The Impact of Mathematical Developments in the
Derivation of p
Daniel Y.
Chong
Contents
The Impact of Mathematical Developments in the Derivation of Pi
Appendix A - Convergence of Pi Indentities
Bibliography
Abstract
Mathematicians
throughout history have been fascinated with p. For millennia, they have focused on finding as many
digits of p as possible. We have
come such a long way from the crude estimations given by ancient mathematicians
that it seems only natural to wonder – how have mathematical advances through
history affected mathematicians’ approach to calculating p? The origin of this constant, defined as the ratio of
a circle’s circumference to its diameter, lies in geometry. Mathematicians such
as Archimedes used geometric principles to work out rudimentary approximations
for p. As new branches of
mathematics have developed, formulae for p have become increasingly sophisticated and
accordingly, have become faster. Trigonometry extended geometric formulae by
allowing mathematicians to work with expressions involving angles, and algebra
allowed for easier manipulation of these expressions. But it was not until the
invention of calculus that the approximation of p truly began to see significant improvement. Newton,
Liebniz, and others came derived many infinite series and products for p. Next came Euler, whose identities and series
provided much better approximations than ever before. Mathematicians have since
continued to find more and more digits; as of 1995, the known digits of p exceeds six billion.
The Impact of Mathematical Developments in the Derivation of Pi
It is defined as the ratio of
the circumference of a circle to its diameter. Denoted by the Greek letter p and approximated by the ever-familiar 3.14159, its
knowledge dates back to pre-history; as early as 2000 BC the ancient civilizations
of Babylon and Egypt knew of its existence (CECM, 1996). However, as
mathematics progressed, the repeated occurrences of p in higher mathematics has both startled and fascinated
mathematicians for centuries. Man’s fascination for p has led him to convert it from an abstract concept to
a concrete string of numbers. Although early attempts at calculating it have
been understandably imperfect, hampered by a dearth of any mathematical
knowledge beyond rudimentary geometry and simple arithmetic, in more recent
years man has had greater success at calculating its digits, which leads one to
wonder: how have developments in the mathematics impacted man’s quest to
uncover the digits of p? With the understanding that as man’s inevitable quest for knowledge
and progress has led to the development of higher mathematics, which in turn
has led to more sophisticated and accurate derivations for p, we will examine, in roughly chronological order, the
impact of the developments of geometry, algebra, trigonometry, and calculus on
the derivation of p.
It can be said that geometry,
next to arithmetic, was the first branch of mathematics to be explored by early
man. Indeed, to the ancient Greeks, mathematics was entirely geometry. This is
apparent when one considers, among other things, the Pythagoreans’ practice of
representing numbers by arranging stones on the ground, the importance placed
in precise construction using a straight-edge and a compass, and Euclid, author
of the Elements, a work which so
profoundly impacted the development of geometry that its effects are evident
even today, nearly two-and-a-half millennia after it was written.
Ancient mathematicians, when
calculating p, relied almost
completely on an underdeveloped system of geometry to aid them, so early approximations
of p were coarse at best. For
example, the Ahmes Papyrus, which, originating from Egypt, is among the oldest
known writing on mathematics, makes the incorrect assumption that a circle with
a radius of 4.5 is equal in area to a square with side 8. (Beckmann, 1971).
Given this, the value Ahmes used for p is clear:
The method of deriving a value
for p used by Archimedes was
much more sophisticated than that used by Ahmes, who seems to have simply made
an invalid assumption. Archimedes reasoned that the perimeter of an n-sided polygon inscribed in a circle
approximates the circumference of the circle. This was nothing new; many
mathematicians before him utilized the same principle in calculating p themselves. But Archimedes took it one step further.
He realized that the approximation given by an inscribed polygon will always
fall short of the actual circumference. To compensate for this, he
circumscribed the same polygon about the circle as well. Thus, instead of
calculating a simple approximation for p, he obtained a range in which p lay (O’Connor and Robertson, 1996). Beginning with a
hexagon (n = 6), he continuously
doubled n until he obtained n = 96, with which he calculated
a
remarkable approximation, especially in light of the fact that Archimedes
accomplished this without the use of trigonometry, whose half-angle formulae
would have greatly simplified and aided his work.
The next branch of mathematics
to be developed was algebra. Although the ancient Greeks used geometric
construction to solve many problems which can be solved algebraically, the
first mathematician to make the transition from a geometric process to an
algebraic one is Diophantus (circa
275 AD). Following Diophantus was the Arab mathematician Mohammed ibn Musa
al-Khwarizmi, who authored a booked titled Hisab
al-jabr w’al-muqbala, or Al-jabr
for short, from which the word algebra
originates (Hollingdale, 1991). In his book, al-Khwarizmi formulated many of
the foundations of algebra, namely the equivalency postulates of arithmetic
(additive/multiplicative properties of equality, etc.), presenting their proofs
in geometric form. Through the next few centuries, improvements in algebra
abound with such discoveries as the quadratic equation, solutions of
higher-degree polynomials, and complex numbers. It was not until about the 17th
century that a universal set of symbols describing algebraic operations and processes
was fully developed and adopted.
The effects of algebra on
derivations of p are not as readily apparent as they were for other branches of
mathematics, such as trigonometry and calculus. This is due in part to the fact
that, around the time of its development, many of the equations and formulae
involving p were not yet complicated
enough to necessitate the use of symbolic representation; geometric or even
verbal representation was often sufficient. However, on a less direct level,
algebra has strongly impacted the derivation of p, at least in the sense that it has served as the
foundation of higher mathematics, namely analytic geometry. The latter is especially
important because it allowed mathematicians to view a figure not as a
collection of lines on a piece of paper, but rather as a relationship of points
on a coordinate plane. The mathematician and philosopher René Descartes is
often credited with satisfactorily blending algebra and geometry to form
analytic geometry (Beckmann, 1971). Indeed, it is from his name that the modern
term Cartesian geometry arises. The
development of analytic geometry has also played an integral role in the
development of calculus; the definitions of the derivative as the slope of a
curve and the integral as the area under a curve are meaningless unless the relation
between algebra and geometry is well understood.
It can be said that the modern
concept of trigonometry originated in the late 16th and early 17th
centuries, through the works of such mathematicians as Viète, Fincke, and
Pitiscus, who began to develop methods of calculations of right triangles using
the six trigonometric functions familiar to us today. Up until that time,
trigonometry was still largely treated as the study of the lengths of arcs and
chords of a given circle. A comprehensive table of chords, corresponding to a
modern table of sines, had been compiled as early the second century AD by
Ptolemy (Smith, 1958), but this approach was not as well suited for working
with angles in the manner required by Archimedes’ inscribed and circumscribed
polygons.
With the advent of trigonometry,
we can now carry Archimedes’ calculations of p even further. We start with a circle with a diameter
of 2 units, in which we insert n
congruent isosceles triangles with a common vertex at the center of the circle.
The following diagram illustrates this for n
= 6.
If we call the non-congruent leg of the
triangle x, the perimeter of the
polygon formed by the triangles is given by nx.
Taking a closer look at the triangles, we see that the two congruent legs, both
being radii of the circle, are each 1 unit long. Furthermore, the angle between
these two legs is equal to 
.
To
avoid dealing with oblique triangles (and thus having to use the Law of
Cosines), we can split the triangle into two congruent right triangles.
From
this diagram, it follows that 
. Solving for x
yields 
. Therefore, the perimeter of the polygon (and hence our
approximation for the circumference) equals 
. Substituting this expression in the equation 
gives our approximation
of p to be 
. Evaluating this expression for n = 6 gives us the simple approximation of 
. From here, continuously doubling the value of n and applying the half-angle identities
(
and 
) produces an interesting result. For n = 12,
with
a decimal approximation of 3.105829. For n
= 24,
a
monster expression which reduces to the relatively tame 
, approximately 3.132629. It can be shown through proof by
mathematical induction that these repeated radicals for p will always appear in the form 
, where the number of nested radicals (that is, the number of
radicals within the first radical) is equal to m. This number corresponds to the number n of sides of the polygon inscribed in the circle by the formula 
. So for m = 6
(384-sided polygon), the approximation is
, or about 3.141558, which is accurate to 5 significant
figures.
Although the preceding
discussion presents a method which is strictly of the author’s invention,
several mathematicians since the development of trigonometry have used similar
approaches to derive an expression for p. François Viète (1540-1603), who among other things
was a pioneer in the development of trigonometry, dealt with the relationship
between an n-sided polygon and a 2n-sided polygon both inscribed in the
same circle. With the aid of trigonometric half-angle identities Viète was able
to derive the infinite product
(Beckmann, 1971)
Historically,
this derivation is significant in that it was the first infinite expression for
calculating p. During Viète’s time,
many mathematicians still believed that perhaps p was rational, that it could be expressed as the
quotient of two integers. As each new digit of p was discovered, these mathematicians’ flickering hope
diminished further. It seems reasonable to suspect that Viète’s expression, an
infinite product of nested radicals, all of which are themselves irrational,
may have seemed very near an actual proof of the irrationality of p, which was finally provided by Johann Heinrich
Lambert (1728-1777) some 150 years later (Beckmann, 1971).
After Viète came countless other
mathematicians who refined the known value of p by using polygons with an ever-increasing number of
sides (one Dutch mathematician’s polygon had 
sides). These
mathematicians had little, other than more and more digits, to add to the
theory of p-derivation. The next
great breakthrough in this field would not come until the invention of
calculus, which would change forever man’s efforts at calculating p.
The invention of calculus is
most often ascribed to Isaac Newton and Gottfried Leibniz. To be sure, these
two men did much for the development of this branch of mathematics, and were two
of the first to realize that the integral and the derivative were inverse
operations, but to give them full credit is grossly unfair to the many groundbreaking
mathematicians that preceded them.
One such mathematician was John
Wallis (1616-1703), who, using what essentially amounted to Riemann sums,
derived, in modern notation, the following infinite summation for p to calculate the area of a quarter-circle: 
. While Wallis lacked the insight to convert this expression
to an integral, he did manage to expand it to the infinite product 
(Beckmann, 1971).
Today, by applying the limit definition of the definite integral, we can easily
see that the previous equation is equivalent to 
. Furthermore, we can calculate this using trigonometric
substitution, but doing so yields the meaningless result 
. Alternatively, we can apply Isaac Newton’s modified
binomial theorem to expand the integrand then integrate the resulting infinite
series term-by-term, yielding a decimal approximation.
One other mathematician to
precede Newton and Leibniz noteworthy in respect to the calculations of p is the Scottish mathematician James Gregory. He is
perhaps best remembered for discovering a power series for the arctangent
function, which he knew to be equal to the area under the curve 
. By use of repeated long division coupled with the power
rule for integration discovered some time earlier by Blaise Pascal, Gregory derived
his namesake power series, 
. Leibniz later substituted x = 1 to yield 
(Beckmann, 1971). The
first infinite series derived for p, it is also one of the least useful since its rate of
convergence is significantly slower than many other series which were
discovered soon afterwards. Practicality aside, the Leibniz series for p is important in that it was the first infinite series
discovered for p, and its derivation marked the start of a totally different approach
to calculating p.
The publication of the Leibniz
series drew much attention from the math community of the time. Anyone
attempting to utilize the series immediately recognized its unforgivably slow
rate of convergence (matching Archimedes’ value would require over 300 terms).
With this in mind, a mathematician named Abraham Sharp (1651-1742) explored the
possibility of substituting a value into the Gregory series other than x = 1, the value for which the Gregory
series experiences the slowest rate of convergence. The value he chose was 
, which yields
(Beckmann, 1971)
It
is easy to see that Sharp did indeed succeed in deriving a series with a
greater rate of convergence than the elegant but slow Leibniz series. In the
Leibniz series, the denominators increase in arithmetic procession (3, 5, 7, 9,
…), whereas successive denominators of Sharp’s series increase exponentially
(9, 45, 189, 729). The difference is striking; after just 1000 terms, Sharp’s
series evaluates to 3.141592653591, a value correct to 12 significant figures,
while evaluating the first 100,000 terms of the Leibniz series yields
3.141582653589, which is correct to a meager 5 significant figures.
Drawing from the vast array of
discoveries made by Wallis, Gregory, and others, Newton and Leibniz
independently set to work systematically organizing the laws of differential
and integral calculus during the mid to late 17th centuries. While
each made many significant accomplishments during their lifetimes, both to
calculus and to mathematics in general, the work of Newton, who placed greater
importance upon power series, applies far more to the calculation of p. Newton’s own approach to this task was similar to
Wallis in that he took an equation of a circle and integrated to find the area
underneath it. The specific curve he used was given by the equation 
, which defines the upper half of the circle with center at
(0, ½) and radius ½. Newton integrated to find the area from x = 0 to x = ¼.
After factoring out a 
, he was left with the following integral: 
, which represents the area under the arc AD in the diagram
below.
This
area is also equivalent to the area of the sector ACD minus the area of
triangle BCD. Since 
, 
, and 
. Using the formulae for the area of a sector, 
, and the area of a triangle, Newton calculated the area
under arc AD to be 
. Newton had by this time discovered, by interpolating coefficients
for fractional powers on Pascal’s Triangle, how to expand a binomial to any
rational power to a power series. Using his modified binomial theorem, he
expanded the integrand of his previous expression to 
. Distributing the 
and integrating,
Newton calculated the area to be 
. After simplifying this expression and setting it equal to
the value obtained earlier, Newton derived the following identity:
(Beckmann, 1971). Although it is neither as simple nor
aesthetically pleasing as the derivations of Wallis and Leibniz, it is more
practical and useful since it converges much faster (note that successive
denominators of the series increase exponentially).
The next significant
mathematician to work extensively with p was Leonard Euler (1707-1783). Euler began his work
by evaluating the infinite summation of inverse squares (
). He managed this by solving the innocent-looking equation 
. He replaced the sine function with its power series to
yield 
. Dividing through by x
and making the substitution 
, he reached the equation 
. Since he knew the expression on the right side was
equivalent to 
, y ¹ 0 because he had divided by x in the original power series, he knew that y must equal p2, 4p2, 9p2, ….
According to a higher-level theorem, the sum of the reciprocals of the zeroes
of an equation is equal to the opposite of the ratio of the linear coefficient
to the constant term. Applying this principle to this equation, Euler derived
the identity 
. Multiplying through by p2 yielded
the result 
. Encouraged by his success, Euler repeated the same process
for the cosine function and derived another series for p2: 
Euler derived many, many
expressions for p, and to list them all would require an entire volume devoted to that
purpose. A few more examples will suffice before moving on. Euler discovered
infinite series for the reciprocals of all even powers up to 26, for which 
. He also derived an infinite product for p2: 
, as well as the following summation of arctangents: 
. (Beckmann, 1971)
Another important contribution
of Euler was the derivation and evaluation (using an infinite series of his own
discovery) of a new arctangent series for p. Inspired by the success of English mathematician
John Machin, Euler adopted a method of deriving an arctangent identity similar
to the one Machin had used some three or four decades ago. To begin with, Euler
supposed that there be two angles, call them a and b, for which 
. Next he arbitrarily picked 
to be the value of
the tangent of a, since he realized that for the identity to converge
quickly when expanded into series the angles would have to be fairly small, and
the arctangent of 
is very small indeed.
Using his arctangent series, he calculated a to be
approximately 0.1419, which divides into 
five times. Thus,
Euler obtained the equation 
and proceeded to
solve for b:
Applying
the sum/difference rule for tangent yields
To
calculate tan 5a Euler once
again used the sum/difference rule:
Back-substituting
into the previous equation,
At
this point, Euler had his two angles. He could have stopped here and left his
identity as 
. But Euler did not relish the thought expanding 
into series. He began
to wonder if perhaps b could be expressed as a multiple of a simpler,
smaller angle f. Setting b equal to 2f, Euler solved for f:
Algebraic
manipulation reduces this equation to a quadratic, which can simply be plugged
into the Quadratic Equation to solve for tan f (happily, the
discriminant turns out to be a perfect square):
Discarding
the negative root in favor of the positive one, Euler thus found the angle he
was looking for. With this newly acquired angle he obtained the equation 
. With this identity and his power series for arctangent, 
, where 
, Euler provided us with one of the fastest converging series
for p ever discovered up to
that time (Beckmann, 1971).
As we have seen, many
mathematicians throughout history have tried their hand at calculating the
value of p. Through their work, we
now have a mass of formulae which converge at infinity. This poses an obvious
question: which formula is ‘best’, and why? Historically there have been two
main factors in determining the effectiveness of a series or a formula – its
rate of convergence and its convenience (does it involve messy radicals and
fractions, etc.?). With the advent of the computer and calculator, the latter
is less of a concern, but the former is as important as ever because a slower
rate of convergence generally means more computing time to calculate to a given
number of decimals.
Through repeated testing of
several of the formulae it can be shown that the fastest by far is Euler’s
arctangent identity paired with his power series for arctangent, which, after
just 1000 terms evaluates to 3.141592653586962, a value accurate to 12
significant figures. By contrast, some of the slowest methods, such as the
Leibniz series, requires over 100,000 terms to obtain just six-digit accuracy.
For a complete list comparing the rates of convergence of several different
methods, see appendix A.
It is interesting to see how the
rates of convergence for the various methods have increased throughout history.
This comes as no surprise; the more contemporary mathematicians had the
benefits of more fully developed mathematics. Euler, for example, drew upon the
work of countless mathematicians before him in deriving his multitude of series
for p – the calculus of Newton
and Leibniz, the trigonometry of Viète and Fincke, the algebra of al-Khwarizmi,
the geometry of the ancient Greeks. However, this increase in rates of
convergence have often come at the expense of elegance and simplicity. Consider
again Euler’s arctangent series. Its evaluation involves both repeated multiplication
and division. By contrast, the Leibniz series requires only a simple
reciprocation. Needless to say, it takes far less time to evaluate a single
term of the latter series. Euler’s series, however, contains many redundant
steps, and from a programming standpoint, could be made much more efficient.
Thus, the true rate of convergence of a series (accuracy as a function of time)
is as much a question of implementation as the complexity of the series.
What now? Formulae for p have been improving for centuries. How much better
could they possibly become? Not too recently, we broke the one billion digit
mark. Does it even matter anymore? Probably not. But that will not stop
mathematicians from searching for more and more digits. Historically, approximating
p has been one of the
quintessential examples of mathematics for its own sake. We will never need so
many digits, but in the process of deriving and applying formulae, we gather
new insights that we can apply to other fields. And that is what mathematics is
all about.
