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MATHEMATICS: SCIENCE OF NUMBERS
Mr. K.NASA/GRC/LTP
1. What does the Title Imply?
To begin,let’s 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.
Science
Theoretical
Applied
Analyzes own
internal structure
Uses information to model the “Real” worldLatin, ‘scientia,’
means ‘knowledge.’
Science
x + iy
x
iy = (-1)y
3,5,6,11
0.5, 0.32, 1.333
2, 3, (any prime), e
Numbers
Of (preposition) = having to do with;
pertaining to.
What does the phrase,“Science of Numbers,” mean when applied to
Mathematics?
What do you think??
Now look at the whole definition: “MATHEMATICS:
SCIENCE of NUMBERS”
HERE’S 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.
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”
HERE’S 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 person’s mind. Concept is necessary for interpretation, and interpretation is necessary for putting mathematics to the best possible use!
2. Interpreting Numbers
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?
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.
Now, back to our regularly scheduled program.
When we use mathematics in research and design,
interpreting our results is VITAL! ! !
BOOM!!!!!
Otherwise ...
Let’s 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? ”
NEAT! But WHAT’S a thought experiment?
?
Einstein asked this question at age 16. He spent the rest of his life answering it ! ! !
SO . . . LET’S PRACTICE INTERPRETING !
How about trying a thought experiment?
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.
Gedanken . . .
V = C
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?
Don’t worry about Relativity! Assume a pre-relativity standpoint.
HINT: The speed of light is about 10,000 times the earth’s speed around
the sun . . .
10
–5 Y
ear
~ 5
3 m
inu
tes .
.
.
,,
HOW ABOUT TAKING A TRIP AROUND THE WORLD????
HOW MANY TRIPS COULD I MAKE IN AN HOUR?
A MINUTE? A SECOND?
Terra Firma
Latin for‘Solid
Ground’; or literally,
“Firm Earth”
v = c
At 300,000 km/sec, I would make 7.3 complete trips per second, or 230 MILLION trips per
year! ! !
Magellan’s voyage took 3 years, and he never lived to see home. . .
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).
3. Making Connections with New Concepts
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 about6 TRILLION miles straight out into space! ! !
c ~ 10,000 VE
VE ~ 30 km/sec
c = 300,000 km/sec
&
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 … )
How old would I beif I lived for a trillion seconds of ordinary
clock time?
?
?
?
WELL, TO BEGIN, HOW MANY SECONDS ARE THERE IN ONE YEAR?
??????
Try . . . 31.6 Million . . .
So . . .
1 10 12 seconds
3.2 10 7 seconds/year~ 32,000 yrs.
190,000 years to count the Debt! ! !
SO ... WHAT DO OUR GEOLOGISTS AND PALEONTOLOGISTS (ETC.) TELL US WAS GOING ON 190,000 YEARS
AGO?????
ANYONE CARE TO TRY?
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!
4. Final Thoughts . . .
Well, we’ve 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! !
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?
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 . . .
Keep on Thinking!!! It’s absolutely the
BEST thing that you can do with
your time!!!
