Normality and nonnormality of mathematicalconstants
David H Bailey
dhbailey@lbl.gov

1Normal numbers
Given an integer b > 1, a real number α is b-normal (or “normal base b”) ifevery m-long string of digits appears in the base-b expansion of α with
precisely the expected limiting frequency 1 / bm.
Base-10 example: Suppose α satisfies all of these conditions:1.
2.

Every digit appears, in the limit, with frequency 1/10.
Every 2-long string of digits (e.g., “23”, “55”, “74”, etc.) appears, in the limit,with frequency 1/100.
3. Every 3-long string (e.g., “234”, “551”, “749”, etc.) appears, in the limit, withfrequency 1/1000.
4. Every 4-long string (e.g., “2345”, “5518”, “7493”, etc.) appears, in the limit,with frequency 1/10000.
Similarly for every m-long string of digits, for every integer m > 1.

Then α is “10-normal” or “normal base 10.”
Using measure theory, it is can be shown that, given an integer b > 1, almost
all real numbers are b-normal. In fact, almost all reals are b-normal for
all integer bases b > 1 simultaneously (i.e., are “absolutely normal”).
2Which specific reals are normal?
These are widely believed to be b-normal, for all integer bases b > 1:
π = 3.1415926535…
e = 2.7182818284…
sqrt(2) = 1.4142135623…
log(2) = 0.6931471805…
Every irrational algebraic number (this conjecture is due to Borel).

But there are no proofs of normality for any of these constants in any base,nor are there any nonnormality results.
Until recently, normality proofs were known only for a few relatively contrived
examples, such as Champernowne’s constant =0.123456789101112131415… (which is 10-normal).
Chaitin’s omega constant (from the theory of computational complexity) has
been shown to be absolutely normal.

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Fascination with π	

Newton (1670):· “I am ashamed to tell you to how many figures I
carried these computations, having no otherbusiness at the time.”
Carl Sagan (1986):· In his book Contact, the lead scientist (played byJodie Foster in the movie version) looked forpatterns in the digits of π.

“Is π normal?” (and why) is one of the most ancientand fundamental of long-standing unsolved
mathematical questions.
Gaining insight on this problem has been a prime
motivation for many of the computations of πthrough the ages.
4Fax to DHB from “The Simpsons”
TV show5

21st century techniques to address thenormality problem· Statistical analyses, using sophisticated models of normality:§D.H. Bailey, J.M. Borwein, C.S. Calude, M.J. Dinneen, M. Dumitrescu and A. Yee, “Normality andthe digits of pi, http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/normality-digits-pi.pdf.

· Graphics and visualization – e.g., viewing digits as a random walk:§F.J. Aragon Artacho, D.H. Bailey, J.M. Borwein and P.B. Borwein, “Walking on real numbers,”
http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tools-walk.pdf.

· Analyses of algebraic irrationals:§§

D.H. Bailey, J.M. Borwein, R.E. Crandall and C. Pomerance, “On the binary expansions ofalgebraic numbers,” Journal of Number Theory Bordeau, vol. 16 (2004), pg. 487-518.
Hajime Kaneko, “On normal numbers and powers of algebraic numbers,” Integers, vol. 10 (2010),
pg. 31–64.

· Connections to the theory of BBP-type constants:§§

D.H. Bailey and R.E. Crandall, "On the random character of fundamental constant expansions,"
Experimental Mathematics, vol. 10, no. 2 (Jun 2001), pg. 175-190.
D.H. Bailey and R.E. Crandall, “Random generators and normal numbers,” ExperimentalMathematics, vol. 11, no. 4 (2002), pg. 527-546.

· Analyses of Stoneham constants:§§

D.H. Bailey and J.M. Borwein, “Nonnormality of Stoneham constants,” Ramanujan Journal,
http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf.
D.H. Bailey and J.M. Borwein, “Normal numbers and pseudorandom generators,”
http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf.
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Random walk on the first two billion bits of π(base-4 digit 0,1,2,3 codes up, down, left, right)

F.J. Aragon Artacho, D.H. Bailey, J.M. Borwein and P.B. Borwein, “Walking on real numbers,” available at
http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tools-walk.pdf.
7Explore the digits of π yourself
A compelling online interactive tool is available to study the random walk on
the first 100 billion binary digits of π:
http://gigapan.com/gigapans?tags=pi
This was constructed by Fran Aragon Artacho, based on binary digits of πprovided by Alex Yee.

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Random walk on α2,3 base 2 (normal)
compared to α2,3 base 6 (nonnormal)

We will discuss the normality and nonnormality of α2,3 later in the talk.
9Poisson model analysis of digits of π	· Recently a Poisson model-based approach was employed to analyze thenormality of π, using the first 15,925,868,541,400 bits (approx. 4 trillionhexadecimal digits) of π, as computed by Alex Yee.
· Based on this analysis, the conclusion “π is not normal” (given the first 4trillion hexadecimal digits) has probability 10-3064.

D.H. Bailey, J.M. Borwein, C.S. Calude, M.J. Dinneen, M. Dumitrescu and A. Yee, “Normality and the digits ofpi,” manuscript, http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/normality-digits-pi.pdf.

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A result for algebraic numbers
If x is algebraic of degree d > 1, then its binary expansion through position nmust have at least C n1/d 1-bits, for all sufficiently large n and some C that
depends on x.
Example: The first n binary digits of sqrt(2) must have at least sqrt(n) ones.
For the special case sqrt(m) for integer m, the result follows by simply notingthat in binary notation, the one-bit count of the product of two integers is
less than or equal to the product of the one-bit counts of the two integers.
A related result was obtained by Hajime Kaneko of Kyoto University.
However, note that these results are still a far cry from a full proof ofnormality, even in the single-digit binary sense.
D.H. Bailey, J.M. Borwein, R.E. Crandall and C. Pomerance, “On the binary expansions of algebraic
numbers,” Journal of Number Theory Bordeau, vol. 16 (2004), pg. 487-518.
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BBP formulas for π and log 2In 1996, a computer program running the PSLQ algorithm discovered thisformula, now known as the BBP formula for π:

✓1
X11
⇡=n16
8n + 1n=0

18n + 418n + 518n + 6◆This formula permits one to directly calculate binary or hexadecimal(base-16) digits of π beginning at an arbitrary starting position n, without
needing to calculate any of the first n-1 digits.
A similar formula for log 2, which also has the arbitrary digit calculation
property, is:1
X1log 2 =nn2
n=1D.H. Bailey, P.B. Borwein and S. Plouffe, “On the rapid computation of various polylogarithmic constants,”
Mathematics of Computation, vol. 66, no. 218 (Apr 1997), pg. 903-913.
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How to calculate individual binary
digits of log 2
Let x mod 1 denote the fractional part of x. Note that we can write the binary
expansion of log 2 beginning after position d as follows:!
!
d1
X 2d nX 2d nd(2 log 2) mod 1 =mod 1 +nn
n=1
n=d+1!
!
d1dn
dn
X2
X 2
mod n
=
mod 1 +nn
n=1
n=d+1

The numerator of first part can be computed very rapidly, using the binary
algorithm for exponentiation (i.e., the observation that an exponentiation canbe accelerated by using the binary expansion of the exponent). Only a few
terms of the second part need be computed, since it is very small.
This “trick” also leads to an interesting connection to normality.

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BBP formulas, pseudorandom numbergenerators and normality
Consider the sequence x0 = 0, and

xn =

✓1
2xn 1 +n◆

mod 1

(The 1/n term comes from BBP formula for log 2). If it can be demonstrated
that this sequence is equidistributed in the unit interval, then this would imply
that log 2 is 2-normal.
Similarly, consider the sequence x0 = 0, andxn =✓16xn 1 +120n
89n + 161024n3 + 712n2 206n + 21
2512n4◆

mod 1(The large term is a combination of the four terms in BBP formula for π). If it
can be shown that this sequence is equidistributed in the unit interval, thenthis would imply that π is 16-normal (and hence 2-normal).
D.H. Bailey and R. E. Crandall, “On the random character of fundamental constant expansions,” Experimental
Mathematics, vol. 10, no. 2 (Jun 2001), pg. 175-190.
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Normality of the Stoneham constants
By extending the BBP approach, normality has been proved for theStoneham numbers, the simplest of which is:2,3=1n 23n3n=1=0.041883680831502985071252898624571682426096 . . .10=

0.0ab8e38f 684bda12f 684bf 35ba781948b0f cd6e9e0 . . .16

This particular constant was proven 2-normal by Stoneham in 1971. Thishas been extended to the case where (2,3) are any pair (b,c) of coprime
integers > 1, and also to an uncountable class (here rn is n-th bit of r in [0,1)):1(r)=2,3n 23n +rn
3n=1
More recently, the 2-normality of α2,3 was proven more simply by means of a“hot spot” lemma proved using ergodic theory.
D.H. Bailey and M. Misiurewicz, “A strong hot spot theorem,” Proceedings of the American MathematicalSociety, vol. 134 (2006), no. 9, pg. 2495-2501.
D.H. Bailey and R.E. Crandall, “Random generators and normal numbers,” Experimental Mathematics, vol. 11,no. 4 (2002), pg. 527-546.
15

The “hot spot” lemma and a proof ofnormality for α2,3Let A(a,y,n,m) denote the count of occurrences where the m-long binary
string y is found to start at position p in the base-b expansion of α, where
1 <= p <= n. Then if x is not b-normal, there is some y in the unit interval s.t.

bm A(x, y, n, m)
lim inf lim sup=1
m!1 n!1nThus if α is not b-normal, there must exist some interval [r1, s1) in which
successive shifts of the base-b expansion of a visit [r1,s1) 10 times more
frequently, in the limit, relative to its length s1 - r1; there must be anotherinterval [r2, s2) that is visited 100 times more often relative to its length; etc.
On the other hand, if it can be established that no subinterval of the unit
interval is visited, say, 1,000 times more often relative to its length bysuccessive shifts of the base-b digits, this suffices to prove α is b-normal.
This can be condition is met for α2,3 in base-2. Thus α2,3 is 2-normal.
D.H. Bailey and M. Misiurewicz, “A strong hot spot theorem,” Proceedings of the American MathematicalSociety, vol. 134 (2006), no. 9, pg. 2495-2501.
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A pseudorandom number generator
based on the binary digits of α2,3· It can be shown that one can generate successive sections of the binary
expansion of α2,3 by means of a simple scheme analogous to commonlyused linear congruential pseudorandom number generators.
· The period of this generator is 3.7 x 1015. Higher periods are possible if
one uses higher-precision arithmetic.
· Floating-point numbers so generated appear to have excellent statisticalproperties, and do not have problems with power-of-two strides.
· The generator has a “jump-ahead” feature, which is useful in a parallel
environment so each processor can generator its own section.
· The generator can be implemented at a speed matching that of othercommonly used pseudorandom generators.

D.H. Bailey and J.M. Borwein, “Normal numbers and pseudorandom generators,”
http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf.
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A nonnormality result
Although α2,3 is provably 2-normal, surprisingly it is NOT 6-normal. Note that
we can write01
blog3 nc
1XXmn m n 3mnn m n 3 A@
mod 1 +326 ↵2,3 mod 1 =32
m=1

m=blog3 nc+1

The first portion of this expression is zero, since all of the terms in thesummation are integers. When n = 3m, the second part is accurately
approximated by the first term of the series. Thus,m63m↵2,3 mod 1 ⇡3 343m+1

Since this is extremely small for large m, the base-6 expansion of α2,3 haslong stretches of zeroes beginning at positions 3m + 1. This observation canbe fashioned into a rigorous proof of nonnormality.
D.H. Bailey and J. M. Borwein, “Normal numbers and pseudorandom generators,” available at "http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf."
D.H. Bailey and J.M. Borwein, “Nonnormality of Stoneham constants,” Ramanujan Journal, "http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf."
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Base-6 expansion of α2,30.
013014043000333425113050213000000124355504543223301150024352532055135234354101
043000000000000000051411300540405554553031442504334351012413452351125142125134
505503545015053522052044340452151505102411552500425130051124454001044131150032
420303213000000000000000000000000000000000000000000142120343111214520135254453
421134122402205253010542044235524110554150155204350414555400310145303033532002
534340401301240104453254343502142020432415025555101004043300045544114501031331
451151014451412344334234124005513133350454235305531511535015334524354502500555
214530542343421530350125024205404135451231323245353031534552304115020154242121
145201542222534340340450530123325534440443103332445332141415014233454542412432
031253400501341502455144043000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000031335054
244443111105553414105201454021341231300142433313311551144002552443344143003224335300102353515310402031320020533130021221200323413302222122521000035254543002
545040035245112205252324300200332523104022522341033520112253412034105440205323
451011020023525150015212545502421545343134553235001545540400023411521342323335
352445254035541452214132514501442052123331155522222331511535031203434310450421
303512422055300241415240113013242241234152033301205305300410045411552323315404124335430412302302244135255011522313421543013205100505502430344422115032003312043000355240351155014141310244412135431134045402354431451021050554450430513450
125125412054431444230051425001123154412512211541355432123414214030443515204024
235524524554300502531314011015041242313530200005202433402032025411301324022320
213143235351510550435543534053314401324313010031034432330333020541411524554102024511252554525045245533541442124051204242044021002315121441215553532044450502551452010241043242055023411434513124341231325032130205521143454221303254322525
001003215023424404353350251050434103131435231504421350241010405301043142100315114422021202242031254124153534242220443313314424101440231132320424442150031332450441322425500055412431400314402120525313133331045531400355154135352044001245
423040225112500244121430424020245242243022325455215430225105432551034135040520
213000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000005304302452321100553442520302

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Counts Zm of consecutive zeroes following
position 3m in the base-6 expansion of α2,3m123456789103m
3927
81
243
729
2187
6561
19683
59049

Zm
1
3616
42
121356
1058
3166
9487

Note that in the first 59049+9487 = 68,536 digits, there are at least 1+3+…
+9487 = 14,256 zeroes (not counting zeroes in the “random” sections), sothat the frequency of zeroes up to here is at least 0.208007… > 1/6). It canbe shown that this ratio also holds in the limit. Thus α2,3 is not 6-normal.
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A general nonnormality result for theStoneham constants
Given co-prime integers b > 1 and c > 1, and integers p, q, r > 0, with neitherb nor c dividing r, let B = bp cq r, and assume this condition holds:

D = cq/p r1/p /bc 1 < 1Then the Stoneham constant1X

1↵b,c =n bcn
cn=1
is not B-normal. In particular, αb,c is b-normal but not bc-normal.
Example: α2,3 is nonnormal base 6, 12, 24, 36, 48, 60, 72, 96, 120, 144, … ""It is not known whether or not this result gives a complete catalog of thebases for which a Stoneham constant is nonnormal.
D.H. Bailey and J.M. Borwein, “Nonnormality of Stoneham constants,” Ramanujan Journal, "http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf."
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Nonnormality of the sum of twoStoneham constants
Assume that αb,c and αd,e are two Stoneham constants, both B-nonnormal as
given in the previous result. Assume further that c and e are notmultiplicatively related -- there are no integers s and t such that cs = et.
Then the sum αb,c + αd,e is also B-nonnormal.

D.H. Bailey and J.M. Borwein, “Nonnormality of Stoneham constants,” Ramanujan Journal,
http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf.
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Open questions· Is α2,3 3-normal, or not? There are many open cases of this type.
· Is the nonnormality result given above a complete catalog of cases?
· α2,3 and α2,5 are each 2-normal. Is α2,3+ α2,5 2-normal? What conditions
ensure that normal + normal is normal?
· Can the normality and nonnormality proofs of Stoneham constants begeneralized to a larger class of real constants? Answer: Yes!
· Can absolute normality (i.e., b-normal for all integer bases b > 1) beestablished for a Stoneham-like constant? Or for any other constant?
· Can normality (to any base) be established for any of the “natural”
irrational constants of mathematics – e, π, sqrt(2), log(2), zeta(3), etc.?
· Can nonnormality be proven for any of the natural irrational constants?
Any normality or nonnormality proof for a well-known mathematical constantwould be a very significant result.
So would any result that helps to better understand why simple mathematicaloperations produce highly complex digit expansions.
23