Download Euler`s Elegant Equation - University of Hawaii Mathematics

Euler’s Elegant Equation
The five most important constants in
mathematics, and the amazing equation
that unites them.
0011 0010 1010 1101 0001 0100 1011
Michelle Manes
Assistant Professor
University of Hawaii at Manoa
[email protected]
23
1
45
iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
•  Each fundamental algebraic operation
appears exactly once.
•  Each of five fundamental mathematical
constants appears exactly once.
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iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
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iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
•  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
0011 0010 1010 1101 0001 0100 1011
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Sumerian tokens
0011 0010 1010 1101 0001 0100 1011
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One what?
0011 0010 1010 1101 0001 0100 1011
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1 in Modern Mathematics
0011 0010 1010 1101 0001 0100 1011
•  “Successor”:
whole number after n is n+1.
•  Multiplicative identity:
a × 1 = 1 × a = a for any number a.
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•  Well ordering principle: every nonempty subset of
(positive whole) numbers has a smallest element.
•  Suppose
is 0100
a number
smaller than 1:
0011
0010 1010 there
1101 0001
1011
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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iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
1
45
iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
•  “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
0011 0010 1010 1101 0001 0100 1011
•  Base 20 positional
system.
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Additive number systems
0011 0010 1010 1101 0001 0100 1011
•  Repeat a symbol to
indicate bigger numbers.
•  No need for “0,” just
omit that symbol.
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Ancient Greece
0011 0010 1010 1101 0001 0100 1011
“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
0011 0010 1010 1101 0001 0100 1011
•  “Arabic” numbers brought to Europe by
Fibonacci (12th century).
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0 as a quantity
Brahmagupta (560 AD)
0011 0010 1010 1101 0001 0100 1011
•  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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0 in Modern Mathematics
0011 0010 1010 1101 0001 0100 1011
•  Additive identity:
a + 0 = 0 + a = a for any number a.
•  Multiplicative behavior:
a × 0 = 0 × a = 0 for any number a.
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0011 0010 1010 1101 0001 0100 1011
•  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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iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
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i
π
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
•  “Imaginary number”.
•  i2 = –1.
•  An Imaginary Tale: The story of i (by Nahin).
1
45
Number Systems
0011 0010 1010 1101 0001 0100 1011
i
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Algebraic numbers
0011 0010 1010 1101 0001 0100 1011
•  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.
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Number Systems
Complex
numbers
0011 0010 1010 1101 0001 0100 1011
Real
numbers
Algebraic
numbers
Rational
numbers
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Transcendental
numbers
i in History
0011 0010 1010 1101 0001 0100 1011
1530s: Tartaglia discovers cubic formula.
For this equation:
3
x = 15x + 4
his method yields a root of
3
€
€
3
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x = 2 + −121 + 2 − −121 .
i in History
0011 0010 1010 1101 0001 0100 1011
Imagine
€
2 + −121 = a + b −1
3
2 − −121 = a − b −1
(
) (
)
Some clever algebra yields x = 4.
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so
x = a + b −1 + a − b −2 = 2a.
€
€
3
i in History
0011 0010 1010 1101 0001 0100 1011
•  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
0011 0010 1010 1101 0001 0100 1011
4i
i
4
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Multiply by i
0011 0010 1010 1101 0001 0100 1011
ai
i
a
23
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iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
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π
i
e
+1=0
0011 0010 1010 1101 0001 0100 1011
•  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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0011 0010 1010 1101 0001 0100 1011
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0011 0010 1010 1101 0001 0100 1011
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π in History
0011 0010 1010 1101 0001 0100 1011
•  “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
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circumference 30
π=
=
=3
diameter
10
π in History
0011 0010 1010 1101 0001 0100 1011
10
1
•  200 BC: Archimedes found 3 < π < 3 .
71
7
€
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π in History
0011 0010 1010 1101 0001 0100 1011
•  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
0011 0010 1010 1101 0001 0100 1011
•  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
π
=
4
0011 0010 1010 1101 0001 0100 1011
2
1
1 1 1
1 1 1 1 1
=
×
+
×
+
+
×
π
2
2 2 2
2 2 2 2 2
2 2×2 4 ×4 6×6 8×8
=
×
×
×
×
π 1× 3 3 × 5 5 × 7 7 × 9
€
32
2+
52
2+
72
2+
2 +
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π
1 1 1 1
= 1− + − + −
4
3 5 7 9
π 2 1 1 1 €1
1
= 2 + 2 + 2 + 2 + 2 +
6 1 2
3
4
5
€
1+
12
12
π
i
e
+1=0
0011 0010 1010 1101 0001 0100 1011
23
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e +1=0
1
iπ
0011 0010 1010 1101 0001 0100 1011
• 
• 
• 
• 
2.718281828459045235360287471…
Irrational (Euler, 1737).
Transcendental (Hermite, 1873).
e: The story of a number (by Maor).
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45
e in History
0011 0010 1010 1101 0001 0100 1011
•  1661: Huygens investigates the area under
the curve y = 1/x.
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e in History
0011 0010 1010 1101 0001 0100 1011
•  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?
0011 0010 1010 1101 0001 0100 1011
•  Base of the natural logarithm.
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So what is e?
0011 0010 1010 1101 0001 0100 1011
•  A limit.
 1
e = lim1+ 
n
n →∞ 
n
23
1
n
(1+1/n)n
1 €
(2)1 = 2
10
(1.1)10 = 2.5937424601
100
(1.01)100
1000
(1.001)1000 = 2.716923932235893…
45
= 2.704813829421526…
Formulas for e
0011 0010 1010 1101 0001 0100 1011
1
1
1
1
1
e = 1+ +
+
+
+
+
1 1× 2 1× 2 × 3 1× 2 × 3 × 4 1× 2 × 3 × 4 × 5
23
1
1
1
1
1
1
1
= 1− +
−
+
−
+
e
1 1× 2 1× 2 × 3 1× 2 × 3 × 4 1× 2 × 3 × 4 × 5
1
e =2+
1
1+
2
2+
3+
3
4
4+
5 +
45
iπ
e
+1=0
0011 0010 1010 1101 0001 0100 1011
•  “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.”
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Making sense of
iπ
e
0011 0010 1010 1101 0001 0100 1011
n

e = lim1+
n →∞ 
1

n


e = lim1+
 n →∞ 
2
n 2

1 
  = lim1+
n 
n →∞ 
n 2
n



2 1
2
e = lim1+ + 2  = lim1+
n n 
n →∞ 
n →∞ 
n

2

n
23
1

1 
  = lim1+
n 
n →∞ 
2 n
1 

n 
45
Making sense of
iπ
e
1+iπ
0011 0010 1010 1101 0001 0100 1011
 x
e = lim1+ 
n
n →∞ 
n
x
 iπ 
e = lim1+ 
n
n →∞ 
iπ
n
23
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Making sense of
iπ
e
0011 0010 1010 1101 0001 0100
1011
2
(1+iπ/2)
 x
e = lim1+ 
n
n →∞ 
n
x
 iπ 
e = lim1+ 
n
n →∞ 
iπ
n
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1+iπ/2
Making sense of
iπ
e
0011 0010 1010 1101 0001 0100 1011
 x
e = lim1+ 
n
n →∞ 
n
 iπ 
e = lim1+ 
n
n →∞ 
23
1
x
n
iπ
(1+iπ/5)5
45
1+iπ/5
Making sense of
iπ
e
0011 0010 1010 1101 0001 0100 1011
 x
e = lim1+ 
n
n →∞ 
23
n
1
x
 iπ 
e = lim1+ 
n
n →∞ 
n
iπ
(1+iπ/10)10
45
1+iπ/10
Making sense of
iπ
e
0011 0010 1010 1101 0001 0100 1011
 iπ 
e = lim1+ 
n
n →∞ 
iπ
n
23
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0011 0010 1010 1101 0001 0100 1011
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