"It is probable that the number π is not even contained among the algebraical irrationalities, i.e., that it cannot be a root of an algebraical equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly." (Adrien-Marie Legendre, "Elements de geometrie", 1794)
"What good is your beautiful investigation regarding π? Why study such problems, since irrational numbers do not exist?" (Leopold Kronecker [letter to Ferdinand von Lindemann] 1882)
"The proof that π is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that π can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots."
"The mysterious and wonderful π is reduced to a gargle that helps computing machines clear their throats." (Philip J Davis, "The Lore of Large Numbers", 1961)
"This mysterious π which comes in at every door and window, and down every chimney." (Augustus De Morgan, "A Budget of Paradoxes", 1872)
“Ten decimal places of π are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope.” (Simon Newcomb)
“Ten decimal places of π are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope.” (Simon Newcomb)
"If we take the geometrical relations, the thousandth decimal of π sleeps there, though no one may ever try to compute it." (William James, "The Meaning of Truth", 1909)
"The number was first studied in respect of its rationality
or irrationality, and it was shown to be really irrational. When the discovery
was made of the fundamental distinction between algebraic and transcendental
numbers, i. e. between those numbers which can be, and those numbers which
cannot be, roots of an algebraical equation with rational coefficients, the
question arose to which of these categories the number π belongs. It was
finally established by a method which involved the use of some of the most
modern of analytical investigation that the number π was transcendental. When
this result was combined with the results of a critical investigation of the
possibilities of a Euclidean determination, the inferences could be made that
the number π, being transcendental, does not admit of a construction either by
a Euclidean determination, or even by a determination in which the use of other
algebraic curves besides the straight line and the circle are permitted." (Ernest
W Hobson, "Squaring the Circle", 1913)
"What does it matter whether π is rational or irrational? A mathematician faced with [this] question is in much the same position as a composer of music being questioned by someone with no ear for music. Why do you select some sets of notes and have them repeated by musicians, and reject others as worthless? It is difficult to answer except to say that there are harmonies in these things which we find that we can enjoy. It is true of course that some mathematics is useful. [...] But the so-called pure mathematicians do not do mathematics for such [practical applications]. It can be of no practical used to know that π is irrational, but if we can know it would surely be intolerable not to know. (Edward C Titchmarsh, "Mathematics for the General Reader", 1948)
"The mysterious and wonderful π is reduced to a gargle that helps computing machines clear their throats." (Philip J Davis, "The Lore of Large Numbers", 1961)
"Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number π." (William L Schaaf, "Nature and History of π", 1967)
"The most remarkable thing about π, however, is the way it turns up all over the place in math, including in calculations that seem to have nothing to do with circles." (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)
"There is more to the calculation of π to a large number of decimal places than just the challenge involved. One reason for doing it is to secure statistical information concerning the 'normalcy' of π. A real number is said to be simply normal if in its decimal expansion all digits occur with equal frequency, and it is said to be normal if all blocks of digits of the same length occur with equal frequency. It is not known if π (or even √2, for that matter) is normal or even simply normal." (Howard Eves, "Mathematical Circles Revisited", 1971)
"The matter of the normalcy or non-normalcy of π will never, of course, be resolved by electronic computers. We have here an example of a theoretical problem which requires profound mathematical talent and cannot be solved by computations alone. The existence of such problems ought to furnish at least a partial antidote to the disease of computeritis, which seems so rampant today." (Howard Eves, "Mathematical Circles Revisited", 1971)
“The digits of π beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere” (Petr Beckmann, “A History of π”, 1976)
“The digits of π beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere” (Petr Beckmann, “A History of π”, 1976)
"Computing π is the ultimate stress test for a computer - a kind of digital cardiogram." (Ivars Peterson, Islands of Truth, 1990)
“The digits of π march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)
“π is not the solution to any equation built from a less than infinite series of whole numbers. If equations are trains threading the landscape of numbers, then no train stops at π.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)
“The digits of π march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)
“π is not the solution to any equation built from a less than infinite series of whole numbers. If equations are trains threading the landscape of numbers, then no train stops at π.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)
"The story of π has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics." (Eli Maor, "e: The Story of a Number", 1994)
"There’s a beauty to π that keeps us looking at it [...] The digits of π are extremely random. They really have no pattern, and in mathematics that’s really the same as saying they have every pattern." (Peter Borwein, 1996)
"When we think of π, let’s not always think of circles. It is related to all the odd whole numbers. It also is connected to all the whole numbers that are not divisible by the square of a prime. And it is part of an important formula in statistics. These are just a few of the many places where it appears, as if by magic. It is through such astonishing connections that mathematics reveals its unique and beguiling charm." (Sherman K Stein, "Strength in Numbers", 1996)
"The story of π reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics. A surprising amount of the most important mathematicians and a significant number of the most important mathematicians have contributed to its unfolding - directly or otherwise." (J Lennart Berggren et al, "π", 2004)
“A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say π.” (Daniel Tammet, “Thinking in Numbers: How Maths Illuminates Our Lives”, 2012)
“A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say π.” (Daniel Tammet, “Thinking in Numbers: How Maths Illuminates Our Lives”, 2012)
"It turns out π is different. Not only is it incapable of being expressed as a fraction, but in fact π fails to satisfy any algebraic relationship whatsoever. What does π do? It doesn’t do anything. It is what it is. Numbers like this are called transcendental (Latin for 'climbing beyond'). Transcendental numbers - and there are lots of them - are simply beyond the power of algebra to describe."
"Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π." (Steven Strogatz, "Why π Matters" 2015)
"Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π." (Steven Strogatz, "Why π Matters" 2015)
"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way."
"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the "i times π" power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Since it’s impossible to express an irrational number such as π as a fraction, the quest for a fraction equal to π could never be successful. Ancient mathematicians didn’t know that, however. As noted above, it wasn’t until the eighteenth century that the irrationality of π was demonstrated. Their labors weren’t in vain, though. While enthusiastically pursuing their fundamentally doomed enterprise, they developed a lot of interesting mathematics as well as impressively accurate approximations of π." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"The geometric interpretation of e^iπ is rich with emblematic potential. You could see its suggestion of a 180-degree spin as standing for a soldier’s about-face, a ballet dancer’s half pirouette, a turnaround jump shot, the movement of someone setting out on a long journey who looks back to wave farewell, the motion of the sun from dawn to dusk, the changing of the seasons from winter to summer, the turning of the tide. You could also associate it with turning the tables on someone, a reversal of fortune, turning one’s life around, the transformation of Dr. Jekyll into Mr. Hyde (and vice versa), the pivoting away from loss or regret to face the future, the ugly duckling becoming a beauty, drought giving way to rain. You might even interpret its highlighting of opposites as an allusion to elemental dualities - shadow and light, birth and death, yin and yang." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"The most remarkable thing about π, however, is the way it turns up all over the place in math, including in calculations that seem to have nothing to do with circles." (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)
"Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature - the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, e^iπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"[…] the story of π is the deeply ironic tale of one thinker after another trying to nail down the size of a number that is fundamentally immeasurable. (Because it’s irrational.)" (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"There’s something so paradoxical about pi. On the one hand, it represents order, as embodied by the shape of a circle, long held to be a symbol of perfection and eternity. On the other hand, pi is unruly, disheveled in appearance, its digits obeying no obvious rule, or at least none that we can perceive. Pi is elusive and mysterious, forever beyond reach. Its mix of order and disorder is what makes it so bewitching." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"With its yin and yang binaries, pi is like all of calculus
in miniature. Pi is a portal between the round and the straight, a single number
yet infinitely complex, a balance of order and chaos. Calculus, for its part,
uses the infinite to study the finite, the unlimited to study the limited, and
the straight to study the curved. The Infinity Principle is the key to
unlocking the mystery of curves, and it arose here first, in the mystery of pi."
"Exploring π is like exploring the universe." (David Chudnovsky)
"God is in the details, for mathematicians have plunged deeper and deeper within π's digits with a religious fervor, hoping to find even a hint of understanding." (Ludwig Mies van der Rohe)
"π is more like exploring underwater. You’re in the mud, and every thing looks the same." (Gregory Chudnovsky)
"π is not just a collection of random digits. π is a journey, an experience; unless you try to see the natural poetry that exists in π, you will find it very difficult to learn." (Antranig Basman)
