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The Power of Zero. Nikki Pinakidis James Scholar Project MATH 405 April 27 th , 2012. USS Yorktown. Missile cruiser designed to withstand the strike of torpedo or the blast of a mine 80,000 horsepower. USS Yorktown New Software. Controlled the engines - PowerPoint PPT Presentation
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Nikki Pinakidis
James Scholar Project
MATH 405
April 27th, 2012
The Power of Zero
Missile cruiser designed to withstand the strike of torpedo or
the blast of a mine80,000 horsepower
USS Yorktown
• Controlled the engines• Zero lurked in the code that engineers
failed to remove while installing software
USS Yorktown New Software
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
Attempt to divide by zero
What other powers does zero have…….?
Equal and oppositeEqually paradoxical and troublingBiggest questions in science &
religion:NOTHINGNESS andETERNITYVOID andINFINITE
Zero’s Twin: Infinity
The birth of zero…
oDesire to count sheep, keep track of property, and passage of time
oToday 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
The beginnings of math….
One vs. Many
Stone Age
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
Zero arose from the Babylonian style of countingMachine to help count: abacusRelies on sliding stones to keep track of
amountsThe words calculus, calculate, and calcium all
come from the Latin word for pebble: calculusAdding=moving stones up and downStones in different columns have different
valuesLook at final position of stones and translate
that into a number
The Abacus
• 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!
Babylonian Counting
The number 1 was written as: The number 60 was written as:
Only difference: the was in the second position rather than the first
PROBLEM
ZERO!Zero finally appeared in the East, in the Fertile
Crescent of present-day Iraq
SOLUTION to the PROBLEM
• 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
300 BC
• 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
Invention of zero being its own
“A number’s value comes from its place on the number line—from its position compared with other numbers”
Seife, 16
Separates the positive numbers from negative numbers
Even number The integer that precedes
one
Today we know zero has a definite numerical value of
its own:
Top of computer board: 0 comes after 9(not before the 1 where it belongs)
Yet it is treated as a nonnumber!
Or on a telephone keypad!
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!
Fear of the number zero
Multiply by number >1, stretches the value
Multiply by number <1, shrinks the value
But multiply by 0, ALWAYS 0shrinks to single point
Multiplication 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 = 0so 1/0 x 0 should equal 1There is no such number that, when multiplied by zero, yields one.
Division by zero
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?
What is 00 ?
Student 1
Student 2
Student 3
Answer
It is undefined (since yx as a function of 2 variables is not continuous at the origin).
• 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
Zeno’s Paradox: “The Achilles”
o In one second, Achilles catches up to where tortoise waso But by the time he reaches that point, the tortoise,
which is also running, has moved ahead by half a footo In half a second, Achilles makes up the half footo But again, the tortoise has moved ahead, this time by a
quarter footo In a quarter second, Achilles has made up the distanceo 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
o Achilles runs and runs but the tortoise scoots ahead each time; no matter how close Achilles gets
o Achilles never catches up, the tortoise is always ahead!
Zeno’s Argument
START
After 1 second
Achilles catches up to where the tortoise was, but now the tortoise has moved half of a foot
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
Same situation!
After a quarter second
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?
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
The heart of Zeno’s paradox:
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
How to approach infinity…
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
• 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
But Greeks resist zero!
• 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
Today…
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:
• 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
Infinity
o 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
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
What kept Greeks from discovering Calculus
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.
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 ratioSum diverges if |r|≥1If |r|≤1, sum converges to:
Geometric Series
• 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:
The importance of ZERO
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.
Works Cited
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