistry, chain reaction explosions, radioactive decayscale (MMS) used to quantify earthquakesand human population growth over a span of ten millennia.

Dr. Albert Bartlett, Professor of Physics andhas called the mathematics of the exponential function "In the pages that follow, we will contrast the “grocerywith the powerful, counterintuitive, misleading,thematics.

The mathematics of our traditionalmultiplication, and division

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We will see that exponential number sequences are powerful,fueling, self-amplifying, and self-intensifying, as well as

misleading, deceptive, counterintuitive, and often dangerous.

Exponential Mathematics - I"The World's Most Important Arithmetic"

Our curricula help develop studentin an ordinary day-to-daycomprises a relatively simple kind of "grocery-store" arithmetic that findsplication in our daily lives.erful type of mathematics exists, however,that has extraordinary impeconomics, and the modhighly literate societies, however, expertisein this exponential, logarithmic, or nonlinear mathematics receivesemphasis.

An understanding of exponentialtics is necessary, for example, to properlyappreciate human population growthtween 8,000 B.C. and todaythematics also applies to phenomena such asnuclear detonations, algal blooms,erating cancer cells, monetary inflation,

istry, chain reaction explosions, radioactive decay rates, the(MMS) used to quantify earthquakes, compounding interest rates, dino

and human population growth over a span of ten millennia.

Dr. Albert Bartlett, Professor of Physics and Astrophysics Emeritus at the Univematics of the exponential function "the world's most important arithmetic

In the pages that follow, we will contrast the “grocery-store” mathematics of our early schooling, counterintuitive, misleading, and deceptive characteristics of exponential ma

Linear Mathematics

e mathematics of our traditional schooling centers on operations such as addition, subtraction,multiplication, and division – those simple and non-sophisticated types of math that we might

Time

We will see that exponential number sequences are powerful,intensifying, as well as

misleading, deceptive, counterintuitive, and often dangerous.

"The World's Most Important Arithmetic"

Our curricula help develop student expertiseday mathematics thatsimple kind of "gro-

store" arithmetic that finds routine ap-our daily lives. A far more pow-

erful type of mathematics exists, however,that has extraordinary importance in science,

the modern world. Even inhighly literate societies, however, expertise

logarithmic, or non-linear mathematics receives too little

xponential mathema-, for example, to properly

human population growth be-today. This same ma-

thematics also applies to phenomena such astonations, algal blooms, prolif-

monetary inflation, thethe moment magnitude

, compounding interest rates, dinoflagellate red-tides,

at the University of Colorado,the world's most important arithmetic."

matics of our early schoolingand deceptive characteristics of exponential ma-

schooling centers on operations such as addition, subtraction,sophisticated types of math that we might

categorize as ordinary arithmetic. It turns out, however, that such simple math can be, in its way,highly dangerous.

How can "2 + 2" or "4 x 12" possibly be dangerous?mathematics of our daily lives.ter? While such mathematics clearly applbecause we teach it so thoroughly and so well

From childhood on, we are taught such “grocerysively, that our minds are literally wired to “interpret the world” using the mathematics of ourdaily lives. If we try, however,ponentially, we arrive at dangerously incorrect answersWith each passing year, and each new homesame mathematics that prepared their great grandparents for life in the 1930s and a different timein history.

Because we place so much emphasis on the skills of addition, subtraction, multiplication, anddivision and other traditional mathematics of the past,deal with (or to even perceive)

the critical REAL

the mathematics of the

Some number sequences grow in a traditional way, by repeatedly adding a like amount to a growing variable, and generate a linear graph

Linear graphs are characteristamounts, such as: 3…6...9…12…15…18…21… etc. In this example, the numbers are growingby repeated additions of three. When numbersgrowing arithmetically and when we graph such a sequence over time, the graph they produce,like the one above, is linear

arithmetic. It turns out, however, that such simple math can be, in its way,

How can "2 + 2" or "4 x 12" possibly be dangerous? Such understandings, after all, make up themathematics of our daily lives. Aren't such calculations a "fundamental" that all of us should mas

While such mathematics clearly applies to our role as consumers, it is dangerous preciselybecause we teach it so thoroughly and so well.

From childhood on, we are taught such “grocery-store” arithmetic so completely and so exclusively, that our minds are literally wired to “interpret the world” using the mathematics of our

If we try, however, to apply this everyday expertise to numbers that behave exdangerously incorrect answers. As a result, one of our dangers is this:

passing year, and each new homework set, we are wiring our students' brains with thesame mathematics that prepared their great grandparents for life in the 1930s and a different time

Because we place so much emphasis on the skills of addition, subtraction, multiplication, andtraditional mathematics of the past, we leave today's students

(or to even perceive)

REAL-WORLD mathematics of the twenty-first centuryhe mathematics of the EXPONENTIAL FUNCTION.

Linear Mathematics

Some number sequences grow in a traditional way, by repeatedly adding a like amount to a growa linear graph like the one below.

graphs are characteristic of number sequences that increase by repeated6...9…12…15…18…21… etc. In this example, the numbers are growing

by repeated additions of three. When numbers grow (or decline) in this way, we say that they areand when we graph such a sequence over time, the graph they produce,

– that is, a straight line.

arithmetic. It turns out, however, that such simple math can be, in its way,

Such understandings, after all, make up theAren't such calculations a "fundamental" that all of us should mas-

it is dangerous precisely

store” arithmetic so completely and so exclu-sively, that our minds are literally wired to “interpret the world” using the mathematics of our

expertise to numbers that behave ex-ult, one of our dangers is this:

work set, we are wiring our students' brains with thesame mathematics that prepared their great grandparents for life in the 1930s and a different time

Because we place so much emphasis on the skills of addition, subtraction, multiplication, andwe leave today's students ill-equipped to

first century -

Some number sequences grow in a traditional way, by repeatedly adding a like amount to a grow-

crease by repeated additions of like6...9…12…15…18…21… etc. In this example, the numbers are growing

in this way, we say that they areand when we graph such a sequence over time, the graph they produce,

Numbers that increase “exponentially” grow by repeatedample of an exponential sequenceeach number has been multiplieda J-curve like the one shown below

trajectories in their proper historical context. Two examples will help us contrast arithmeticversus exponential patterns of growth.

Example

Linear mathematics is illustrated by the following example: Suppose one is offered a contractwith a salary for thirty days ofcrease each day as follows: On day one we receive $1000 and on day two(Notice that we have just earned three thousand dollars over the first two days.) On day three ourpay is $3000; on day four we earn $4000, etc. Sinceadditions of like amounts ($1000) we ssalary grows steadily and predictably larger at a straightdollars per day. If this proceeds over a period of thirty days$30,000 while the total for all thirty days will amount to $465,000.

Example

Suppose that a mathematician suggests asking for a more humble salary that grows largerEXPONENTIALLY. In this case, one's initial salaryary of just one cent. However, for one’s work on day two, this increases to two cents. On daythree the pay for the entire day is four cents. (Notice that the total for the first three days is onlyseven cents.) Then, on day four, the pay climbs to eight cents and jumps again to sixteen cents on

Exponential Mathematics

Numbers that increase “exponentially” grow by repeated multiplications by like amounts. An exexponential sequence is 1...2...4...8...16...32...64.. etc. In this example, notice that

multiplied by two. When we graph an exponential progression, we obtaine the one shown below.

Notice that an exponential graph is shaped a little likethe letter “J” so that initially, it rises slowly from thex-axis like an airplane rising from a runway. Towardthe end of the data set, however,exponentially turn sharply upwardwe graph human population growth between 8000B.C. and 2000 A.D. (see addenda at close of thisdocument), we generate a graph like the one shownleft.

The mathematics that produces graphs of thiour topic in the four “Exponential” PDFsIV) in this set. We will see that exponential numbersequences are misleading, deceptivetive, exceedingly powerful, and frequently dangerous. To properly evaluate today's demographic andenvironmental issues, we must be able to analyzenumbers in an exponential context.appreciate the special properties of exponenprogressions in order to place humanity’s current

trajectories in their proper historical context. Two examples will help us contrast arithmeticversus exponential patterns of growth.

Example 1 - An "Arithmetic" Progression

mathematics is illustrated by the following example: Suppose one is offered a contractwith a salary for thirty days of work over a one-month period. Furthermore, this pay is to

each day as follows: On day one we receive $1000 and on day two(Notice that we have just earned three thousand dollars over the first two days.) On day three ourpay is $3000; on day four we earn $4000, etc. Since this salary increases by a pattern of

($1000) we say that it is growing ARITHMETICALLY

salary grows steadily and predictably larger at a straight-forward rate of an extrIf this proceeds over a period of thirty days, the pay for day thirty alone will be

$30,000 while the total for all thirty days will amount to $465,000.

Example 2 - An Exponential Alternative

Suppose that a mathematician suggests asking for a more humble salary that grows largerIn this case, one's initial salary will be painfully small with a total day one sal

ary of just one cent. However, for one’s work on day two, this increases to two cents. On daythree the pay for the entire day is four cents. (Notice that the total for the first three days is only

nts.) Then, on day four, the pay climbs to eight cents and jumps again to sixteen cents on

by like amounts. An ex-is 1...2...4...8...16...32...64.. etc. In this example, notice that

ponential progression, we obtain

Notice that an exponential graph is shaped a little likeit rises slowly from the

like an airplane rising from a runway. Towardthe end of the data set, however, numbers that grow

turn sharply upward like a rocket. Ifwe graph human population growth between 8000

(see addenda at close of thisa graph like the one shown

The mathematics that produces graphs of this sort is“Exponential” PDFs (I, II, III,

We will see that exponential numberdeceptive, counterintui-

, and frequently danger-To properly evaluate today's demographic and

environmental issues, we must be able to analyzenumbers in an exponential context. We must also

preciate the special properties of exponentialprogressions in order to place humanity’s current

trajectories in their proper historical context. Two examples will help us contrast arithmetic

mathematics is illustrated by the following example: Suppose one is offered a contractmonth period. Furthermore, this pay is to in-

each day as follows: On day one we receive $1000 and on day two we receive $2000.(Notice that we have just earned three thousand dollars over the first two days.) On day three our

this salary increases by a pattern of repeatedARITHMETICALLY. In this case, the

forward rate of an extra one thousand, the pay for day thirty alone will be

Suppose that a mathematician suggests asking for a more humble salary that grows largerwill be painfully small with a total day one sal-

ary of just one cent. However, for one’s work on day two, this increases to two cents. On daythree the pay for the entire day is four cents. (Notice that the total for the first three days is only

nts.) Then, on day four, the pay climbs to eight cents and jumps again to sixteen cents on

day five. Because the numbers in this progression are growing by repeated multiplications bylike amounts, we say that the pattern of growth is exponential.

Contrasting the Options

Let us now contrast the two options. We want to know which option will be more rewarding.How much will each pattern generate in the effective month? The linear or arithmetic progressionis exceptionally generous at the outset, but will it remain most generous by month's end? Sincesome months are thirty-one days long, how much difference will a day make?

During the first seven days, the exponential option generates a total of just $1.27 – or about twocents an hour. During the same seven days, however, the arithmetic option generates $28,000.Thus, by the end of the first week, the net result is: arithmetic growth = $28,000; exponentialgrowth = $1.27.

This disparity continues during week two. By the end of the first fourteen days, the arithmeticoption amounts to $105,000 while the exponential salary generates a total income of just $163.83– (averaging a little less than $1.50 an hour). Thus, nearly halfway through the month, the tallystands at: arithmetic growth = $105,000; exponential growth = $163.83.

At this point, the exponential progression seems to be a poor choice. Anyone who chose a salarythat grows exponentially, has, so far, very little to show for it. During the remainder of themonth, however, the results will be quite different.

The Remainder of the Month

The reader is invited to use a calculator to compare the two salaries over the remaining days ofthe month. Knowing that most of us are busy, however, we have calculated some of the results asfollows: On day twenty-five, the exponential option generates more than $167,000 as the single-day salary due on that day alone. Then:

Day 26 335,544.32Day 27 671,088.64Day 28 1,342,177.28Day 29 2,684,354.56Day 30 5,368,709.12

Initially the arithmetic alternative, with a thirty-day total of $465,000 looked quite attractive.Now, however, it becomes clear that the exponential option (that began with such tiny amounts)will reward one with more than ten million dollars ($10,737,418.23) during the same period.When was most of this exponential salary generated? When did most of the growth take place?

The values peaked explosively inthe closing stages of the sequence

In the mid-twentieth century, there was a popular blues piece entitled “What difference does aday make?” The song was about a love affair, but its title presents an interesting question in thecontext of a salary that grows exponentially. If a set of numbers grows exponentially, what dif-ference does a day make? If we choose a month that is thirty-one days long, the extra day willdouble our salary again to $10,737,418.24. (This will also cause the thirty-one-day TOTAL toALSO DOUBLE, growing to more than $21,000,000 dollars.)

Summarizing Exponential Patterns

Let us summarize several key observations: First, notice that although the numbers making upthe LINEAR sequence seem attractively large at the outset and present themselves in obviousterms, the initial numbers of the EXPONENTIAL data set are so small that they seem to be harmlessor unimportant.

This means that numbers that grow exponentiallycan be DECEPTIVE and misleading. Becausethe original numbers can be quite small, theyinvite us to suppose that they are innocuous andrequire little attention. We have just seen,however, a number set begin with one cent andthen turn itself into more than twenty-one milliondollars in just thirty-one days. Numbers thatbehave in this way can be exceedingly dangerousbecause first they lure us into complacency, andthen they hammer us.

We should also notice that the exponential salaryis still exceptionally modest even after two fullweeks. This fact underscores a second crucialcharacteristic: Most of the growth in an expo-nential progression occurs toward the end of theprogression. As the sequence proceeds, its num-bers grow explosively larger, piling up astronom-ically in the closing stages of the sequence.

Finally, we see that exponential sequences are exceedingly POWERFUL. We just saw one centtransformed into more than $21,000,000 in just thirty-one days, with most of these dollars pilingup in the closing stages of the progression. (And the sample exercise also showed a startlingtransformation from modest values to utter calamity in just two weeks.) This illustrates that

exponential growth can turn tiny numbers into exceedinglylarge numbers in an unexpectedly short time.

Because exponential sequences are both powerful and deceptive, they can also be extremelydangerous. Thus, three key adjectives serve as our summary so far: Exponential numbers behavein a way that is, simultaneously, (a) powerful, (b) deceptive, and (c) dangerous.

Events inside explosions and nuclear detonations are not linear / arithmetic events. Instead, theyare EXPONENTIAL events that, when graphed, produce a J-curve like those depicted earlier. Thusthe nuclear fission events inside the atomic weapon that was dropped on Hiroshima, Japan at theclose of World War II followed an exponential pattern, releasing enough energy to suddenly des-troy the city and kill tens of thousands of people. In one sense, the energy released by the atomicfission destroyed the city.

In another sense, however, it was the exponentialnature of the fission events that destroyed the city.

These characteristics should alarm us because human population growth over the millenniabetween 8000 B.C. and 2000 A.D. has been exponential (or even, as Joel Cohen points out, "hy-

perexponential"). In some of our presentations and PDFs we sketch humanjourney through history. When wepopulation in 8000 B.C. and ending in 2000, we obtain a classical JMitchell (1999) described it this way:

"The exponential growth model... descriOurs is a singular case; it is unlikehas ever sustained so much growth for so long."

We should find this disquieting because, in the strictly mathematical sense of the term, we may bein the closing stages of a detonation that is not unlike the detonation that fJapan in World War II. As this is written,treated from this exponential track, but in more than fifty nations, especially in the Middle Eastparts of Asia, and parts of Africa, popu

Graph above depicts human history to2011, and then shows U.N. mediumfertility population projections of 10billion by the end of this century

Notice that BOTH of these graphs of human population growth arenumbers are rocketing up-ward along the explosive yhas taken place in the last two hundred years, and with the bulk of that growth having ocpopulation mil-stone of two billioour numbers will have reachedby 2100. These graphs should be all the more sobering if oneon Hiroshima, Japan.

In some of our presentations and PDFs we sketch humanWhen we graph those data sets, beginning with earth'sand ending in 2000, we obtain a classical J-curve. Campbell, Reece, and

Mitchell (1999) described it this way:

"The exponential growth model... describes the population explosion in humans.Ours is a singular case; it is unlikely that any other population of large animalshas ever sustained so much growth for so long."

We should find this disquieting because, in the strictly mathematical sense of the term, we may bein the closing stages of a detonation that is not unlike the detonation that f

As this is written, most of the industrialized world appears to have retreated from this exponential track, but in more than fifty nations, especially in the Middle East

Africa, populations are still rocketing upward.

Graph above depicts human history to2011, and then shows U.N. medium-fertility population projections of 10billion by the end of this century.

This larger graph also depicts human history to2011, and then shows U.N. highlation projections to 15.8 billion

of these graphs of human population growth are. J-CURVES and that in both cases, ourward along the explosive y-axis. Also notice that essentially all of our growth

has taken place in the last two hundred years, and with the bulk of that growth having octone of two billion in 1930. United Nations medium projections in May 2011 estimate that

our numbers will have reached NINE billion by 2043 and high-fertility projections send us toby 2100. These graphs should be all the more sobering if one considers the effect that a J

In some of our presentations and PDFs we sketch humankind's demographic, beginning with earth's worldwide human

curve. Campbell, Reece, and

bes the population explosion in humans.large animals

We should find this disquieting because, in the strictly mathematical sense of the term, we may bein the closing stages of a detonation that is not unlike the detonation that flattened Hiroshima,

the industrialized world appears to have re-treated from this exponential track, but in more than fifty nations, especially in the Middle East,

This larger graph also depicts human history to2011, and then shows U.N. high-fertility popu-

15.8 billion by 2100.

and that in both cases, ouraxis. Also notice that essentially all of our growth

has taken place in the last two hundred years, and with the bulk of that growth having occurred since ourn in 1930. United Nations medium projections in May 2011 estimate that

fertility projections send us to 15.8 billionconsiders the effect that a J-curve once had

In recent decades, top scientists from around the world have written papers and issued formalwarnings about human population growthtrophic events share characteristic nonlinear behaviors [that] result in surprises that cannot be easily predicted” (Peters, et al., 2004). And a decade earlier, more than 1500 top scientists, including99 recipients of the Nobel prize isfollows:

tion of our environment, we must ac

“No more than one or a few decades remainfront will be lost and the prospects for huma

“We, the undersigned, senior members of the world's scientific community, hereby warn all humanity of what lies ahead. A great change in our stewardship of the earth and the life on it isrequired if vast human misery is ttrievably mutilated" (Kendall, et al., 1992).

* In a similar way, scholars repeatedly tried to warn government officials and reand the insufficiency of the city’s levee system for more than a decade before hurricane Katrina struck in 2005perhaps, the fact that governments and institutions have something less than a 100% track record in their responses to warnings.

In the years since the aboveadditional people to our planetand ninth billions are on-track to arrive between now andbillion by 2100.

A Collision Course

In recent decades, top scientists from around the world have written papers and issued formalwarnings about human population growth and our impacts on our planet. Fortrophic events share characteristic nonlinear behaviors [that] result in surprises that cannot be easily predicted” (Peters, et al., 2004). And a decade earlier, more than 1500 top scientists, including99 recipients of the Nobel prize issued an "Urgent warning to humanity" which we excerpt as

"Human beings and the natural world are on acollision course. Human activities inflict harshand often irreversible damage on the environment and on critical resources.many of our current practices put at seriousthe future that we wish for human society andthe plant and animal kingdoms, and may so alter the living world that it will be unable to sustain life in the manner that wemental changes are urgent if we are to avoid thecollision our present course will bring

The signatories then warn that "The earth is finite. Its ability to absorb wastes and destructiveeffluent is finite. Its ability to provide food andenergy is finite. Its ability to provide fing numbers of people is finite.approaching many of the earth's limits."

“Pressures resulting from unrestrained population growth put demands on the natural worldthat can overwhelm any esustainable future. If we are to halt the destruc

tion of our environment, we must accept limits to that growth.”

No more than one or a few decades remain before the chance to avert thefront will be lost and the prospects for humanity immeasurably diminished.”

We, the undersigned, senior members of the world's scientific community, hereby warn all humanity of what lies ahead. A great change in our stewardship of the earth and the life on it isrequired if vast human misery is to be avoided and our global home on this planet is not to be

(Kendall, et al., 1992).

ly tried to warn government officials and residents of New Orleans about category five hurricanesnsufficiency of the city’s levee system for more than a decade before hurricane Katrina struck in 2005

fact that governments and institutions have something less than a 100% track record in their responses to warnings.

above Kendall, et al. warning, we have added more thanpeople to our planet - and according to current U.N. projections, our seventh, eighth

track to arrive between now and 2041, with projections of

In recent decades, top scientists from around the world have written papers and issued formalFor example, “catas-

trophic events share characteristic nonlinear behaviors [that] result in surprises that cannot be eas-ily predicted” (Peters, et al., 2004). And a decade earlier, more than 1500 top scientists, including

" which we excerpt as

Human beings and the natural world are on acollision course. Human activities inflict harshand often irreversible damage on the environ-ment and on critical resources. If not checked,many of our current practices put at serious risk

for human society andand animal kingdoms, and may so al-

ter the living world that it will be unable to sus-tain life in the manner that we know. Funda-

urgent if we are to avoid then our present course will bring about.”

The signatories then warn that "The earth is fi-nite. Its ability to absorb wastes and destructive

Its ability to provide food andIts ability to provide for grow-

finite. And we are fastapproaching many of the earth's limits."

Pressures resulting from unrestrained popula-tion growth put demands on the natural worldthat can overwhelm any efforts to achieve a

If we are to halt the destruc-

before the chance to avert the threats we now con-

We, the undersigned, senior members of the world's scientific community, hereby warn all hu-manity of what lies ahead. A great change in our stewardship of the earth and the life on it is

o be avoided and our global home on this planet is not to be ire-

about category five hurricanesnsufficiency of the city’s levee system for more than a decade before hurricane Katrina struck in 2005 – underscoring,

fact that governments and institutions have something less than a 100% track record in their responses to warnings.

we have added more than 1.5 BILLION

and according to current U.N. projections, our seventh, eighth,projections of TEN to 15.8

A continuation of today’s demographic tidal wave may constitutethe greatest single risk that our species has ever undertaken.

Excerpted fromWhat Every Citizen Should Know About Our Planet

Used with permission.

Copyright 2011. Randolph Femmer.

This document is entirely free for use by scientists,students, and educator anywhere in the world.

The paperback version of Wecskaopis available from

M. Arman Publishing Fax: 386-951-1101

Sources and Cited References…pending…

Anson, 2009.Anson, 1996.

BARTLETT, A. 2005. The Essential Exponential.Campbell, et al., 1999Cohen, 1995Cohen and Tilman, 1996.Dobson, et al., 1997Duggins, 1980;Estes and Palmisano, 1974Mader, 1996Mill, J.S., 1848.Pimm, 2001.Prescott, et al., 1999.Raven, et al., 1986Wilson, 2002.

Former United States CIA director Jim Woolsey once observedthat “nature is not always going to behave in linear fashion [just]because our minds think that way.”

(From: Friedman, 2008)

Biospheric Literacy: Five PowerPoints / Five DaysOpen-courseware presentation and PDF resources are accessible at

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Biospheric Literacy: Five PowerPoints / Five Dayscourseware presentation and PDF resources are accessible at

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