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