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𝐏𝐢 and Integration
by Okkyung Cho
Abstract
Pi(π) is one of the most widely known mathematical
constants. There have been a lot of efforts for the
computation of the mathematical constant π = 3.14 ⋯
through the ages. With the discovery of Calculus in 1600s,
a number of formulas for π were discovered. During this
time one motivation for computations of π was to see if
the decimal expansion of π repeats. This talk gives
several formulas for π related to the integration and a
proof that π is irrational.
1
Pi and Popular culture
Star Trek (1967 episode)
Kirk asks:
“Aren’t there some mathematical problems that
simply can’t be solved?”
Spock answers by telling to a rogue computer:
“Compute to the last digit the value of Pi.”
2
Pi and Popular culture
Life of Pi (2001 Yann Martel’s book)
“My name is Piscine Molitor Patel known to all
Pi Patel. For good measure I added
𝜋 = 3.14∗
and I then drew a large circle which I sliced in
two with a diameter, to evoke that basic lesson
of geometry.”
 The Notation of 𝜋 was introduced by Euler in 1737.
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𝟐𝟐
𝐖𝐡𝐲 𝝅 𝐢𝐬 𝐧𝐨𝐭
𝟕
Even Maple or Mathematica `knows' this since
1(
1 − 𝑥 )4 𝑥 4
22
(1)
0<∫
𝑑𝑥 =
−𝜋
2
1+𝑥
7
0
The integrand is strictly positive on (0, 1), and
the answer in (1) is an area and so strictly
positive.
In this case, the indefinite integral provides immediate reassurance. We obtain
𝑡(
1 − 𝑥)4 𝑥 4
(2 )
∫
𝑑𝑥
2
1+𝑥
0
1 7 2 6
4 3
5
= 𝑡 − 𝑡 + 𝑡 − 𝑡 + 4𝑡 − 4 arctan(𝑡)
7
3
3
and the FTOC proves (1).
 22
is one of the early continued fraction approximations.
7
4
One can take this idea a bit further. Note that
1
1
4
4
(3)
∫ (1 − 𝑥) 𝑥 𝑑𝑥 =
,
630
0
and we observe that
1(
4 4
)
1 1
1
−
𝑥
𝑥
4
4
(4 )
∫ (1 − 𝑥) 𝑥 𝑑𝑥 < ∫
𝑑𝑥
2
2 0
1+𝑥
0
1
< ∫ (1 − 𝑥)4 𝑥 4 𝑑𝑥
0
Combine this with (1) and (3) to derive:
223 22
1
22
1
22
<
−
<𝜋<
−
<
71
7 630
7 1260
7
and so obtain Archimedes famous computation:
10
10
3
<𝜋<3 .
71
70
 The derivation above seems first to have been written
down in Eureka, the Cambridge student journal in 1971.
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The Childhood of Pi
About 2000 BC, the Babylonians used the
1
approximation 3 = 3.125. At this same time or
8
earlier, according to an ancient papyrus,
Egyptians assumed a circle with diameter nine
has the same area as a square of side eight,
256
which implies 𝜋 =
= 3.1604 ⋯
81
Some have argued that the ancient Hebrews
used 𝜋 = 3. (I Kings 7:23)
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Uniqueness of 𝝅
Archimedes (250 BC) was the first to show that
the ‘two Pi’s’ are the same:
𝑨𝒓𝒆𝒂 = 𝜋1 𝑟 2 and 𝑷𝒆𝒓𝒊𝒎𝒆𝒕𝒆𝒓 = 2𝜋2 𝑟
 Archimedes' construction for the uniqueness of 𝜋,
taken from his Measurement of a Circle.
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Archimedes Method
The first rigorous mathematical calculation of 𝜋
was also due to Archimedes, who used brilliant
scheme on building inscribed and circumscribed
polygons 6 ↦ 12 ↦ 24 ↦ 48 ↦ 96
𝟏𝟎
𝟏
to obtain the bounds: 𝟑 < 𝝅 < 3 .
𝟕𝟏
𝟕
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Precalculus 𝝅 Calculation
Variations of Archimedes’ geometric scheme
were the basis for all high-accuracy calculations
of 𝜋 for the next 1800 years.
Name
Babylonians
Egyptians
Hebrews (1 Kings7:23)
Archimedes
Ptolemy
Liu Hui
Tsu Ch’ung Chi
Al-Kashi
Romanus
Van Ceulen (Ludolph’s number* )
Yea
2000?r BC
2000? BC
550? BC
250? BC
150
263
480?
1429
1593
1615
Digits
1
1
1
3
3
5
7
14
15
35
* The last great Archimedean calculation, performed by
Van Ceulen using 262 -gons - to 39 places with 35
correct - was published posthumously.
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Pi’s Adult Life with Calculus
In the later 17th century, Newton and Leibniz
founded the calculus, and this powerful tool was
quickly exploited to find new formulae for π.
One early calculus-based formula comes from
the integral:
𝑥
1
−1
tan 𝑥 = ∫
𝑑𝑡
2
0 1+𝑡
𝑥
= ∫ (1 − 𝑡 2 + 𝑡 4 − 𝑡 6 + ⋯ )𝑑𝑡
0
𝑥3 𝑥5 𝑥7
=𝑥− + − + ⋯
3
5
7
Substituting 𝑥 = 1 formally proves the wellknown Leibniz formula (1671-74):
π
1 1 1 1 1
= 1− + − + −
+⋯
4
3 5 7 9 11
10
Newton discovered a different more effective
formula, considering the area A of the left most
Red region in the figure:
Now, A is the integral
1/4
𝐴=∫
√𝑥 − 𝑥 2 𝑑𝑥
0
Also, A is the area of the circular sector, 𝜋/24,
less the area of the triangle, √3/32.
11
Newton used his binomial theorem,
1
4
𝐴=∫
1
𝑥 2 (1
0
1/4
=∫
0
1/4
=∫
1
𝑥2
−
1
𝑥)2
𝑑𝑥
𝑥 𝑥 2 𝑥 3 5𝑥 4
(1 − − −
−
− ⋯ ) 𝑑𝑥
2 8 16 128
1
(𝑥 2
−
0
3
𝑥2
2
−
5
𝑥2
8
−
7
𝑥2
16
−
9
5𝑥 2
128
− ⋯ ) 𝑑𝑥
Integrate term-by-term and combining the
above produces
3√3
1
1
1
𝜋=
+ 24 (
−
−
⋯ ).
4
3 ⋅ 4 5 ⋅ 32 28 ⋅ 128
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The Irrationality of 𝝅
Let 𝜋 = 𝑎/𝑏, the quotient of positive integers.
We define the polynomials
𝑥 𝑛 (𝑎 − 𝑏𝑥)𝑛
𝑓 (𝑥) =
,
𝑛!
𝐹 (𝑥) = 𝑓(𝑥) − 𝑓 (2) (𝑥) + 𝑓 (4) (𝑥) − ⋯
+ (−1)𝑛 𝑓 (2𝑛) (𝑥),
the positive integer 𝑛 being specified later.
Since 𝑛! 𝑓(𝑥) has integral coefficients and
terms in 𝑥 of degree not less than 𝑛, 𝑓(𝑥)
and its derivatives 𝑓 (𝑗) (𝑥) have integral values
for 𝑥 = 0; also for 𝑥 = 𝜋 = 𝑎/𝑏, since 𝑓(𝑥) =
𝑓(𝑎/𝑏 − 𝑥). By elementary calculus we have
𝑑
{𝐹 ′ (𝑥) sin 𝑥 − 𝐹 (𝑥) cos 𝑥}
𝑑𝑥
= 𝐹 ′′ (𝑥) sin 𝑥 + 𝐹 (𝑥) sin 𝑥 = 𝑓 (𝑥) sin 𝑥
13
and
(5 )
𝜋
∫ 𝑓(𝑥) sin 𝑥 𝑑𝑥
0
= [𝐹 ′ (𝑥) sin 𝑥 − 𝐹 (𝑥) cos 𝑥]𝜋0 = 𝐹 (𝜋) + 𝐹(0)
Now 𝐹 (𝜋) + 𝐹(0) is an integer, since 𝑓 (𝑗) (0)
and 𝑓 (𝑗) (𝜋) are integers. But for 0 < 𝑥 < 𝜋,
𝜋 𝑛 𝑎𝑛
0 < 𝑓 (𝑥) sin 𝑥 <
,
𝑛!
so that the integral above is positive, but
arbitrarily small for n sufficiently large. Thus (5)
is false, and so is our assumption that 𝜋 is
rational.
 In 1761 Lambert proved that 𝜋 is irrational. This is
Niven’s 1947 short proof.
 In 1882, Lindemann proved that 𝜋 is transcendental.
14
Why Pi?
 One motivation is the raw challenge of
harnessing the stupendous power of modern
computer systems.
 There have been substantial practical spinoffs. For example, some new techniques for
performing the Fast Fourier transform (FFT),
heavily used in modern science and
engineering computing, had their roots in
attempts to accelerate computations of 𝜋.
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References
 D. Bailey, J. M. Borwein, P. Borwein and S. Plouffe,
The Quest for 𝜋
 J. Borwein, The Life of 𝜋: From Archimedes to Eniac
and Beyond
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