Integer Arithmetic Floating Point Representation Floating Point Arithmetic
Part Four: The World’s Most Important Arithmetic What Every Citizen Should Know About Our Planet.
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Transcript of Part Four: The World’s Most Important Arithmetic What Every Citizen Should Know About Our Planet.
Part Four:
The World’s Most Important Arithmetic
What Every Citizen Should
Know About Our Planet
Copyright Randolph Femmer, 1999.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 Coloradohas 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 :
ApplicationsApplications
� 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 thatdescribes a nuclear detonation also
describeshuman population growth.
GrowthGrowth
A graph of the fission events inside
a nuclear detonation
has a shape which isnearly 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 mathematicshave planet-wide impacts?
Exponential number sequences
help usbetter 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.
� Example: 3...6...9...12...15...18...21...etc.� Example: 5...10...15...20...25...30...35...etc.
Arithmetic number sequences grow larger by repeated additions of like amounts.
Grocery-Store Arithmetic
� Example: 3...6...9...12...15...18...21...etc.� Example: 5...10...15...20...25...30...35...etc.
Arithmetic number sequences grow larger by repeated additions of like amounts.
Grocery-Store Arithmetic
Push Here
Now:
By repeated additions of$1000 per day.
You will only receive the salary for 30 days.
2005Suppose you are offered
a salary that grows LARGER arithmetically.
An Riddle An Riddle
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 SalaryAn 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 SalaryAn Salary
How much will you earn on day 30?
Answer: Push Here
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 sequenceare easy to understand.
We are usedto them.
using a salary that grows
The same riddleagain
Now:
Exponential number sequences grow by repeated multiplications by like amounts.
Example: 1...10...100...1000...10,000...etc.
Example: 1...2...4...8...16...32...64...etc.( Notice we multiplied by two each time.)
( Notice we multiplied by ten each time.)
An Salary An Salary
00
An SalaryAn Salary
Push Here
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....
2005
An SalaryAn Salary
Initial Numbers Are Deceptively Small
Day one: 1 centDay two: 2 centsDay three: 4 centsDay four: 8 cents
How much will you earn on day seven?
Answer:
Initial Numbers Are Deceptively Small
Day one: 1 centDay two: 2 centsDay three: 4 centsDay 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 1 - 14)….?
Click Here
This exponential salary begins with exceptionally-small numbers.
Deceptive and MisleadingDeceptive and Misleading
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 increaseafter 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 earnon 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.
...Never......Never...
(Most employers would never accept such a salary arrangement.)
An Salary? An Salary?
It at least seems possible that someone’s busy and distracted boss
-- somewhere
An Salary?
mightagree to an exponential salary
Why?
...MaybeMaybe...
Because exponential growth isBecause exponential growth isnot only powerful, it is alsonot only powerful, it is also
and
The initial numbers in an exponentialnumber sequence are so small that they
harmlessor
unimportant.
Push Here
first grow slowly
then suddenly explode into
enormous values.
Numbers that are extremely small at the outset...
The devastating effects of numbers that grow exponentiallyexponentially occur LATELATE in the sequence.
By the time danger becomes apparent, it can be
like the detonation of the Hiroshima bomb.
like thesecond salary...
Human population growth over the
past 10,000 years has been
Like The Hiroshima Bomb
Push Here
A Linear GraphA Linear Graph
Arithmetic number sequences producegraphs which are straight lines...
( “linear” ).
The ‘ J-Curve: ’An Exponential Graph
number sequences producegraphs called
J-curves
19991999
18001800
16501650
190019001 1 A.DA.D..
Year
In Billions
Year
In Billions
Click Here
A Mathematical Fire-AlarmA Mathematical Fire-Alarm
A J-curve is the mathematical equivalent of a fire-alarm going
off in a burning building.
It warns usof
potentially-devastating
effects
no matter how small the
numbers may seem at first.
A Mathematical Fire-AlarmA Mathematical Fire-Alarm
Click Here: 1
Click Here: 2
A Mathematical Fire-AlarmA Mathematical Fire-Alarm
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
