Square root of 2

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"Pythagoras's constant" redirects here; not to be confused with Pythagoras number
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1.

The square root of 2, often known as root 2, radical (rad) 2, or Pythagoras' constant, and written as 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value, truncated to 65 decimal places, is:

1.41421356237309504880168872420969807856967187537694807317667973799... (sequence A002193 in OEIS).
The square root of 2 (number line not to scale).

The quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10−5). The approximation 665857/470832 is valid to within 1.13 x 1012: its square is 2.0000000000045....

Binary 1.0110101000001001111...
Decimal 1.4142135623730950488...
Hexadecimal 1.6A09E667F3BCC908B2F...
Continued fraction 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}

History[edit]

Babylonian clay tablet YBC 7289 with annotations. Besides showing the square root of 2 in sexagesimal (1 24 51 10), the tablet also gives an example where one side of the square is 30 and the diagonal then is 42 25 35. The sexagesimal digit 30 can also stand for 1/2, in which case 42 25 35 is approximately 0.7071065.

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of 2 in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits,[1] and is the closest possible three-place sexagesimal representation of 2:

1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = \frac{30547}{21600} = 1.41421\overline{296}.

Another early close approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC) as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth.[2] That is,

1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot4 \cdot 34} = \frac{577}{408} = 1.414\overline{2156862745098039}.

This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of 2. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.[3][4][5] The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example Conway & Guy (1996).[6]

Computation algorithms[edit]

There are a number of algorithms for approximating 2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method[7] of computing square roots, which is one of many methods of computing square roots. It goes as follows:

First, pick a guess, a_0 > 0; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:

a_{n+1} = \frac{a_n + \frac{2}{a_n}}{2}=\frac{a_n}{2}+\frac{1}{a_n}.

The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits. Starting with a0 = 1 the next approximations are

  • 3/2 = 1.5
  • 17/12 = 1.416...
  • 577/408 = 1.414215...
  • 665857/470832 = 1.4142135623746....

The value of 2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997. In February 2006 the record for the calculation of \sqrt{2} was eclipsed with the use of a home computer. Shigeru Kondo calculated 200,000,000,000 decimal places in slightly over 13 days and 14 hours using a 3.6 GHz PC with 16 GiB of memory.[8] Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely.[9] Such computations aim to check empirically whether such numbers are normal.

Proofs of irrationality[edit]

A short proof of the irrationality of 2 can be obtained from the rational root theorem, that is, if p(x) is a monic polynomial with integer coefficients, then any rational root of p(x) is necessarily an integer. Applying this to the polynomial p(x) = x^2 - 2, it follows that 2is either an integer or irrational. Because 2 is not an integer (2 is not a perfect square), 2 must therefore be irrational. This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational.

See quadratic irrational or infinite descent#Irrationality of √k if it is not an integer for a proof that the square root of any non-square natural number is irrational.

Proof by infinite descent[edit]

One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true.

  1. Assume that 2 is a rational number, meaning that there exists a pair of integers whose ratio is 2.
  2. If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
  3. Then 2 can be written as an irreducible fraction a/b such that a and b are coprime integers (having no common factor).
  4. It follows that a^2 / b^2 = 2 and a^2 = 2 b^2.   ( (a/b)^n = a^n/b^n  )
  5. Therefore a^2 is even because it is equal to 2 b^2. (2 b^2 is necessarily even because it is 2 times another whole number and multiples of 2 are even.)
  6. It follows that a must be even (as squares of odd integers are never even).
  7. Because a is even, there exists an integer k that fulfills: a = 2k.
  8. Substituting 2k from step 7 for a in the second equation of step 4: 2 b^2 = (2k)^2 is equivalent to 2b^2 = 4k^2, which is equivalent to b^2 = 2k^2.
  9. Because 2k^2 is divisible by two and therefore even, and because 2k^2 = b^2, it follows that b^2 is also even which means that b is even.
  10. By steps 5 and 8 a and b are both even, which contradicts that a/b is irreducible as stated in step 2.
Q.E.D.

Because there is a contradiction, the assumption (1) that \sqrt{2} is a rational number must be false. This means that \sqrt{2} is not a rational number; i.e., \sqrt{2} is irrational.

This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.[10] It appeared first as a full proof in Euclid's Elements, as proposition 117 of Book X. However, since the early 19th century historians have agreed that this proof is an interpolation and not attributable to Euclid.[11]

Proof by unique factorization[edit]

An alternative proof uses the same approach with the fundamental theorem of arithmetic which says every integer greater than 1 has a unique factorization into powers of primes.

  1. Assume that \sqrt{2} is a rational number. Then there are integers a and b such that a is coprime to b and \sqrt{2} =a/b. In other words, \sqrt{2} can be written as an irreducible fraction.
  2. The value of b cannot be 1 as there is no integer a the square of which is 2.
  3. There must be a prime p which divides b and which does not divide a, otherwise the fraction would not be irreducible.
  4. The square of a can be factored as the product of the primes into which a is factored but with each power doubled.
  5. Therefore by unique factorization the prime p which divides b, and also its square, cannot divide the square of a.
  6. Therefore the square of an irreducible fraction cannot be reduced to an integer.
  7. Therefore the square root of 2 cannot be a rational number.

This proof can be generalized to show that if an integer is not an exact kth power of another integer then its kth root is irrational. For a proof of the same result which does not rely on the fundamental theorem of arithmetic, see: quadratic irrational.

Proof by infinite descent, not involving factoring[edit]

The following reductio ad absurdum argument showing the irrationality of \sqrt{2} is less well-known. It uses the additional information 2 > \sqrt{2} > 1 so that 1 > \sqrt{2} - 1 > 0.[12]

  1. Assume that \sqrt{2} is a rational number. This would mean that there exist positive integers m and n with n \ne 0 such that m/n = \sqrt{2}. Then m = n \sqrt{2} and m\sqrt{2} = 2n.
  2. We may assume that n is the smallest integer so that n\sqrt{2} is an integer. That is, that the fraction m/n is in lowest terms.
  3. Then \sqrt{2} = \frac{m}{n}=\frac{m(\sqrt{2}-1)}{n(\sqrt{2}-1)}=\frac{2n-m}{m-n}
  4. Because 1 > \sqrt{2} - 1 > 0, it follows that n > n(\sqrt{2} - 1) = m - n > 0.
  5. So the fraction m/n for \sqrt{2}, which according to (2) is already in lowest terms, is represented by (3) in strictly lower terms. This is a contradiction, so the assumption that \sqrt{2} is rational must be false.

Geometric proof[edit]

Irrationality of sqrt2.svg

Another reductio ad absurdum showing that \sqrt{2} is irrational is less well-known.[13] It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the previous proof viewed geometrically.

Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = \sqrt{2}. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.

Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and the ∠BAC and ∠DAE coincide. Therefore the triangles ABC and ADE are congruent by SAS.

Because ∠EBF is a right angle and ∠BEF is half a right angle, BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n. By symmetry, DF = m − n, and FDC is also a right isosceles triangle. It also follows that FC = n − (m − n) = 2n − m.

Hence we have an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore m and n cannot be both integers, hence \sqrt{2} is irrational.

Pythagorean Theorem Proof[edit]

Suppose \sqrt{2} is rational.

That means that we can make a right isosceles triangle where the side lengths are natural numbers and the legs and the hypotenuse do not share any common factors (except 1). {1}

Since the legs are equal, so are their squares. So in order for the Pythagorean Theorem to work for this special right triangle, the square of the hypotenuse has to be an even number (and if we cut it in half once then we have the area of the square of the leg).

Recall that the square of an even number is even and the square of an odd number is odd. So if the square of the hypotenuse is even the hypotenuse is even as well. {2}

Remember that a square is a quadrilateral with 2 pairs of parallel sides which are equal in length and has 4 right angles. So both sides of the square of the hypotenuse are even.

So the square of the hypotenuse of this right triangle can be cut in half twice and still have integer area. Since we only want to cut it in half once, then we'll get an even number.

So the square of the leg is even. Now according to {2} the leg must be even

This contradicts our assumption at {1} that the leg and hypotenuse have no common factors (except 1). Because if they're both even they share a common factor of 2. So the assumption that \sqrt{2} was rational has to be false. Or in other words \sqrt{2} is an irrational number. Q. E. D.

Analytic proof[edit]

  • Lemma: let \alpha \in \mathbb{R}^+ and p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N} such that \left|\alpha q_n - p_n \right| \neq 0 for all n \in \mathbb{N} and
\lim_{n \rightarrow \infty} p_n = \lim_{n \rightarrow \infty} q_n = \infty\,
\lim_{n \rightarrow \infty} \left| \alpha q_n - p_n \right| = 0.\,
Then α is irrational.

Proof: suppose α = a/b with ab ∈ N+.

For sufficiently big n,

0 < \left| \alpha q_n - p_n \right| < \frac{1}{b}

then

0 < \left| a q_n /\ \! b - p_n \right| < \frac{1}{b}
0 < \left| a q_n - b p_n \right| < 1 \,

but aq_n - bp_n is an integer, absurd, then α is irrational.

  • \sqrt{2} is irrational.

Proof: let p_1  = q_1 = 1 and

p_{n+1} = p_n^2 + 2 q_n^2 \,
q_{n+1} = 2 p_n q_n \,\!

for all n \in \mathbf{N}.

By induction,

0 < \left| \sqrt{2} q_n - p_n \right| < \frac{1}{2^{2^{n-1}}}

for all n \in \mathbf{N}. For n = 1,

0 < \left| \sqrt{2} q_1 - p_1 \right| < \frac{1}{2}

and if is true for n then is true for n + 1. In fact

0 < \left| \sqrt{2} q_n - p_n \right|^2 < \frac{1}{2^{2^n}}
0 < \left| \sqrt{2} (2 p_n q_n) - (p_n^2 + 2 q_n^2) \right| < \frac{1}{2^{2^n}}
0 < \left| \sqrt{2} q_{n + 1} - p_{n + 1} \right| < \frac{1}{2^{2^n}}.

By application of the lemma, \sqrt{2} is irrational.

Constructive proof[edit]

In a constructive approach, one distinguishes between on the one hand not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given positive integers a and b, because the valuation (i.e., highest power of 2 dividing a number) of 2b2 is odd, while the valuation of a2 is even, they must be distinct integers; thus |2 b^2 - a^2| \geq 1. Then[14]

\left|\sqrt2 - \frac{a}{b}\right| = \frac{|2b^2-a^2|}{b^2(\sqrt{2}+a/b)} \ge \frac{1}{b^2(\sqrt2 + a / b)} \ge \frac{1}{3b^2},

the latter inequality being true because we assume \tfrac{a}{b} \le 3- \sqrt{2} (otherwise the quantitative apartness can be trivially established). This gives a lower bound of \frac{1}{3b^2} for the difference |\sqrt2 - a / b|, yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop (1985, p. 18). This proof constructively exhibits a discrepancy between \sqrt{2} and any rational.

Properties of the square root of two[edit]

Angle size and sector area are the same when the conic radius is square root of two. This diagram illustrates the circular and hyperbolic functions based on sector areas u.

One-half of \sqrt{2}, also 1 divided by the square root of 2, approximately 0.70710 67811 86548, is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates

\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).

This number satisfies

\tfrac{1}{2}\sqrt{2} = \sqrt{\tfrac{1}{2}} = \frac{1}{\sqrt{2}} = \cos(45^{\circ}) = \sin(45^{\circ}).

One interesting property of the square root of 2 is as follows:

 \!\ {1 \over {\sqrt{2} - 1}} = \sqrt{2} + 1

since (\sqrt{2}+1)(\sqrt{2}-1)=2-1=1. This is related to the property of silver ratios.

The square root of 2 can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations:

\frac{\sqrt{i}+i \sqrt{i}}{i}\text{ and }\frac{\sqrt{-i}-i \sqrt{-i}}{-i}

if the square root symbol is interpreted suitably for the complex numbers i and -i.

The square root of 2 is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: If for c > 1 we define x1 = c and xn+1 = cxn for n > 1, we will call the limit of xn as n → ∞, if this limit exists, by the name f(c). Then sqrt(2) is the only number c > 1 for which f(c) = c2. Or symbolically:

\sqrt{2}^ {(\sqrt{2}^ {(\sqrt{2}^ {(\ \cdot^ {\cdot^ \cdot)))}}}} = 2.

The square root of 2 appears in Viète's formula for π:

2^m\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}} \to \pi\text{ as }m \to \infty\,

for m square roots and only one minus sign.[15]

Similar in appearance but with a finite number of terms, the square root of 2 appears in various trigonometric constants:[16]

\sin(5\tfrac58 ^\circ) = \frac12\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}};
\sin(11\tfrac14 ^\circ) = \frac12\sqrt{2-\sqrt{2+\sqrt{2}}};
\sin(16\tfrac78 ^\circ) = \frac12\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}};
\sin(22\tfrac12 ^\circ) = \frac12\sqrt{2-\sqrt{2}};
\sin(28\tfrac18 ^\circ) = \frac12\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}};
\sin(33\tfrac34 ^\circ) = \frac12\sqrt{2-\sqrt{2-\sqrt{2}}};
\sin(39\tfrac38 ^\circ) = \frac12\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}};
\sin(45 ^\circ) = \frac12\sqrt{2};
\sin(50\tfrac58 ^\circ) = \frac12\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}};
\sin(56\tfrac14 ^\circ) = \frac12\sqrt{2+\sqrt{2-\sqrt{2}}};
\sin(61\tfrac78 ^\circ) = \frac12\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}};
\sin(67\tfrac12 ^\circ) = \frac12\sqrt{2+\sqrt{2}};
\sin(73\tfrac18 ^\circ) = \frac12\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}};
\sin(78\tfrac34 ^\circ) = \frac12\sqrt{2+\sqrt{2+\sqrt{2}}};
\sin(84\tfrac38 ^\circ) = \frac12\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}.

It is not known whether 2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.[17]

Series and product representations[edit]

The identity \cos(\pi/4) = \sin(\pi/4) = 1/\sqrt{2}, along with the infinite product representations for the sine and cosine, leads to products such as

\frac{1}{\sqrt 2} = \prod_{k=0}^\infty
\left(1-\frac{1}{(4k+2)^2}\right) =
\left(1-\frac{1}{4}\right)
\left(1-\frac{1}{36}\right)
\left(1-\frac{1}{100}\right) \cdots

and

\sqrt{2} =
\prod_{k=0}^\infty
\frac{(4k+2)^2}{(4k+1)(4k+3)} =
\left(\frac{2 \cdot 2}{1 \cdot 3}\right)
\left(\frac{6 \cdot 6}{5 \cdot 7}\right)
\left(\frac{10 \cdot 10}{9 \cdot 11}\right)
\left(\frac{14 \cdot 14}{13 \cdot 15}\right) \cdots

or equivalently,

\sqrt{2} =
\prod_{k=0}^\infty
\left(1+\frac{1}{4k+1}\right)
\left(1-\frac{1}{4k+3}\right)
=
\left(1+\frac{1}{1}\right)
\left(1-\frac{1}{3}\right)
\left(1+\frac{1}{5}\right)
\left(1-\frac{1}{7}\right) \cdots.

The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for \cos(\pi/4) gives

\frac{1}{\sqrt{2}} = \sum_{k=0}^\infty \frac{(-1)^k \left(\frac{\pi}{4}\right)^{2k}}{(2k)!}.

The Taylor series of \sqrt{(1 + x)} with x = 1 and using the double factorial n!! gives

\sqrt{2} = \sum_{k=0}^\infty (-1)^{k+1} \frac{(2k-3)!!}{(2k)!!} =
1 + \frac{1}{2} - \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} -
\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots.

The convergence of this series can be accelerated with an Euler transform, producing

\sqrt{2} = \sum_{k=0}^\infty \frac{(2k+1)!}{(k!)^2 2^{3k+1}} = \frac{1}{2} +\frac{3}{8} +
\frac{15}{64} + \frac{35}{256} + \frac{315}{4096} + \frac{693}{16384} + \cdots.

It is not known whether \sqrt{2} can be represented with a BBP-type formula. BBP-type formulas are known for \pi\sqrt 2 and \sqrt{2} \ln(1 + \sqrt{2}), however.[18]

Continued fraction representation[edit]

The square root of two has the following continued fraction representation:

 \!\ \sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.

The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408. The convergent p/q differs from the square root of 2 by almost exactly \scriptstyle{\frac{1}{2 q^2\sqrt{2}}}[citation needed] and then the next convergent is (p + 2q)/(p + q).

Paper size[edit]

The approximate aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1:2. This ratio guarantees that cutting a sheet in half along a line parallel to its short side results in the smaller sheets having the same ratio as the original sheet.

See also[edit]

Notes[edit]

  1. ^ Fowler and Robson, p. 368.
    Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
    High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
  2. ^ Henderson.
  3. ^ Stephanie J. Morris, "The Pythagorean Theorem", Dept. of Math. Ed., University of Georgia.
  4. ^ Brian Clegg, "The Dangerous Ratio ...", Nrich.org, November 2004.
  5. ^ Kurt von Fritz, "The discovery of incommensurability by Hippasus of Metapontum", Annals of Mathematics, 1945.
  6. ^ Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, p. 25 
  7. ^ Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of 2 seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures.
    Fowler and Robson, p. 376. Flannery, p. 32, 158.
  8. ^ "Constants and Records of Computation". Numbers.computation.free.fr. 2010-08-12. Retrieved 2012-09-07. 
  9. ^ "Number of known digits". Numbers.computation.free.fr. 2010-08-12. Retrieved 2012-09-07. 
  10. ^ All that Aristotle says, while writing about proofs by contradiction, is that “the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate”.
  11. ^ The edition of the Greek text of the Elements published by E. F. August in Berlin in 1826–1829 already relegates this proof to an Appendix. The same thing occurs with J. L. Heiberg's edition (1883–1888).
  12. ^ Gardner, Martin (2001), A Gardner's workout: training the mind and entertaining the spirit, A K Peters, Ltd., ISBN 978-1-56881-120-8 , p. 16
  13. ^ Apostol (2000), p. 841
  14. ^ See Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica 56 (2): 223–302 (see esp. Section 2.3, footnote 15), arXiv:1110.5456 
  15. ^ Courant, Richard; Robbins, Herbert (1941), What is mathematics? An Elementary Approach to Ideas and Methods, London: Oxford University Press, p. 124 
  16. ^ Julian D. A. Wiseman Sin and cos in surds
  17. ^ Good & Gover (1967).
  18. ^ http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf

References[edit]

External links[edit]