The Power of Zero

1. The Power of Zero Nikki Pinakidis James Scholar Project MATH 405 April 27th, 2012

2. USS Yorktown • Missile cruiser designed to withstand the strike of torpedo or the blast of a mine • 80,000 horsepower

3. USS Yorktown New Software • Controlled the engines • Zero lurked in the code that engineers failed to remove while installing software

4. Attempt to divide by zero • When the Yorktown’s computer system tried to divide by 0, the ship came to a halt • Took 3 hours to attach emergency controls to the engines and to bring it back into port • Took engineers 2 days to get rid of the zero, repair the engines, and ship the Yorktown back out to sea

5. What other powers does zero have…….?

6. Zero’s Twin: Infinity • Equal and opposite • Equally paradoxical and troubling • Biggest questions in science &religion: • NOTHINGNESS and • ETERNITY • VOID and • INFINITE

7. The birth of zero…

8. The beginnings of math…. • Desire to count sheep, keep track of property, and passage of time • Today it is hard to imagine life without the number zero, but a few centuries before the birth of Christ, life functioned perfectly fine without this number

9. Stone Age • One vs. Many Didn’t need a word to express the lack of something, didn’t assign a symbol to the absence of objects->simply just didn’t have any

10. The Abacus • Zero arose from the Babylonian style of counting • Machine to help count: abacus • Relies on sliding stones to keep track of amounts • The words calculus, calculate, and calcium all come from the Latin word for pebble: calculus • Adding=moving stones up and down • Stones in different columns have different values • Look at final position of stones and translate that into a number

11. Babylonian Counting • Sexagesimal system-based on the number 60 • Their system of numbering was an abacus inscribed symbolically onto a clay tablet • Each grouping of symbols represented a certain number of stones that had been moved on the abacus • Like each column on the abacus, each grouping had a different value, depending on its position • Each symbol could represent a multitude of different numbers--PROBLEM!

12. PROBLEM • The number 1 was written as: • The number 60 was written as: • Only difference: the was in the second position rather than the first

13. SOLUTIONto the PROBLEM • ZERO! • Zero finally appeared in the East, in the Fertile Crescent of present-day Iraq

15. 300 BC • Two slanted wedges represented an empty space,an empty column on the abacus • Acted as a placeholder • Symbol for blank space in abacus, a column where all the stones were at the bottom • No numerical value of its own yet • DIGIT, not a number • NO VALUE

16. Invention of zero being its own • Abacus way of counting spread • Eventually an unknown Hindu invented a symbol of his own to represent a column in which there were no beads • A dot called sunya (empty) • Not zero but represented space

17. “A number’s value comes from its place on the number line—from its position compared with other numbers”Seife, 16

18. Today we know zero has a definite numerical value of its own: • Separates the positive numbers from negative numbers • Even number • The integer that precedes one

19. Yet it is treated as a nonnumber! • Top of computer board: 0 comes after 9 • (not before the 1 where it belongs)

20. Or on a telephone keypad!

21. Fear of the number zero • Primal fear of void, chaos, and zero • Properties different from all other numbers: • Add a number to itself and it changes (1+1=2) • Violated Axiom of Archimedes: If you add something to itself enough times, it will exceed any other number in magnitude • But zero and zero is zero!

22. Multiplication by zero • Multiply by number >1, stretches the value • Multiply by number <1, shrinks the value • But multiply by 0, • ALWAYS 0 • shrinks to single point

23. Division by zero • Dividing by a number “undoes” the multiplication, but even though multiplying by zero shrinks the number to zero, dividing by zero is NOT possible! • Ex: 1 x 0 = 0 so 1/0 x 0 should equal 1 There is no such number that, when multiplied by zero, yields one.

24. What is 00 ? • It is commonly taught that any number to the zero power is 1, and zero to any power is 0. But if that is the case, what is zero to the zero power?

25. Student 1 Student 2

26. Student 3

28. Answer It is undefined (since yx as a function of 2 variables is not continuous at the origin).

31. Zeno’s Paradox: “The Achilles” • Swift Achilles can never catch up with a lumbering tortoise! • 3 CONDITIONS: • Achilles runs at 1 ft/sec • Tortoise runs at half that speed • The tortoise starts off a foot ahead of Achilles

32. Zeno’s Argument • In one second, Achilles catches up to where tortoise was • But by the time he reaches that point, the tortoise, which is also running, has moved ahead by half a foot • In half a second, Achilles makes up the half foot • But again, the tortoise has moved ahead, this time by a quarter foot • In a quarter second, Achilles has made up the distance • But the tortoise moves ahead in that time by an eighth of a foot… a sixteenth of a foot…a thirty-second of a foot… smaller and smaller distances • Achilles runs and runs but the tortoise scoots ahead each time; no matter how close Achilles gets • Achilles never catches up, the tortoise is always ahead!

33. START

34. After 1 second Achilles catches up to where the tortoise was, but now the tortoise has moved half of a foot

35. After a half second Achilles again catches up to where the tortoise was, but the tortoise also moves ahead by a quarter of a foot

36. After a quarter second • Same situation!

37. Zeno’s argument seemed to prove that Achilles would never catch up, but we know in the real world that Achilles would quickly run past the tortoise. But WHY? And WHEN?

39. The heart of Zeno’s paradox: • THE INFINITE! • Zeno takes continuous motion and divides it into an infinite number of tiny steps • Greeks assumed race would go on forever! • Even though steps get smaller and smaller • Race would never finish in finite time

40. How to approach infinity… • Greeks did not have zero! • It is possible to add infinite terms together to get a finite result • (if the terms being added together approach zero) • Converges to a finite number

41. Add up the distance Achilles runs • We see that the terms get closer to zero • Each term is like a step along a journey where the destination is zero

42. But Greeks resist zero! • Couldn’t understand this journey could ever have an end • To them, the numbers , , , , ,…… aren’t approaching anything; destination doesn’t exist • Terms are just getting smaller and smaller

43. Today… • We know the terms have a limit (approaches zero) • The journey has a destination • Once the journey has a destination, it is easy to ask how far away it is and how long it will take to get there

44. Definition of a Limit • If f is a function and b and L are numbers such that: • As x gets closer and closer to b but not equal to b then f(x) gets closer and closer to L • We say that the limit of f(x) as x approaches b is L • Notation:

45. Infinity • Indefinitely large number or amount • In our case, represents unbounded time • An idea that something never ends • Treated as a number but not a real number

46. In the same way that the steps that Achilles takes get smaller and smaller, and closer and closer to zero, the sum of these steps gets closer and closer to…. 2

47. What kept Greeks from discovering Calculus • Infinity, zero, and the concept of limits are all tied together in a bundle • Greek philosophers unable to untie this bundle couldn’t solve Zeno’s puzzle Greeks had a geometric way of thinking. Zero was a number that didn’t seem to make any geometric sense, so to include it, the Greeks would have to revamp their entire way of doing mathematics.

48. Geometric Series • Achilles puzzle is an example of a Geometric Series! • We have a sum, S= a+ ar2 + ar3 + ar4 + ar5 + … • Where a is our first term and r is the ratio • Sum diverges if |r|≥1 • If |r|≤1, sum converges to:

49. The importance of ZERO • So, as we can see, zero played an important role in the development of Calculus, as well as many other every day things we use today:

50. Works Cited • Reid, Constance. (1992). From Zero to Infinity: What Makes Numbers Interesting. Mathematical Association of America. • Seife, Charles. (2000). Zero: The Biography of a Dangerous Idea. New York: Penguin Group. • Su, Francis. "Zero to the Zero Power." Math Fun Facts. Harvey Mudd College Math Department. • What Does 0^0 (zero Raised to the Zeroth Power) Equal? Why Do Mathematicians and High School Teachers Disagree?" Ask a Mathematician / Ask a Physicist. WordPress, 4 Dec. 2010.