A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes.

In base 10, all the permutable primes with fewer than 49,081 digits are known

Of the above, there are 16 unique permutation sets, with smallest elements

Note

is a repunit, a number consisting only of n ones (in base 10). Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proven that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.

There is no n-digit permutable prime for

which is not a repunit. It is conjectured that there are no non-repunit permutable primes other than those listed above.]]>Narcissistic Number is a number that is the sum of its own digits each raised to the power of the number of digits:

Examples :

153

Explanation:

1634

Explanation:

In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base b is a number that is the sum of its own digits each raised to the power of the number of digits.

**Definition**

Let n be a natural number. We define the narcissistic function for base

to be the following:where is the number of digits in the number in base b, and

is the value of each digit of the number. A natural number n is a narcissistic number if it is a fixed point for , which occurs if . The natural numbers are trivial narcissistic numbers for all b, all other narcissistic numbers are nontrivial narcissistic numbers.

For example, the number 153 in base

is a narcissistic number, because k=3 and .]]>In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written

, where . The conjugate of z is . Since , the product of a number z with its conjugate is , an isotropic quadratic form, .The collection D of all split complex numbers z = x + y j for x, y ∈ R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N(wz) = N(w)N(z). This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.

A similar algebra based on

and component-wise operations of addition and multiplication, , where xy is the quadratic form on , also forms a quadratic space. The ring isomorphismrelates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of R2 is at a distance √2 from 0, which is normalized in D.

Split-complex numbers have many other names;

In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition.

For example,

Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis.

**Definition**

A split-complex number is an ordered pair of real numbers, written in the form

where x and y are real numbers and the quantity j satisfies

Choosing

results in the complex numbers. It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity j here is not a real number but an independent quantity.The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

This multiplication is commutative, associative and distributes over addition.

]]>In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

Starting from n = 1, the sequence of harmonic numbers begins:

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.

Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.

The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.

Bertrand's postulate implies that, except for the case n = 1, the harmonic numbers are never integers.

]]>Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first-order statements about R to be reinterpreted as true first-order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

(for any finite number of terms).Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since

for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955.Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes

for an infinitesimal , where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.]]>For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". Nevertheless, the term "transfinite" also remains in use.

**Definition**

Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. The most notable ordinal and cardinal numbers are, respectively:

(Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering. (Aleph-null): the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, . If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one.The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between

and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proven.Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:

is a transfinite cardinal. That is, there is a Dedekind infinite set such that the cardinality of isThere is a cardinal

such thatAlthough transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers.

a) In set theory, **an ordinal number**, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another.

Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. The basic idea of ordinal numbers is to generalize this process to possibly infinite collections and to provide a "label" for each step in the process. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.

An ordinal number is used to describe the order type of a well-ordered set (though this does not work for a well-ordered proper class). A well-ordered set is a set with a relation < such that:

* (Trichotomy) For any elements x and y, exactly one of these statements is true:

* x < y

* y < x

* x = y

* (Transitivity) For any elements x, y, z, if x < y and y < z, then x < z.

* (Well-foundedness) Every nonempty subset has a least element, that is, it has an element x such that there is no other element y in the subset where y < x.

Two well-ordered sets have the same order type, if and only if there is a bijection from one set to the other that converts the relation in the first set, to the relation in the second set.

b) In mathematics, **cardinal numbers**, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

]]>Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called hypercomplex numbers. They include the quaternions H, introduced by Sir William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative, and the sedenions, in which multiplication is not alternative, neither associative nor commutative.

In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

A definition of a hypercomplex number is given by Kantor & Solodovnikov (1989) as an element of a finite-dimensional algebra over the real numbers that is unital but not necessarily associative or commutative. Elements are generated with real number coefficients

for a basis . Where possible, it is conventional to choose the basis so that . A technical approach to hypercomplex numbers directs attention first to those of dimension two.]]>In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

**Definition**

Stated mathematically, let X be a positive integer with m digits when written in base n, and let the digits be

(It follows that must be either zero or a positive integer up to . X can be expressed asX is a harshad number in base n if:

A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and 6. The number 12 is a harshad number in all bases except octal.

**Examples**

The number 18 is a harshad number in base 10, because the sum of the digits 1 and 8 is 9 (1 + 8 = 9), and 18 is divisible by 9.

The Hardy–Ramanujan number (1729) is a harshad number in base 10, since it is divisible by 19, the sum of its digits (1729 = 19 × 91).

The number 19 is not a harshad number in base 10, because the sum of the digits 1 and 9 is 10 (1 + 9 = 10), and 19 is not divisible by 10.

6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following rule:

* Take any four-digit number, using at least two different digits (leading zeros are allowed).

* Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.

* Subtract the smaller number from the bigger number.

* Go back to step 2 and repeat.

The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1495:

9541 – 1459 = 8082

8820 – 0288 = 8532

8532 – 2358 = 6174

7641 – 1467 = 6174

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4.

**Other "Kaprekar's constants"**

There can be analogous fixed points for digit lengths other than four, for instance if we use 3-digit numbers then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Kaprekar constants".

**Other properties**

* 6174 is a Harshad number, since it is divisible by the sum of its digits.

* 6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7.

* 6174 can be written as the sum of the first three degrees of 18:

The sum of squares of the prime factors of 6174 is a square:

]]>In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

]]>Pierre de Fermat (between 31 October and 6 December 1607 - 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' *Arithmetica*. He was also a lawyer at the *Parlement* of Toulouse, France.

No. of known terms : 5

Conjectured no. of terms : 5

Subsequence of Fermat numbers

First terms : 3, 5, 17, 257, 65537

Largest known term : 65537

In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form

where n is a non-negative integer. The first few Fermat numbers are:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ...

If

**Basic properties**

The Fermat numbers satisfy the following recurrence relations:

for n ≥ 1,

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

and Fj; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd.

**Further properties**

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers

are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that]]>In mathematics, Cahen's constant is defined as an infinite series of unit fractions with alternating signs:

Here

denotes Sylvester's sequence, which is defined recursively by andCombining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:

This constant is named after Eugène Cahen (also known for the Cahen-Mellin integral), who was the first to introduce it and prove its irrationality.

]]>The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:

= 2.6651441426902251886502972498731...which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general Gelfond–Schneider theorem, which solved the part of Hilbert's seventh problem described below.

**Properties**

The square root of the Gelfond–Schneider constant is the transcendental number

1.63252691943815284477....This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either

is rational, which proves the theorem, or it is irrational (as it turns out to be), and thenis an irrational to an irrational power that is rational, which proves the theorem. The proof is not constructive, as it does not say which of the two cases is true, but it is much simpler than Kuzmin's proof.

]]>In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is

, that is, e raised to the power . Like both e and , this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting thatwhere i is the imaginary unit. Since −i is algebraic but not rational, is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is , known as the Gelfond–Schneider constant. The related value is also irrational.

The decimal expansion of Gelfond's constant begins

23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492196....]]>You know what's cooler than ONE pie? … TWO pies. In other words, two times pi, or the number "tau," which is roughly 6.28.

"Using tau makes every formula clearer and more logical than using pi," said John Baez, a mathematician at the University of California, Riverside. "Our focus on pi rather than 2pi is a historical accident."

Tau is what shows up in the most important formulas, he said.

While pi relates a circle's circumference to its diameter, tau relates a circle's circumference to its radius — and many mathematicians argue that this relationship is much more important. Tau also makes seemingly unrelated equations nicely symmetrical, such as the one for a circle's area and an equation describing kinetic and elastic energy.

**Tau – A Mathematical Constant**

*What is Tau?*

The constant is numerically equal to 2*pi (2 times pi), and with value approximately 6.28. The ratio equates to 2*C/D. Where C is circumference and D is diameter of circle.

**Applications of Tau**

* There are many expressions that actually require “2*pi” calculation, having tau being equal to that simplifies them to great extent, for e.g Circumference of circle = 2*pi*r = tau*r.

* Concept of tau can be useful in angular measurements like angles in radians, representing as a complete “one-turn” and cos,sine functions in trigonometry have period of tau.

* These concepts can be useful for teaching geometry as would reduce the confusion of using “pi” and “2*pi” at many applications and would help get rid of factor of 2.

* Tau simplifies euler’s identity by eradicating the factor of 2.

* It is useful at many places where “2*pi” are used such as fourier transforms, cauchy integral formula’s etc.

**Criticism against Tau**

* Since it contradicts with the symbols of torque, shear stress and time, this symbol has been a lot of criticism.

* We already had a ratio of “C/D” equal to pi, having another circle ratio with factor of two will create confusion in choice.

* There exist formulas which look more elegant as expression of “pi” rather than tau, for example, area of circle = pi*r*r = (tau*r*r)/2, introducing an extra factor of “1/2”.

**Coding Prospects**

Since Programming has always been trying to match up with mathematical advancements, symbol of tau has been introduced as a constant in recent python 3.6 under the math module.

**Output:**

The value of tau (using 2*pi) is : 6.283185307179586

The value of tau (using in-built tau) is : 6.283185307179586.

**Let's Use Tau--It's Easier Than Pi**

There aren't many things that Congress can agree on, but in early 2009 it passed a bipartisan resolution designating March 14th of each year as "Pi Day." Pi, the mathematical constant that students first encounter with the geometry of circles, equals about 3.14, hence its celebration on March 14. The math holiday had been a staple of geeks and teachers for years—festivities include eating pie the pastry while talking about pi the number—but dissent began to appear from an unexpected quarter: a vocal and growing minority of mathematicians who rally around the radical proposition that pi is wrong.

They don't mean anything has been miscalculated. Pi (π) still equals the same infinite string of never-repeating digits. Rather, according to *The Tau Manifesto*, "pi is a confusing and unnatural choice for the circle constant." Far more relevant, according to the algebraic apostates, is 2π, aka tau.

*Manifesto* author Michael Hartl received his PhD in theoretical physics from the California Institute of Technology and is only one in a string of established players beginning to question the orthodoxy. Last year the University of Oxford hosted a daylong conference titled "Tau versus Pi: Fixing a 250-Year-Old Mistake." In 2012 the Massachusetts Institute of Technology modified its practice of letting applicants know admissions decisions on Pi Day by further specifying that it will happen at tau time—that is, at 6:28 P.M. The Internet glommed onto the topic as well, with its traditional fervor for whimsical causes. YouTube videos on the subject abound with millions of views and feisty comment sections—hardly a common occurrence in mathematical debates.

The crux of the argument is that pi is a ratio comparing a circle’s circumference with its diameter, which is not a quantity mathematicians generally care about. In fact, almost every mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that connects a circumference to that quantity.

But usage of pi extends far beyond the geometry of circles. Critical mathematical applications such as Fourier transforms, Riemann zeta functions, Gaussian distributions, roots of unity, integrating over polar coordinates and pretty much anything involving trigonometry employs pi. And throughout these diverse mathematical areas the constant π is preceded by the number 2 more often than not. Tauists (yes, they call themselves tauists) have compiled exhaustively long lists of equations—both common and esoteric, in both mathematics and physics—with 2π holding a central place. If 2π is the perennial theme, the almost magically recurring number across myriad branches of mathematics, shouldn’t that be the fundamental constant we name and celebrate?

If that’s all there was, the tau movement would likely be a curiosity and nothing more. But reasons for switching to tau are deeply rooted in pedagogy as well. University of Utah mathematics professor Robert Palais, who is considered the founding father of the movement, started the "pi is wrong" ruckus with an article of the same name in 2001[pdf]. The article, which should be required reading for all advanced high school students, creates a tantalizing picture of how much easier certain fundamental concepts of trigonometry could be in an alternate universe where we use tau. For example, with pi-based thinking, if you want to designate a point one third of the way around the circle, you say it has gone two thirds pi radians. Three quarters around the same circle has gone one and a half pi radians. Everything is distorted by a confusing factor of two. By contrast, a third of a circle is a third of tau. Three quarters of a circle is three quarters tau. As a result of pi, Palais says, "the opportunity to impress students with a beautiful and natural simplification is turned into an absurd exercise in memorization and dogma."

At its heart, pi refers to a semicircle, whereas tau refers to the circle in its entirety. Mathematician and poet Mike Keith once wrote a 10,000 word poem dedicated to the first 10,000 digits of pi. He is now a proponent of tau. According to a PBS article from last year, he said that thinking in terms of pi is like reaching your destination and saying you're twice halfway there.

For mathematicians, pi obscures some of the underlying symmetries of mathematics and muddies up what should be elegant with extraneous factors of two. There’s an admittedly grandiose idea that mathematics is the language with which we express and see certain underpinning truths to the universe. To clutter that language with superfluous twos would be as bad as littering a Shakespearean monologue with “likes” and “umms” and “whatevers.” As the Bard nearly wrote, “Knowledge is two of the half-wings wherewith we fly to heaven.”

We Americans have almost a proud tradition of using poorly chosen units because of inertia: Fahrenheit instead of Celsius, miles instead of kilometers. Even the great Benjamin Franklin inadvertently established the convention of calling positive charge negative and vice-versa as a result of his experiments with electricity.

Indeed, the whole problem began as a historical accident, tauists say. In early civilizations a diameter was an easier quantity to measure than a radius. So when the Babylonians or Egyptians wanted rules of thumb for their architecture, a ratio of circumference to diameter is what they turned to. (The two civilizations estimated it to be 3.125 and 3.16, respectively.) Even the Bible specifies the ratio of a circle’s diameter to its circumference: “And [Hiram] made a molten sea, 10 cubits from the one brim to the other: it was round all about, and…a line of thirty cubits did compass it round about” (1 Kings 7:23).

The Greeks used formal geometric proofs to estimate the circumference-to-diameter ratio. Archimedes (he of the lever and shouts of "Eureka!") found strict lower and upper bounds of 3.1408 and 3.1429. Yet his choice of comparing the circumference with diameter was arbitrary; he could just as easily have used radius instead. (Interestingly, Archimedes did not use the Greek letter π. That didn't come until Swiss mathematician Leonhard Euler popularized the convention in 1736, and even he seemed to be ambivalent about whether to define π as 3.14 or as the 6.28 we now write as τ.)

Although switching to tau when all the textbooks and academic papers use pi may sound daunting, it doesn’t need to be. There could be a transitional period of using both mathematical constants while we phase out the old and humor the intransigents who can’t or won’t change.

Asked in an e-mail about the reaction his original piece has received, Palais is humbled. "I never would have imagined the scale of the discussion," he says. And given that it's already far exceeded his expectations, he expresses optimism that it could continue even further.

]]>