Part Four: The World’s Most Important Arithmetic What Every Citizen Should Know About Our Planet
Copyright Randolph Femmer, All rights reserved.
that we literally interpret the world using this Unfortunately, we are so thoroughly trained in this type of mathematics In virtually all our public schooling we are taught a mathematics that applies to our daily lives. Grocery-Store Arithmetic GROCERY-STORE ARITHMETIC
the world’s most important arithmetic Dr. Albert Bartlett of the University of Colorado has called exponential mathematics There is a more powerful kind of mathematics called ‘ exponential mathematics. ‘
explosions nuclear detonations monetary inflation runaway growth of cancer cells and compounding interest rates Exponential mathematics describes :
the pH scale in chemistry the Richter earthquake scale in geology nuclear decay rates in radioactive atoms rates of resource consumption human population growth and dinoflagellate red-tides in the sea. Exponential mathematics also applies to:
Unfortunately, the same mathematics that describes a nuclear detonation also describes human population growth.
A graph of the fission events inside a nuclear detonation has a shape which is nearly identical to a graph of human population growth over the past 10,000 years. Is This Important?
Exponential mathematics is extremely For example: Exponential fission events inside an atomic bomb destroyed the city of Hiroshima, Japan at the end of World War II... and runaway monetary inflation can destroy an economic system... or topple a government.
Might exponential mathematics have planet-wide impacts?
Exponential number sequences help us better understand human impacts on the natural world in the decades ahead.
Fortunately, understanding the behavior of exponential number sequences can be achieved easily. Two or three riddles will be our key.
EExample: etc. EExample: etc. Arithmetic number sequences grow larger by repeated additions of like amounts. Grocery-Store Arithmetic
Example: etc. Example: etc. Arithmetic number sequences grow larger by repeated additions of like amounts. Grocery-Store Arithmetic
Now:
By repeated additions of $1000 per day. You will only receive the salary for 30 days Suppose you are offered a salary that grows LARGER arithmetically.
Like this What will be your total earnings for days 1 - 7? Answer: Day one: $1,000 Day two: $2,000 Day three: $3,000 Day four: $4,000
An Salary Day one: $1,000 Day two: $2,000 Day three: $3,000 Day four: $4,000 How much will you earn on day 30? Answer:
An Salary How much will you earn on day 30? Answer: Day one: $1,000 Day two: $2,000 Day three: $3,000 Day four: $4,000
What We Are Used To How much will you earn during the 30 days of your employ? Answer:
What We Are Used To We use them almost every day. Numbers in an arithmetic number sequence are easy to understand. We are used to them.
using a salary that grows The same riddle again Now:
Exponential number sequences grow by repeated multiplications by like amounts. Example: ,000...etc. Example: etc. ( Notice we multiplied by two each time.) ( Notice we multiplied by ten each time.)
An Salary
but your salary grows exponentially… by doubling each day. Imagine that you are offered a starting salary of one cent per day. Assume your employ lasts only 30 days An Salary
Initial Numbers Are Deceptively Small Day one: 1 cent Day two: 2 cents Day three: 4 cents Day four: 8 cents How much will you earn on day seven? Answer:
Initial Numbers Are Deceptively Small Day one: 1 cent Day two: 2 cents Day three: 4 cents Day four: 8 cents What will be your total earnings for the first week? Answer:
The Second Week Answer: How much will you earn on day ten?
The Second Week Answer: What will be your total earnings after two full weeks...(days )….?
This exponential salary begins with exceptionally-small numbers.
The growth of the numbers using “grocery-store” arithmetic was large -- and straight-forward -- right from the outset. The exponential salary, however, begins with numbers that are so small that they seem harmless or unimportant.
Suddenly Larger What is your salary for day 16? Answer: Notice the numbers are now somewhat larger.
Suddenly Answer: Notice the sudden increase after three weeks. How much are you paid for your work on day 21?
Disaster occurs in week four. What is your pay for day 28? Disaster In Week Four Answer:
How much do you earn on day 30? Disaster In Week Four Answer:
and that salary grows exponentially by doubling each day for thirty days.... If you are given a salary of one cent per day.... what is your total salary for the month?
What Is Your Month’s Total? Answer:
One cent, growing exponentially (by doubling each day for 30 days) will result in in one month.
Only a lucky few will ever earn a salary that grows by $1,000 per day. (Most employers would never accept such a salary arrangement.) An Salary?
It at least seems possible that someone’s busy and distracted boss -- somewhere An Salary? might agree to an exponential salary Why?
and
The initial numbers in an exponential number sequence are so small that they harmless or unimportant.
first grow slowly then suddenly explode into enormous values. Numbers that are extremely small at the outset...
By the time danger becomes apparent, it can be
like the detonation of the Hiroshima bomb. like the second salary... Human population growth over the past 10,000 years has been
Like The Hiroshima Bomb
Arithmetic number sequences produce graphs which are straight lines... ( “linear” ).
The ‘ J-Curve: ’ An Exponential Graph number sequences produce graphs called J-curves
Year In Billions
Year In Billions
A J-curve is the mathematical equivalent of a fire-alarm going off in a burning building.
It warns us of potentially- devastating effects no matter how small the numbers may seem at first.
A Million vs. a Billion Arithmetic number sequences: repeated additions Exponential number sequences: repeated multiplications Powerful, misleading, and deceptive Linear graphs vs. J-curves A mathematical “fire alarm”
wishes to thank and acknowledge Randolph Femmer who authored and developed this presentation