[e-web] mathematics story | talk. 2 | infinity concept | eng. sarah hassan kamel

7/31/2019 [E-WEB] Mathematics Story | Talk. 2 | Infinity Concept | Eng. Sarah Hassan Kamel

1/36

Speaker:

Sara Hassan KamelTeaching Assistant

Engineering Mathematics Dept.


2/36

The Concept of Infinity

How would YOU describe INFINITY?


3/36

The number of sand grains?

Nope!


4/36

The Universe perhaps?

Maybe!


5/36

Time?


6/36

The number of points on a

straight line


7/36

The number of harmonics needed

to represent a square functionexactly?


8/36

The Story of Infinity


9/36

The Ancient Greeks

Greece (600 B.C.)

The Greeks were the first to acknowledge

the notion of infinity.

There is no smallest among the small andno largest among the large; but always

something still smaller and something still

larger. Anaxagorus (500 428 B.C)

Although the Greeks acknowledged theinfinite, they could not confrontit!

It brought about


10/36


11/36

Zenos Paradox

Zeno argued that:

Motion is impossible!

How so?

If an object moves in a straight line

from point 0 to point A, then it

must first reach A/2. But before it

can reach A/2, it must first reachA/4. Ad infinitum.

Latin for: to infinity

Zeno of Elea490 430 B.C.


12/36

Zenos Paradox

0 11/21/4

1/8


13/36

Zenos Paradox

Heres another perception of the paradox:


14/36

Zenos Paradox

The Greeks failed to resolve the paradox.

They regarded the limit process

1/2 + 1/4 + 1/8 + . = 1

as a potentially infinite process.

There are two kinds of infinite:

Potential Infinite: Something that couldcontinue on, were effort to be applied.

Actual Infinite: Something that is actuallyinfinite in extent. Like the set {1,2,3,} which

actually contains an infinite number of digits.


15/36

Zenos Paradox

The Greeks could not accept the

existence of an actual infinite.

The infinite has potentialexistenceThere will not be an

actual infinite - Aristotle


16/36

The Symbol of Infinity

John Wallis (1616 1703)

English Mathematician

Introduced the symbol for infinity

in his book De sectionibus conicis The symbol is said to be derived

from the Etruscan symbol for 1000,

sometimes used to mean many.


17/36


18/36

Georg Cantor (1845 1918)

Born in St. Petersburg, Russia

Moved to Germany in 1865

University of Berlin

Studied with Weierstrass, Kronecker

and others.

Graduated in 1867 with a doctorate in

Number Theory Worked as a Math teacher in a girls

school in 1869


19/36

Georg Cantor (1845 1918)

1869: Appointed a position at the University of

Halle

1873: Promoted to Extraordinary Professor

Started his work on Infinity

Communicated with Richard Dedekind

1874: Married Vally Guttmann

Spent part of his honeymoon in Switzerland discussing

mathematics with Dedekind


20/36

Cantors work on Infinity

Cantor discovered that there wasnt just

oneinfinity, but manyinfinities!

1234

5...

.

10203040

50...

.

...

.

So although

the set {10,20, 30,}seemssmaller, bothsets are

infinitely theSAME SIZE


21/36

Cantors work on Infinity

What about the set of fractions? Is it the same size or larger?


22/36

Cantors Set Theory

Unlike other areas of mathematics, the

Set Theorydid not evolve through theideas of many mathematicians, but is

the creation of one manalone: Cantor!


23/36

Cantors Set Theory

Cantor was the first to mathematically

prove that the set of Real Numbers was

larger than any other countable set

(like the Natural Numbers). The question that Cantor was unable to

answer, was whether there was a set

whose size was in between these twoThe Continuum Hypothesis


24/36

Cantors Struggle

Cantors work received harsh criticism

from his contemporaries.

Poincar:Grave disease!

Kronecker: Corruption of youth!

Kronecker Weyl Poincar Wittgenstein


25/36

Cantors Struggle

Cantor eventually fell into depression.

He spent time at Halle Universitys

sanatorium.


26/36

Cantors Commemoration

Cantor was awarded the Sylvester

Medal by the Royal Society


27/36

Cantors Commemoration

David Hilbert defended Cantor from his

critics saying:

"No one shall expel us from the Paradise

that Cantor has created."


28/36

Gerhard Geyers Science

Cube Monument Placed on the Eastern side of

Nietlebener St., Halle, Germany

Each side of the cube features one of

four famous scientists: Friedrich August Wolf

Georg Cantor

Georg Ernst Stahl Victor Klemperer


29/36


30/36

Gerhard Geyers Monument


31/36

A World Without Infinity


32/36

Uses of Infinity (to name a few):

Sometimes infinite result of a physical

quantity may point out the limitations of a

certain theory.

Many approximations,equations use infinite

series.

Infinitesimal Calculus(Leibniz & Newton)

Differentiation &Integration!


33/36


34/36

Cosmology

Black Hole:

A point of zero volume,

hence infinite density.

They are invisible, but

detected by the effect of

gravity on the stars andgases around the black

hole.


35/36

References

An Episodic History of Mathematics,Mathematical Culture through ProblemSolving by Steven G. Krantz, 2006

To Infinity and Beyond, A CulturalHistory of the Infinite by Eli Maor, 1987

The Story of Maths, BBC Series

presented by Prof. Marcus de Sautoy


36/36

[e-web] mathematics story | talk. 2 | infinity concept | eng. sarah hassan kamel

Documents