Indian
Journal of Science,
ARTICLEVol. 24, No. 91, March, 2017
RESEARCH
ISSN
2319–7730
EISSN
2319–7749
RESEARCH
Indian Journal of Science
An International Journal
Consideration of manual estimation of values of Pi (π):
improvement of Tan 15° over Tan 30°
Olademo Johnson O Ani, Lawal MO, Akinmuyise MF
Department of Mathematics, Adeyemi College of Education, Ondo, Nigeria
Publication History
Received: 11 January 2017
Accepted: 3 February 2017
Published: 1 March 2017
Citation
Olademo Johnson O Ani, Lawal MO, Akinmuyise MF. Consideration of manual estimation of values of Pi (π): improvement of Tan 15°
over Tan 30°. Indian Journal of Science, 2017, 24(91), 147-151
Publication License
This work is licensed under a Creative Commons Attribution 4.0 International License.
General Note
Article is recommended to print as digital color version in recycled paper.
ABSTRACT
schools of thought about the discovery of this magical and golden number call (π). This paper focused on more effective way of
discovering the value of π with a rapid convergence rate through manual computation. Some existing methods discussed include
Olademo et al.
Consideration of manual estimation of values of Pi (π): improvement of Tan 15° over Tan 30°,
Indian Journal of Science, 2017, 24(91), 147-151,
www.discoveryjournals.com
Page
the algebraic numerical method, the old fashion ways of using different sizes of circular tins, Isaac Barrow’s method involving
147
The use of the value of Pi (π) cannot be swept under the carpet in Mathematics and other related fields. There had been different
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0
0
trigonometrical ratios and the use of tan30 . The application of tan15 is an improved method that showed the value of π correct to
seven significant figures after the first five terms, showing a faster rate of convergence over other methods.
Keywords: π, convergence rate, algebraic numerical method, trigonometrical ratio, significant figures.
1. INTRODUCTION
Some likely answers to what π is, include “Circumference equals π times diameter” or π is approximately
or 3.14 or 3.142. Many
Mathematicians were devoted to studying this constant π. Some Mathematicians even spent their whole lives investigating this
constant.
Alfred (2004) said that the first mathematician to use the Greek letter π to represent the ratio of a circle’s circumference to its
diameter was William Jones who used it in 1706 in his work “Synopsis.” i.e. Palmsriorium Matheseos or a New Introduction to the
Mathematics. It was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used
letters such as C or P instead.
Beckman (2012) stressed that Leonhard Euler popularised the use of the Greek letter π in a work he published in 1748. The
circumference of a circle is slightly greater than three times as long as its diameter. The exact ratio is called π (a Greek letter
pronounced Pi). He defined π as the ratio of a circle’s circumference ‘C’ to its diameter, ‘d’. i.e.
π = . The
ratio
is constant,
regardless of the circle’s size.
1.1. Properties of π
According to Alfred (2004), π is an irrational number, meaning that it cannot be written as the ratio of two integers, such as
or
other fractions that are commonly used to approximate π. Since π is irrational, it has an infinite number of digits in its decimal
representation, and it does not end with an infinitely repeating pattern of digits. He said further that π is a transcendental number,
−
which means that it is not the solution of any non-constant polynomial with rational coefficients such as
+
= 0.
2. SOME EXISTING METHODS
There have been various methods used to discover the value of π. These include use of polygons; use of different sizes of circular
tins and the use of first arc-tangent series (Isaac Barrow’s Method).
2.1. Polygon Method
Arthur (1972) stressed that the ratio of the perimeter of an inscribed 96-sided polygon to the diameter of the circle is greater than
3
and the same ratio for a circumscribed 96-sided polygon is less than 3
. Thus 3
<
< 3 .
Jorg and Christoph (2000) pointed out that the Babylonians believed that the perimeter of a regular hexagon equals
(i.e.
)
times the circumference of the circumscribed circle. He did not show the proof, but it was noted that the sides of the hexagon is 1
and by the property of regular hexagon, the diameter is 2. Thus perimeter of regular hexagon = (
we have P = (
×
×
×
). From this,
× 2), leading to π = 3.125
radii and the polygons to scale, starting with the side of a regular hexagon which is known (i.e. 1 cm). If the angles are dissected, we
have a 12-gon and so on. The result in the second method is not very good, and the second method is not very good, and the origin
not known.
Olademo et al.
Consideration of manual estimation of values of Pi (π): improvement of Tan 15° over Tan 30°,
Indian Journal of Science, 2017, 24(91), 147-151,
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of
148
The first method gives a good approximation of π, but the procedures is very cumbersome, as it involves drawing of circles of same
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2.2. Different Sizes of Circular Tins
Jonathan M. Borwein (2012) used laboratory approach to estimate the value of π. This involved different sizes of circular tins, tread
and ruler. The circumferences of circular tins as well as the diameter were measured. Results were tabulated and the value of the
circumference for each tin was divided by the diameter. The results were very close to each other (i.e. converging to a value of 3.1…).
This value was then called constant π. This method gave a good estimate of π but it was time consuming. In addition, the values
were only rounded up or down to have a constant value. This was due to error in measurements.
3. THE IMPROVEMENT OVER ISAAC BARROW’S DISCOVERY
Since the result above showed a slow convergence rate, there is need to find another method that can make the rate convergence
rapid. Since the Maclaurin’s series became useful in doing this, we need to first discuss this series.
3.1. The Maclaurin’s Series
( ) about 0. He stated the series of
Weistein (2001) stressed that a Maclaurin series is a Taylor Series expansion of a function
( ) about 0 as
( ) = (0) +
(0) +
(0)
2!
(0)
3!
+
+ ⋯+
(0)
!
+ ⋯(3.1)
Furthermore, Maclaurin series are a type of series expansion in which all terms are non-negative integer powers of the variable. In
addition, he said that other more general types of series include the Laurent series and Puiseux series. Among the series given are
tan
=
−
1
3
+
1
5
−
1
7
+ ⋯ (−1 <
< 1),
or
tan
(−1)
2 −1
=
.(3.3)
3.2. Application of Tan 30°
Consider that 30°
Now, tan
=
tan
√
And
=
Radian, then, tan
√
30° = tan
(3.1)
=
(3.2)
Applying the Maclaurin series to expand, we have that
3
3

3
3
 13
   
3
3
3
1
5
3
3
5
 17
 
3
3
7


...   n1 2( n1)1
n 1
 
3
3
2 n 1
(3.3)
149
tan 1
Olademo et al.
Consideration of manual estimation of values of Pi (π): improvement of Tan 15° over Tan 30°,
Indian Journal of Science, 2017, 24(91), 147-151,
www.discoveryjournals.com
Page
Using eqn (3.2) in eqn (3.3), we have
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6
=6
i.e.
√
=
√
−
√3 1 √3
−
3
3 3
+
√
√
+
So, the sum of the first four terms gives π
<
−
1 √3
7 3
+⋯
+⋯
= 3.137760244
= 3.142555814.
While the sum of the first five terms gives
Thus 3.137760244
−
1 √3
5 3
< 3.142555814
3.3. The Improvement: Application of Tan 15°
Let’s consider Tan 15°,
Hence, tan
radians.
15° = tan
15° = Now
15° =
°
(45° − 30°) =
°
°
√
=
°
.
√
= 1 −
1
÷ 1+
1
√3
√3
√3 − 1
=
= 2 − √3
√3 + 1
Thus, tan
= 2 − √3
And so, tan
(2 − √3) =
tan (2 − √3) = 2 − √3 −
Now,

 
 
tan 1 2  3  2  3  13 2  3


5

2 − √3
+
2 − √3
−
2 − √3
+⋯

3

7
 15 2  3  17 2  3  ...
Or
= 2 − √3 −
2 − √3
= 12 2 − √3 −
+
2 − √3
1
2 − √3
3
+
−
2 − √3
1
2 − √3
5
−
+⋯
1
2 − √3
7
+⋯
150
The sum of the first 3 terms gives 3.141753687 …
Olademo et al.
Consideration of manual estimation of values of Pi (π): improvement of Tan 15° over Tan 30°,
Indian Journal of Science, 2017, 24(91), 147-151,
www.discoveryjournals.com
Page
While the sum of the first 5 terms gives 3.141593176 … correct to 6 decimal places.
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This is a great improvement over tan 30° which has the sum of its first five terms correct to only 3 decimal places.
4. CONCLUSION
Man has been searching for the values of π for over 4000 years. From the discovery of the fact that circumference is directly
proportional to diameter, using polygons to approach circles to find Pi, discovering series of formulae for the evaluation of Pi, even
up to the use of computers and technology, we have been getting more and more digits of π. In the Palais de la Decouverte (a
science Museum in Paris), there is a circular room known as the “Pi room.” On its wall are inscribed 707 digits of π. In order to set
new records and test the limit of man, people are willing to spend time and effort to find the infinite and magic value of Pi and
appreciate the beauty of Mathematics. As if the 707 digits are not enough, the http://www.genetics.com/walterhung/misc/pi.htm
gave
3.141592653589793238462643383279502884197163399375105820974944. .. up to 1000 digits as
the value of π. This paper had developed a manual approach to estimate the values of π which threw more light to the fact that π is
abstract.
It is only hoped that this closer value will be appreciated while we continue to struggle for the accurate value of π.
REFERENCES
1. Arthur Gittleman (1972). History of Mathematics. Merill
Publishing Company, Syracuse. 176-177
2. Beckmann Petr (2012). A History of Pi. Eve’s Rating
3. David Blatner (2010). The Joy of Pi. Eve’s Rating.
4. Hardy G.H. (2008). An introduction to the Theory of Numbers.
Oxford University Press. 287-301.
5. Jonathan M. Borwein (2012), The Life of Pi: From Archimedes
to Eniac and beyond. Berggren Festschrift.
6. Jorg, Arndt and Christoph Haenel (2000). π – Unleashed
Springer. Verleg Berlin Heidelberg, New York. 165-208
7. Pasamentier Alfred S. and Lehnann Linger (2004). Pi: A
Biography
of
the
World’s
Most
Mysterious
Number.
Prometherus Books. 5-23
8. Weistein, Eric W. (2001). “Maclaurin Series” from Mathworld.
A wolfram web Resource. http://mathworld.wolfram.com/
Olademo et al.
Consideration of manual estimation of values of Pi (π): improvement of Tan 15° over Tan 30°,
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