Euler’s Elegant Equation mathematics, and the amazing equation...
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4 2 5 1 0011 0010 1010 1101 0001 0100 1011 Euler’s Elegant Equation The five most important constants in mathematics, and the amazing equation that unites them. Michelle Manes Assistant Professor University of Hawaii at Manoa [email protected]
Transcript of Euler’s Elegant Equation mathematics, and the amazing equation...
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Euler’s Elegant Equation The five most important constants in mathematics, and the amazing equation that unites them.
Michelle Manes Assistant Professor University of Hawaii at Manoa [email protected]
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• Each fundamental algebraic operation appears exactly once.
• Each of five fundamental mathematical constants appears exactly once.
eiπ + 1 = 0
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• The first number. • At least 20,000 years ago, people were
counting by adding up ones. • Story of one: PBS documentary.
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Ishango bone
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Sumerian tokens
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One what?
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• “Successor”: whole number after n is n+1.
• Multiplicative identity: a × 1 = 1 × a = a for any number a.
1 in Modern Mathematics
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• Well ordering principle: every nonempty subset of (positive whole) numbers has a smallest element.
• Suppose there is a number smaller than 1: 0 < r < 1.
• Then r × r < r × 1 = r < 1. • Continue with that reasoning:
…r5 < r4 < r3 < r2 < r < 1. • The set of numbers less than one
has no smallest element! • If there’s no number less than 1,
there’s no number between n and n+1.
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• “Natural number”? • Digit in positional number system.
(India ~500 BC, Yucatan peninsula) • As a quantity. (Not until much later!) • The Nothing That Is: A natural history
of zero (by Kaplan and Kaplan). • Zero: The biography of a dangerous
idea (by Seife).
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Mayan numbers
• Base 20 positional system.
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Additive number systems
• Repeat a symbol to indicate bigger numbers.
• No need for “0,” just omit that symbol.
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Ancient Greece
“How could he have missed it? To what heights science would have risen by now, if only he had made that discovery!”
- Gauss, about Archimedes
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0 in positional number system
• “Arabic” numbers brought to Europe by Fibonacci (12th century).
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0 as a quantity Brahmagupta (560 AD)
• Sum of zero and positive is positive, sum of zero and zero is zero.
• A number multiplied by 0 is 0. • A number remains unchanged when 0 is
subtracted from it. • Zero divided by zero is zero. • Widespread use in western world not until
17th century!
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• Additive identity: a + 0 = 0 + a = a for any number a.
• Multiplicative behavior: a × 0 = 0 × a = 0 for any number a.
0 in Modern Mathematics
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• 0 + 0 = 0 • Multiply by some number a:
a × (0 + 0) = a × 0 • Distributive law:
(a × 0) + (a × 0) = a × 0 • Subtract a × 0 from each side:
a × 0 = 0
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0011 0010 1010 1101 0001 0100 1011eiπ + 1 = 0 • “Imaginary number”. • i2 = –1. • An Imaginary Tale: The story of i (by Nahin).
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Number Systems
i
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• Take a polynomial equation with integer coefficients. Its solutions are algebraic numbers.
• All integers are algebraic: 5 is a solution of x = 5. • All rational numbers are algebraic: ½ is a solution of
2x = 1. • i is algebraic: It is a solution of x2 = –1. • A number that is not algebraic is transcendental.
Algebraic numbers
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Number Systems
Real numbers
Complex numbers
Algebraic numbers
Transcendental numbers Rational
numbers
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i in History
€
x = 2 + −1213 + 2 − −1213 .
1530s: Tartaglia discovers cubic formula. For this equation:
his method yields a root of
€
x 3 =15x + 4
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i in History
€
2 + −1213 = a + b −1
Imagine
so
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x = a + b −1( ) + a − b −2( ) = 2a.
€
2 − −1213 = a − b −1
Some clever algebra yields x = 4.
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i in History • 1600s: Descartes argues the physical
impossibility of complex numbers. • 1600s: Wallis tries to picture them as a
vertical motion. • 1700s: Wessel describes the complex plane
and says multiplying by i is the same as rotating 90 degrees.
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Multiply by i
i
4
4i
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Multiply by i
i
a
ai
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0011 0010 1010 1101 0001 0100 1011eiπ + 1 = 0 • 3.1415926535897932384626433… • Ratio of circumference to diameter of any
circle. • Irrational (Lambert, 1761). • Transcendental (von Lindemann, 1882). • A History of π (by Beckman).
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π in History
• “And he made a molten sea, ten cubits from one brim to the other; it was round all about, and his height was five cubits, and a line of thirty cubits did encompass it all around.” - Kings 7:23
€
π =circumferencediameter
=3010
= 3
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π in History
• 200 BC: Archimedes found
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31071
< π < 317.
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π in History
• 16th century: Ludolph van Ceulen calculated π to 35 decimal places and had the result carved on his tombstone.
• Germans still refer to π as die Ludolphsche Zahl.
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π in History • 1706: First appearance of the symbol π. • 1873: Shanks spent 20 years calculating π to
707 decimal places. Mistake (found in 1945) in the 528th decimal place.
• 1897: Indiana bill #246. • 1949: Computer took 70 hours to calculate π to 2,000 decimal places.
• Current record: 5 trillion digits.
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Approximations to π
€
2π
=12×
12
+1212×
12
+1212
+1212×
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2π
=2 × 21× 3
×4 × 43× 5
×6 × 65 × 7
×8 × 87 × 9
×
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π4
=1− 13
+15−17
+19−
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π4
=12
1+12
2 +32
2 +52
2 +72
2 +
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π 2
6=112
+122
+132
+142
+152
+
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0011 0010 1010 1101 0001 0100 1011eiπ + 1 = 0 • 2.718281828459045235360287471… • Irrational (Euler, 1737). • Transcendental (Hermite, 1873). • e: The story of a number (by Maor).
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e in History • 1661: Huygens investigates the area under
the curve y = 1/x.
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e in History
• 1690: Leibniz working on the calculus investigates the function f(x) = ex.
• Writes to Huygens about it, naming the constant b.
• 1854: Shanks calculates 205 decimal places of e.
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So what is e?
• Base of the natural logarithm.
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So what is e? • A limit.
€
e =n→∞lim 1+
1n
n
n (1+1/n)n
1 (2)1 = 2
10 (1.1)10 = 2.5937424601
100 (1.01)100 = 2.704813829421526…
1000 (1.001)1000 = 2.716923932235893…
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Formulas for e
€
1e
=1− 11
+11× 2
−1
1× 2 × 3+
11× 2 × 3× 4
−1
1× 2 × 3× 4 × 5+
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e =1+11
+11× 2
+1
1× 2 × 3+
11× 2 × 3× 4
+1
1× 2 × 3× 4 × 5+
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e = 2 +1
1+1
2 +2
3+3
4 +4
5 +
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• “The most beautiful theorem in mathematics”.
• History is unclear. • Certainly known to Euler (1707-1783). • Stigler’s Law: “No scientific discovery
is named after its original discoverer.”
eiπ + 1 = 0
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Making sense of eiπ
€
e =n→∞lim 1+
1n
n
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e2 =n→∞lim 1+
1n
n
2
=n→∞lim 1+
1n
n
2
=n→∞lim 1+
1n
2
n
€
e2 =n→∞lim 1+
2n
+1n2
n
=n→∞lim 1+
2n
n
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Making sense of eiπ
€
ex =n→∞lim 1+
xn
n
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eiπ =n→∞lim 1+
iπn
n
1+iπ
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Making sense of eiπ
€
ex =n→∞lim 1+
xn
n
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eiπ =n→∞lim 1+
iπn
n
1+iπ/2
(1+iπ/2)2
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Making sense of eiπ
€
ex =n→∞lim 1+
xn
n
€
eiπ =n→∞lim 1+
iπn
n
1+iπ/5 (1+iπ/5)5
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Making sense of eiπ
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ex =n→∞lim 1+
xn
n
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eiπ =n→∞lim 1+
iπn
n
1+iπ/10 (1+iπ/10)10
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Making sense of eiπ
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eiπ =n→∞lim 1+
iπn
n
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