# Presentation: 257759

## MATHEMATICS: SCIENCE OF NUMBERS

***MATHEMATICS: SCIENCE OF NUMBERS***

***Mr. K.***
***NASA/GRC/LTP***

## 1. What does the Title Imply?

## To begin,

***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

***Science***

***Theoretical***

***Applied***

***Analyzes own internal structure***

***Uses information to model the *****“*****Real” world***

***Latin, ‘scientia,’ means ‘knowledge.’***

***Science***

## x + iy

***x + iy***

***x***

***iy = (************-1)y***

***3,5,6,11***

***0.5, 0.32, 1.333***

*********2, ************3, ************(any prime)***

*********, e ***

***Numbers***

- Complex

- Imaginary

- Real

- Etc.

## Of (preposition) = having to do with; pertaining to.

_***Of***_*** (preposition) = having to do with; pertaining to.***

## What does the phrase,

_***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:

***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

***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:

***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!

***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:

***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.***

## When we use mathematics in research and design, interpreting our results is VITAL! ! !

***Now, back to our regularly scheduled program.***

***When we use mathematics in research and design, ***_***interpreting***_*** our results is ***_***VITAL***_***! ! !***

## BOOM!!!!!

***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***_***?***

## Gedanken . . .

***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***

## ,

***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****** Year ~ 53 minutes . . .***

***,***

***,***

## HOW ABOUT TAKING A TRIP AROUND THE WORLD????

***HOW ABOUT TAKING A TRIP AROUND THE WORLD????***

## Terra Firma

***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***

## Magellan’s voyage took 3 years, and he never lived to see home. . .

***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 . . .

_***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 about***
***6 TRILLION miles****** straight out into space! ! !***

***c ~ 10,000 ************ V******E***

***V******E****** ~ 30 km/sec***

***c = 300,000 km/sec***

**& ******

****

## WOW! ! SO ... HOW BIG IS A TRILLION??

***In dollars, about as big as 1/6******th****** 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 be***
***if 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 . . .

***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?????

***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****** 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.

***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

_***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

_***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!!!

***Keep on Thinking!!! It’s ***_***absolutely***_*** the ***_***BEST***_*** thing that you can do with your time!!!***

## For those interested in talking more, contact me at:

***For those interested in talking more, contact me at:***
***joseph.c.kolecki@grc.nasa.gov ***