Euler`s Crown Jewel - Biblical Christian World View
Download
Report
Transcript Euler`s Crown Jewel - Biblical Christian World View
A Mathematical Ode to Euler:
Proving Euler’s Identity
ei + 1 = 0
or
ei = -1
By James D. Nickel
Copyright 2010
www.biblicalchristianworldview.net
Leonhard Euler: A gift of God to the world of
mathematics
• To get a glimpse into the amazing mind of
Euler, it is necessary to follow his proofs.
Copyright 2010
www.biblicalchristianworldview.net
Euler’s Identity
• There is a famous formula–perhaps the
most compact and famous of all formulas
–developed by Euler from a discovery of
De Moivre: ei + 1 = 0…. It appeals
equally to the mystic, the scientist, the
philosopher, the mathematician…
Edward Kasner & James Newman, Mathematics and
the Imagination (1940).
Copyright 2010
www.biblicalchristianworldview.net
Starting Point
1
e lim 1
n
n
n
100
1
1
100
?
• e is the base of the natural logarithms.
• e 2.718…
• Euler used this definition to develop a power series for e.
Copyright 2010
www.biblicalchristianworldview.net
Power Series
1 1 1 1
, , , ,
2 4 8 16
1 1 1 1
2 4 8 16
1
1
2
1
2
2
1
2
3
1
2
4
Copyright 2010
www.biblicalchristianworldview.net
Groundwork: The Binomial Formula
1
e lim 1
n
n
n
• He first recognized that he could apply the
binomial formula to this definition.
• This formula is based upon “Pascal’s triangle”
(1623-1662) and derived by Isaac Newton
(1643-1727).
1707-1783
Copyright 2010
www.biblicalchristianworldview.net
Amazing Connection: Combination Formula
• He began with the combination formula (studied
in statistics and finite mathematics courses) for
deriving the terms of a binomial expansion:
n!
n
1. n Cj
j (n j)! j!
6! = 654321 = 720
5! =54321 = 120
4! = 4321 = 24
3! = 321 = 6
2! = 21 = 2
1! = 1
0! = 1
Copyright 2010
www.biblicalchristianworldview.net
Note the
Pattern!
Combination Formula
• n! = n(n - 1)(n - 2)321 or n! = 123n
• Since n! = 123n and …
• (n – j)! = 123(n – j), then, by substitution:
n!
1 2 3 n
n
2. n Cj
j (n j)!j! 1 2 3 n j j!
2
2
6!
1 2 3 4 5 6 1 2 3 4 5 6
6
20
6 C3
3 (6 3)!3! 1 2 3 1 2 3 1 2 3 1 2 3
1
Copyright 2010
www.biblicalchristianworldview.net
1
Combination Formula
6!
1 2 3 4 5 6 4 5 6 6(6 1)(6 2)
6
6 C3
3!
3 (6 3)!3! 1 2 3 1 2 3 1 2 3
• Note that all the factors from 1 to (n – j) in the numerator
cancel with those in the denominator, leaving only these
factors in the numerator:
• n(n – 1)(n – 2)…(n – j + 1).
• In our example, we have 6(6 – 1)(6 – 3 + 1).
• Hence:
n(n 1)(n 2)
1 2 3 n
3. n Cj
1 2 3 n j j!
j!
Copyright 2010
www.biblicalchristianworldview.net
(n j 1)
Combination Formula
• We can now rewrite the combination formula as:
n n(n 1)(n 2)
4. n Cj
j!
j
(n j 1)
Copyright 2010
www.biblicalchristianworldview.net
1
e lim 1
n
n
n
Diversion for Explanation
• Note this instance (n = 4) of the binomial expansion (from
the binomial formula):
4
0
1
2
3
4
1
4 1
3 1
2 1
1 1
0 1
1
(1)
1
(4)
1
(6)
1
(4)
1
(1)
1
4
4
4
4
4
4
0
1
2
3
4
1
1
1
1
1
( 4 C0 ) 14 ( 4 C1 ) 13 ( 4 C2 ) 12 ( 4 C3 ) 11 ( 4 C4 ) 10
4
4
4
4
4
2
3
4
1
1
1
1
1 (4) (6) (4) (1)
4
4
4
4
3 1
1
1 1
2.44140625
8 16 256
Copyright 2010
www.biblicalchristianworldview.net
Diversion for Explanation
• According to Newton’s binomial formula, (x + y)4 =
(1) x4
y (4) x3 y (6) x 2
0
4
3
1
2 2
y (4) x1 y (1) x0 y
2
3
x 4x y 6x y 4xy y
4
Copyright 2010
www.biblicalchristianworldview.net
3
4
Blaise Pascal
• Blaise Pascal (1623-1662), a French
mathematician and amateur theologian.
Copyright 2010
www.biblicalchristianworldview.net
Book
• Wrote a famous book entitled Pensées,
trans. W. F. Trotter (New York: E. P. Dutton,
1958).
Copyright 2010
www.biblicalchristianworldview.net
Some Pascalinian Words of
Wisdom
• “The eternal silence of these infinite
spaces frightens me” (p. 61).
• “Two extremes: to exclude reason, to admit
reason only” (p. 74).
• “The last proceeding of reason is to
recognize that there is an infinity of things
which are beyond it” (p. 77).
Copyright 2010
www.biblicalchristianworldview.net
Pascal’s Triangle
Note the patterned symmetry
Copyright 2010
www.biblicalchristianworldview.net
The Pattern
•
Each number within the triangle (the
numbers 1) is found by adding the
pairs of numbers directly above it at the
left and right.
Copyright 2010
www.biblicalchristianworldview.net
The Pattern
•
The sum of each row is the binary
sequence (powers of 2).
–
–
–
–
–
Row 0: 1 = 20
Row 1: 1 + 1 = 2 = 21
Row 2: 1 + 2 + 1 = 4 = 22
Row 3: 1 + 3 + 3 + 1 = 8 = 23
Row 4: 1 + 4 + 6 + 4 + 1 = 16 = 24
Copyright 2010
www.biblicalchristianworldview.net
Pascal’s Triangle
2
3
4
1
1
1
1
1 (4) (6) (4) (1)
4
4
4
4
Copyright 2010
www.biblicalchristianworldview.net
An Amazing Connection
• The Combination formula:
n!
n Cj
(n j)! j!
• and Pascal’s Triangle
Copyright 2010
www.biblicalchristianworldview.net
Application of Binomial Formula
1
e lim 1
n
n
•
n
Euler applied the binomial formula as follows:
n
0
1
2
3
1
1
1 n (n 1) 1 n (n 1)(n 2) 1
5. 1 1 n
n
2!
3!
n
n
n
n
n
C0
n
C1
n Cj
n
C2
n
n(n 1)(n 2)
n!
(n j)!j!
j!
C3
(n j 1)
Copyright 2010
www.biblicalchristianworldview.net
1
(1)
n
n
Cn
n
Euler Expansion
n
0
1
2
3
1
1
1 n (n 1) 1 n (n 1)(n 2) 1
5. 1 1 n
n
2!
3!
n
n
n
n
1
(1)
n
• Euler expanded the third term and got:
2
n (n 1) 1
n (n 1) 1 n 1 n 1 1
6.
2
2!
2!
2!n n 2!
n
n
Copyright 2010
www.biblicalchristianworldview.net
n
Euler’s Rewrite
2
n (n 1) 1
n (n 1) 1 n 1 n 1 1
6.
2
2!
2!
2!n n 2!
n
n
• Note that 9/10 = 1 – 1/10
• Euler rewrote (n – 1)/n as follows:
n 1 n 1
1
1
n
n n
n
Copyright 2010
www.biblicalchristianworldview.net
Euler’s Rewrite
2
n (n 1) 1
n (n 1) 1 n 1 n 1 1
6.
2
2!
2!
2!n n 2!
n
n
n 1 n 1
1
1
n
n n
n
• Therefore:
1
n (n 1) 1
1 1
n
7.
1
2!
2!
n
n 2!
2
1
Copyright 2010
www.biblicalchristianworldview.net
n 1 n 1
1
1
n
n n
n
Euler’s Rewrite
n2 n 2
2
1
n
n n
n
1
1
n
3
n
(n
1)(n
2)
1
1
1
n
5. 1 1 n
n
n
2!
3!
n
1
n
n (n 1)(n 2) 1 n 1 n 2 1
3
3!
n
n
n
3!
1
1
2
1 1 1
n
1
n n n
8. 1 1 1
n
2!
3!
Copyright 2010
www.biblicalchristianworldview.net
1
nn
n
A Fine Piece of Thinking
• When a baseball player works the count, gets
his pitch, and drives a single into the outfield,
seasoned baseball experts usually remark,
“That was a fine piece of hitting.”
• From equation 2 to equation 8 we have followed
Euler’s “fine piece of thinking.”
• But, there are more of the “amazing thoughts” of
Euler to come.
Copyright 2010
www.biblicalchristianworldview.net
Limits
1
1
2
1 1 1
n
1
n n n
8. e lim 1 1 1
n
n
2!
3!
1
n
n
• Since we are looking for the limit of (1 + 1/n)n as n ,
we must let n increase without bound.
• Our expansion will have more and more terms.
• At the same time, the expression within each pair of
parentheses will tend to 1 …
• since the limits of 1/n, 2/n, … 0 as n .
Copyright 2010
www.biblicalchristianworldview.net
Limits
1
1
2
1 1 1
n
1
n n n
8. e lim 1 1 1
n
n
2!
3!
• Since the limits of 1/n, 2/n, … 0 as n , Euler
produced the following:
n
1
1 1 1 1 1
9. e lim 1
n
n
0! 1! 2! 3! 4 !
Copyright 2010
www.biblicalchristianworldview.net
1
n
n
Substitution
n
1
1 1 1 1 1
9. e lim 1
n
n
0! 1! 2! 3! 4 !
10 11 12 13 14
0! 1! 2! 3! 4 !
• This is a wonderful and beautiful series of e.
• He then replaced 1/n by x/n and got:
n
0
1
2
3
4
x
x
x
x
x
x
10. e x lim 1
n
n
0! 1! 2! 3! 4 !
x x 2 x3 x4
1
1! 2! 3! 4 !
Copyright 2010
www.biblicalchristianworldview.net
Convergence
n
2
3
4
x
x x
x
x
10. e lim 1 1
n
n
1! 2! 3! 4 !
x
• It can be shown that this series converges (i.e., reaches a
limit) for all real values of x.
• In fact, the rapidly increasing denominators cause the
series to converge very quickly.
• It is from this series that numerical values of ex are
calculated.
• This series is programmed into modern scientific
calculators when the ex key is punched.
Copyright 2010
www.biblicalchristianworldview.net
Euler Nerve
n
2
3
4
x
x x
x
x
10. e lim 1 1
n
n
1! 2! 3! 4 !
x
• Euler continued to experiment with this power series for
ex.
• He decided to see what happened to this series if he
replaced x with the imaginary expression ix where i = -1.
• This substitution took some nerve on Euler’s part since ex
had always represented a real number.
• Now Euler was going to see what would happen to the
expression eix.
Copyright 2010
www.biblicalchristianworldview.net
Enter Imaginaries
n
2
3
4
x
x x
x
x
10. e lim 1 1
n
n
1! 2! 3! 4 !
x
• Replacing x with ix, Euler got:
11. e ix 1 ix
ix
2!
2
ix
3!
3
ix
4!
4
ix
5
5!
Copyright 2010
www.biblicalchristianworldview.net
ix
6
6!
Enter Imaginaries
11. e ix 1 ix
ix
2!
2
ix
3
3!
ix
4!
4
ix
5
5!
ix
6
6!
• Knowing the properties of i, i2 = -1, i3 = -i, and i4 = 1, etc.,
he now got:
2
3
4
5
6
x
ix
x
ix
x
12. e 1 ix
2! 3! 4 ! 5! 6!
ix
Copyright 2010
www.biblicalchristianworldview.net
Euler’s Leap
2
3
4
5
6
x
ix
x
ix
x
12. e ix 1 ix
2! 3! 4 ! 5! 6!
• Now Euler did what today’s mathematicians consider
unthinkable.
• He changed the order of the terms collecting all the
real terms separately from the imaginary terms.
• With an infinite series, this could create trouble.
• Instead of converging to a limit, it may diverge. He got:
2
4
6
x
x
x
13. e ix 1
2! 4 ! 6!
ix3 ix5
ix
3! 5!
Copyright 2010
www.biblicalchristianworldview.net
Factoring i
2
4
6
x
x
x
13. e ix 1
2! 4 ! 6!
ix3 ix5
ix
3! 5!
• Factoring out i from the imaginary terms, he now got:
2
4
6
x
x
x
14. e ix 1
2! 4 ! 6!
x 3 x5 x7
i x
3! 5! 7!
This was an aha moment for Euler!
Why?
Copyright 2010
www.biblicalchristianworldview.net
A Remarkable Connection
2
4
6
x
x
x
14. e ix 1
2! 4 ! 6!
x 3 x5 x7
i x
3! 5! 7!
• In Euler’s time, thanks to the work of Brook Taylor (16851731), the power series of sin x and cos x (x in radians)
was well known. Here it is:
x 3 x5 x7
15. sin x x
3! 5! 7!
x 2 x 4 x6
16. cos x 1
2! 4 ! 6!
Copyright 2010
www.biblicalchristianworldview.net
A Grand Substitution
2
4
6
3
5
7
x
x
x
x
x
x
ix
14. e 1
i x
2! 4 ! 6!
3! 5! 7!
x 3 x5 x7
15. sin x x
3! 5! 7!
x 2 x 4 x6
16. cos x 1
2! 4 ! 6!
• Given these definitions, Euler made a grand substitution.
• He got …
drum roll …
17. eix cos x i sin x
eix is a complex
number (has the
form a + bi)
Copyright 2010
www.biblicalchristianworldview.net
Startling …
ix
17. e cos x i sin x
• This equation expresses a startling, indeed, incredible
link between the exponential function (albeit raised to
an imaginary power) and basic trigonometry ratios.
• Euler now replaced x by -x and using the trigonometric
identities cos (-x) = cos x and sin (-x) = -sin x, he rewrote
the above equation as follows:
18. eix cos x i sin x
Copyright 2010
www.biblicalchristianworldview.net
Some Addition
ix
17. e cos x i sin x
18. eix cos x i sin x
• Euler added equation 17 and equation 18 and got:
19. eix eix cos x i sin x cos x i sin x
Copyright 2010
www.biblicalchristianworldview.net
Like Terms
19. eix eix cos x i sin x cos x i sin x
• Combining like terms, he got:
ix
20. e e
ix
2cos x
Copyright 2010
www.biblicalchristianworldview.net
Division
20. eix eix 2cos x
• Dividing both sides of equation 20 by 2, he got:
e ix e ix
21. cos x
2
Copyright 2010
www.biblicalchristianworldview.net
Subtraction
ix
e e
21. cos x
2
ix
17. eix cos x i sin x
18. eix cos x i sin x
• Then, he subtracted equation 18 from equation 17 and
got:
22. eix eix cos x i sin x cos x i sin x
Copyright 2010
www.biblicalchristianworldview.net
Like Terms
e ix e ix
21. cos x
2
22. eix eix cos x i sin x cos x i sin x
• Combining like terms in Equation 22, he got:
ix
23. e e
ix
2i sin x
Copyright 2010
www.biblicalchristianworldview.net
Division
e ix e ix
21. cos x
2
23. eix eix 2i sin x
• Dividing both sides of equation 23 by 2i, he got:
e ix e ix
24. sin x
2i
Copyright 2010
www.biblicalchristianworldview.net
Interlude
• Every step that Euler has taken so far (and some steps
were giant intuitive leaps) has been confirmed by the
rigors of analysis in the 19th and 20th centuries.
• Euler, like many of his time, was a pioneer.
• He blazed many trials and left it for others to confirm his
steps.
Copyright 2010
www.biblicalchristianworldview.net
More Substitutions
e ix e ix
21. cos x
2
e ix e ix
24. sin x
2i
• Since x is in radians, then Euler let x = = 180.
• cos = ?
• Since cos = -1, Equation 21 becomes:
e i e i
25. 1
2
Copyright 2010
www.biblicalchristianworldview.net
More Substitutions
e i e i
25. 1
2
e ix e ix
24. sin x
2i
• sin = ?
• Since sin = 0, Equation 24 becomes:
e i e i
26. 0
2i
Copyright 2010
www.biblicalchristianworldview.net
Multiplication
e i e i
25. 1
2
e i e i
26. 0
2i
• Multiplying both sides of equation 25 by 2, Euler got:
27. 2 ei ei
Copyright 2010
www.biblicalchristianworldview.net
Multiplication
27. 2 ei ei
e i e i
26. 0
2i
• Multiplying both sides of equation 26 by 2i, Euler got:
i
28. 0 e e
i
Copyright 2010
www.biblicalchristianworldview.net
Addition
27. 2 ei ei
28. 0 ei ei
• Adding e-i to both sides of equation 28, he got:
29. e
i
e
i
Copyright 2010
www.biblicalchristianworldview.net
Substitution
27. 2 ei ei
29. e i e i
• Now, Euler substituted equation 29 into equation 27
and got:
30. 2 2e
i
Copyright 2010
www.biblicalchristianworldview.net
Division
30. 2 2e
i
• Dividing both sides of equation 30 by 2, he got:
i
31. e 1
Copyright 2010
www.biblicalchristianworldview.net
Addition
i
31. e 1
• Adding 1 to both sides of equation 31, he got:
i
32. e 1 0
QED and magnifico!
Copyright 2010
www.biblicalchristianworldview.net
Hats off!
• Mathematicians have
described this equation using
words like “remarkable,”
“beautiful,” “a revelation,”
“absolutely paradoxical,”
“certainly true,”
“incomprehensible,” “blew a
few circuits in my head,” and
“cannot be explained in
words.”
• Hats off and thank you,
Professor Euler!
Euler, “It sometimes seems
to me that my pencil is
smarter than I.”
You are
welcome!
Copyright 2010
www.biblicalchristianworldview.net
ei + 1 = 0
• Here we have an equation that connects the five
most important constants of mathematics (0, 1,
i, , e) and three of the most important
mathematical operations (addition,
multiplication, and exponentiation).
• The five constants connect four major branches
of mathematics: arithmetic (0 and 1), algebra (i),
geometry (), and analysis (e).
Copyright 2010
www.biblicalchristianworldview.net
Unity in Diversity
• For the Biblical Christian, this formula
serves as another resounding echo that
can be traced back to the ultimate One
and the Many, the Author and Sustainer of
this striking connection … this astonishing
and wondrous unity and diversity in
mathematics.
Copyright 2010
www.biblicalchristianworldview.net