irrational numbers. incommensurability egyptiona and babylonians calculated square roots these were...
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IRRATIONAL NUMBERS
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INCOMMENSURABILITY
• Egyptiona and Babylonians calculated square roots• These were approximated• Not appreciated
• Hippasus of Metapontum • Told the Pythagorean Secret• Died for revealing the discovery
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INCOMMENSURABILITY
Babylonians method of calculating square roots
Try to figure out which one works sometimes and which one is completely wrong.
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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
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THE HISTORY OF PI
Definition of Pi
• Ratio of
circumference of circle
diameter of circle
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THE HISTORY OF PI
Definition of Pi
• Ratio of
circumference of circle
diameter of circle
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THE HISTORY OF PI
Approximation of Pi
• 1650 BC: Rhind Papyrus x = 3.16045
• 950 BC Temple of Solomon: π = 3
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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
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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?
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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
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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
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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