irrational numbers. incommensurability egyptiona and babylonians calculated square roots these were...

IRRATIONAL NUMBERS

INCOMMENSURABILITY

• Egyptiona and Babylonians calculated square roots• These were approximated• Not appreciated

• Hippasus of Metapontum • Told the Pythagorean Secret• Died for revealing the discovery

INCOMMENSURABILITY

Babylonians method of calculating square roots

Try to figure out which one works sometimes and which one is completely wrong.

INCOMMENSURABILITY• First recorded proof that

• Euclid’s Elements• Here is the most popular proof

2

2 2

2

Suppose 2 is rational.

Then 2 , where and have no common factors.

Then 2 .

This means is even.

Therefore, is even.

How does this lead to a contradiction? Discuss.

pp q

q

q p

p

p

THE HISTORY OF PI

Definition of Pi

• Ratio of

circumference of circle

diameter of circle

THE HISTORY OF PI

Definition of Pi

• Ratio of

circumference of circle

diameter of circle

THE HISTORY OF PI

Approximation of Pi

• 1650 BC: Rhind Papyrus x = 3.16045

• 950 BC Temple of Solomon: π = 3

THE HISTORY OF PI

Approximation of Pi

• 250 BC: Archimedes 3.1418

• 150 CE: Ptolemy used a 360 – gon• 3.14166

• 263 CE: Liu Hiu used a 192 regular inscribed polygon• 3.14159

• 480 CE: Zu Chongzhi used a 24576-gon• 3.1415929265

THE HISTORY OF PI

Approximation of Pi

What do you have to plug into x to get this to equal pi/6? Why can’t you just plug in the x that will give you pi?

THE HISTORY OF Will discuss more in length later

• 150 BCE: Euclid constructed • line divided into extreme and mean ratio

by a point C • AB▪AC = AC▪CB • Not called golden ratio

• Golden ratio is seen throughout history MANY times

• Heron• Proclus• Abu Kamil• Al-Khwarizmi• Bernoulli• Fibonacci

THE HISTORY OF Relatively new

• 1610 CE: Napier wrote table of logs

• 1610 – 1670: e peaked around but not explicitly found• Area under hyperbola• Logarithmic curve• Series expansion of log(1+x)

• 1683 CE: Bernoulli compound interest limit

e

History of Negative Numbers: http://nrich.maths.org/5961

https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/

https://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html

MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk

SOURCES