MATHEMATICS: SCIENCE OF NUMBERS

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MATHEMATICS SCIENCE OF NUMBERS
Mr. K. NASA/GRC/LTP
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1. What does the Title Imply?
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To begin, lets first think about each of the
words in the title. They form a type of
definition.
Definitions state the unknown in terms of the
known.
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Analyzes own internal structure
Theoretical
Science
Science
Uses information to model the Real world
Applied
Latin, scientia, means knowledge.
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3,5,6,11
0.5, 0.32, 1.333
x
?2, ?3, ?(any prime)
Real
?, e
Complex
x iy
Etc.
Imaginary
Numbers
iy (?-1)y
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Of (preposition) having to do with pertaining
to.
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Now look at the whole definition MATHEMATICS
SCIENCE of NUMBERS
What does the phrase, Science of Numbers, mean
when applied to Mathematics? What do you think??
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HERES WHAT I CAME UP WITH A science of numbers
is a body of knowledge that represents what we
know about numbers, the operations between them,
and their various properties. It also includes
techniques for adding new knowledge, and a system
of concepts that allows for interpretation.
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QUESTIONS What do you think is meant by concept,
interpretation, and technique? Is there an
analogy with language --- e.g., When I say a
word, such as, TREE, does anything come to
mind? What? Then --- When I write ?? ?, does
anything come to mind? Why or why not?
Latin exempli gratia
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HERES WHAT I THINK Technique is what we learn
in school. It is the know-how for writing and
solving equations. Concept refers to an
underlying idea, and interpretation refers to an
application of that idea to a real-world
situation. Concept and interpretation are also
taught, but take much longer to develop in a
persons mind. Concept is necessary
for interpretation, and interpretation is
necessary for putting mathematics to the best
possible use!
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2. Interpreting Numbers
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I now want to focus on interpretation. Solving an
equation but not being able to interpret the
result is like writing a grammatically correct
statement without knowing what you mean!
Does this give you a hint as to why the Rosetta
Stone is so important?
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We pause for an Historical Commercial The
Rosetta Stone
The Rosetta Stone was discovered in 1799 at
Rashid near the Nile Delta during a Napoleonic
military campaign. The stone appears to have been
part of a much older structure, demolished to
make way for new construction.
It contains copies of a decree in honour of
Ptolemy V Epiphanes, passed by the priests of
Egypt at Memphis around 200 B.C. The inscription
is duplicated in Egyptian hieroglyphic and
demotic characters, and in Greek. The Rosetta
Stone was the key in deciphering ancient Egyptian
writing. The Rosetta Stone is currently in the
British Museum, London.
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Now, back to our regularly scheduled program.
When we use mathematics in research and design,
interpreting our results is VITAL! ! !
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Otherwise ...
BOOM!!!!!
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SO . . . LETS PRACTICE INTERPRETING !
Lets begin with c 300,000 km/sec 186,000
mi./sec HOW MIGHT I GAIN SOME CONCEPT OF WHAT IT
WOULD BE LIKE TO MOVE THIS FAST?
?
How about trying a thought experiment?
NEAT! But WHATS a thought experiment?
Einstein asked this question at age 16. He spent
the rest of his life answering it ! ! !
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A thought experiment is a device invented by
Albert Einstein to enable the thinker to gain
some CONCEPT about situations that could not be
directly realized via experiment.
Poof!
Gedanken . . .
V C
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CAN YOU SUGGEST A THOUGHT EXPERIMENT THAT WOULD
ENABLE US TO GAIN A CONCEPT OF WHAT IT WOULD BE
LIKE TO TRAVEL AT THE SPEED OF LIGHT?
,
,
10 5 Year 53 minutes . . .
Dont worry about Relativity! Assume a
pre-relativity standpoint. HINT The speed of
light is about 10,000 times the earths speed
around the sun . . .
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HOW ABOUT TAKING A TRIP AROUND THE WORLD????
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HOW MANY TRIPS COULD I MAKE IN AN HOUR? A
MINUTE? A SECOND?
Terra Firma
v c
Latin for Solid Ground or literally, Firm
Earth
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At 300,000 km/sec, I would make 7.3 complete
trips per second, or 230 MILLION trips per year!
! !

Magellans voyage took 3 years, and he never
lived to see home. . .
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FOR COMPARISON . . . In a car (97 km/hr (60
mph)), you would circumnavigate the earth in 17.4
days. In a jet aircraft (800 km/hr (500 mph)),
you would circumnavigate the earth in 2.1
days. In the Space Shuttle (28,800 km/hr (18,000
mph)), you would circumnavigate the earth in 1.4
hours (85 minutes).
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3. Making Connections with New Concepts
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c 10,000 ? VE
?
c 300,000 km/sec
?
VE 30 km/sec
ANOTHER WAY to understand the speed of light is
to recognize that, in one year, I could travel a
distance of 9.5 TRILLION km, or about 6 TRILLION
miles straight out into space! ! !
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WOW! ! SO HOW BIG IS A TRILLION??
In dollars, about as big as 1/6th of the current
U.S. National Debt!
Can you do a thought experiment to get some
concept of just how big a trillion is?????
(Hint In the U.S., 1 trillion is 10 12, but in
the U.K., it is 10 18. Use the U.S. value )
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?
?
?
How old would I be if I lived for a trillion
seconds of ordinary clock time?
?
?
?
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WELL, TO BEGIN, HOW MANY SECONDS ARE THERE IN ONE
YEAR?
?
? ? ? ? ? ?
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Try . . . 31.6 Million . . .
So . . .
1 ? 10 12 seconds
32,000 yrs.
3.2 ? 10 7 seconds/year
190,000 years to count the Debt! ! !
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SO ... WHAT DO OUR GEOLOGISTS AND PALEONTOLOGISTS
(ETC.) TELL US WAS GOING ON 190,000 YEARS
AGO????? ANYONE CARE TO TRY?
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190,000 years ago, we are told that the Northern
Hemisphere was in the throes of a major Ice Age.
The Neanderthal People were still living in parts
of Europe. Wooly mammoths and sabertooth tigers
were roaming about the continental U.S. And NONE
of our so-called modern civilizations were even
being dreamt about yet, let alone being
built! 190,000 years into the future, who knows?
Maybe the Debt will finally be paid off!
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4. Final Thoughts . . .
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Well, weve come a LONG way From the concept of
a concept to the concept of a light year to
how-big-is-a-trillion to the National Debt and
into past geological time.
Good conceptual thinking often leads to new and
unexpected connections. This is one of the ways
that science and engineering progress! !
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THINK ABOUT IT What aspects of your schoolwork or
your daily life can you think of in this way? Can
you list some of the concepts that you regularly
use? Can you extend them and make new connections?
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EXERCISES 1. Make a scale model of the earth-moon
system. 2. Make a scale model of the solar
system. 3. Make a scale model of the entire
universe. At least, do the calculations. If you
run into difficulties, be certain that you can
explain them. 4. Make a scale model of geological
time. 5. Follow some ideas of your own, only . .
.
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Keep on Thinking!!! Its absolutely the BEST
thing that you can do with your time!!!
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For those interested in talking more, contact me
at joseph.c.kolecki_at_grc.nasa.gov