Liouville number

In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (p, q) with q > 1 such that

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}.}$

Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. They are precisely the transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.

The existence of Liouville numbers (Liouville's constant)

Here we show that Liouville numbers exist by exhibiting a construction that produces such numbers.

For any integer b ≥ 2, and any sequence of integers (a₁, a₂, …, ), such that a_k ∈ {0, 1, 2, …, b − 1} for all k ∈ {1, 2, 3, …} and there are infinitely many k with a_k ≠ 0, define the number

${\displaystyle x=\sum _{k=1}^{\infty }{\frac {a_{k}}{b^{k!}}}}$

In the special case when b = 10, and a_k = 1, for all k, the resulting number x is called Liouville's constant:

L = 0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001...

It follows from the definition of x that its base-b representation is

${\displaystyle x=\left(0.a_{1}a_{2}000a_{3}00000000000000000a_{4}0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a_{5}\ldots \right)_{b}\;}$

where the nth term is in the (n!)th decimal place.

Since this base-b representation is non-repeating it follows that x cannot be rational. Therefore, for any rational number p/q, we have |x − p/q | > 0.

Now, for any integer n ≥ 1, define q_n and p_n as follows:

${\displaystyle q_{n}=b^{n!}\,;\quad p_{n}=q_{n}\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}=\sum _{k=1}^{n}{a_{k}}{b^{n!-k!}}\;.}$

Then

${\displaystyle {\begin{aligned}0<\left|x-{\frac {p_{n}}{q_{n}}}\right|&=\left|x-{\frac {q_{n}\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}}{q_{n}}}\right|=\left|x-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\left|\sum _{k=1}^{\infty }{\frac {a_{k}}{b^{k!}}}-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\left|\left(\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}+\sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\right)-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\\[6pt]&\leq \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=(n+1)!}^{\infty }{\frac {b-1}{b^{k}}}={\frac {b-1}{b^{(n+1)!}}}+{\frac {b-1}{b^{(n+1)!+1}}}+{\frac {b-1}{b^{(n+1)!+2}}}+...={\frac {b-1}{b^{(n+1)!}b^{0}}}+{\frac {b-1}{b^{(n+1)!}b^{1}}}+{\frac {b-1}{b^{(n+1)!}b^{2}}}+...={\frac {b-1}{b^{(n+1)!}}}\sum _{k=0}^{\infty }{\frac {1}{b^{k}}}\\[6pt]&={\frac {b-1}{b^{(n+1)!}}}\cdot {\frac {b}{b-1}}={\frac {b}{b^{(n+1)!}}}\leq {\frac {b^{n!}}{b^{(n+1)!}}}={\frac {1}{b^{(n+1)!-n!}}}={\frac {1}{b^{(n+1)n!-n!}}}={\frac {1}{b^{n(n!)+n!-n!}}}={\frac {1}{b^{(n!)n}}}={\frac {1}{{q_{n}}^{n}}}\end{aligned}}}$

Therefore, we conclude that any such x is a Liouville number.

Notes on the proof

1. The inequality ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\leq \sum _{k=n}^{\infty }{\frac {b-1}{b^{k!}}}}$follows from the fact that there exists k, a_k ∈ {0, 1, 2, …, b−1}. Therefore, at most, a_k = b-1. The largest possible sum would occur if the sequence of integers, (a₁, a₂, …), were (b-1, b-1, ...) where a_k = b-1, for all k. ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}}$will thus be less than, or equal to, this largest possible sum.
2. The strong inequality ${\displaystyle {\begin{aligned}\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=(n+1)!}^{\infty }{\frac {b-1}{b^{k}}}\end{aligned}}}$follows from our motivation to eliminate the series by way of reducing it to a series for which we know a formula. In the proof so far, the purpose for introducing the inequality in 1. comes from intuition that ${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{b^{k}}}={\frac {b}{b-1}}}$(the geometric series formula); therefore, if we can find an inequality from ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}}$that introduces a series with (b-1) in the numerator, and if we can work to further reduce the denominator term ${\displaystyle b^{k!}}$to ${\displaystyle b^{k}}$, as well as shifting the series indices from 0 to ${\displaystyle \infty }$, then we will be able to eliminate both series and (b-1) terms, getting us closer to a fraction of the form ${\displaystyle {\frac {1}{b^{(exponent)*n}}}}$, which is the end-goal of the proof. We further this motivation here by selecting now from the sum ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}}$a partial sum. Observe that, for any term in ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}}$, since b ≥ 2, then ${\displaystyle {\frac {b-1}{b^{k!}}}<{\frac {b-1}{b^{k}}}}$, for all k (except for when n=1). Therefore, ${\displaystyle {\begin{aligned}\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k}}}\end{aligned}}}$(since, even if n=1, all subsequent terms are smaller). In order to manipulate the indices so that k starts at 0, we select a partial sum from within ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k}}}}$(also less than the total value since it's a partial sum from a series whose terms are all positive). We will choose the partial sum formed by starting at k = (n+1)! which follows from our motivation to write a new series with k=0, namely by noticing that ${\displaystyle b^{(n+1)!}=b^{(n+1)!}b^{0}}$.
3. For the final inequality ${\displaystyle {\frac {b}{b^{(n+1)!}}}\leq {\frac {b^{n!}}{b^{(n+1)!}}}}$, we have chosen this particular inequality (true because b ≥ 2, where equality follows if and only if n=1) because we wish to manipulate ${\displaystyle {\frac {b}{b^{(n+1)!}}}}$into something of the form ${\displaystyle {\frac {1}{b^{(exponent)*n}}}}$. This particular inequality allows us to eliminate (n+1)! and the numerator, using the property that (n+1)! - n! = (n!)n, thus putting the denominator in ideal form for the substitution ${\displaystyle q_{n}=b^{n!}}$.

Irrationality

Here we will show that the number x = c/d, where c and d are integers and d > 0, cannot satisfy the inequalities that define a Liouville number. Since every rational number can be represented as such c/d, we will have proven that no Liouville number can be rational.

More specifically, we show that for any positive integer n large enough that 2^n − 1 > d > 0 (that is, for any integer n > 1 + log₂(d ) ) no pair of integers (p, q ) exists that simultaneously satisfies the two inequalities

${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}\,.}$

From this the claimed conclusion follows.

Let p and q be any integers with q > 1. Then we have,

${\displaystyle \left|x-{\frac {p}{q}}\right|=\left|{\frac {c}{d}}-{\frac {p}{q}}\right|={\frac {|cq-dp|}{dq}}}$

If |cq − dp | = 0, we would have

${\displaystyle \left|x-{\frac {p}{q}}\right|={\frac {|cq-dp|}{dq}}=0\,,}$

meaning that such pair of integers (p, q ) would violate the first inequality in the definition of a Liouville number, irrespective of any choice of n.

If, on the other hand, |cq − dp | > 0, then, since cq − dp is an integer, we can assert the sharper inequality |cq − dp | ≥ 1. From this it follows that

${\displaystyle \left|x-{\frac {p}{q}}\right|={\frac {|cq-dp|}{dq}}\geq {\frac {1}{dq}}}$

Now for any integer n > 1 + log₂(d ), the last inequality above implies

${\displaystyle \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{dq}}>{\frac {1}{2^{n-1}q}}\geq {\frac {1}{q^{n}}}\,.}$

Therefore, in the case |cq − dp | > 0 such pair of integers (p, q ) would violate the second inequality in the definition of a Liouville number, for some positive integer n.

We conclude that there is no pair of integers (p, q ), with q >1, that would qualify such an x = c/d as a Liouville number.

Hence a Liouville number, if it exists, cannot be rational.

(The section on Liouville's constant proves that Liouville numbers exist by exhibiting the construction of one. The proof given in this section implies that this number must be irrational.)

Uncountability

Consider, for example, the number

3.1400010000000000000000050000....

3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2(4319 zeros)6...

where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of π.

As shown in the section on the existence of Liouville numbers, this number, as well as any other non-terminating decimal with its non-zero digits similarly situated, satisfies the definition of a Liouville number. Since the set of all sequences of non-null digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers.

Moreover, the Liouville numbers form a dense subset of the set of real numbers.

Liouville numbers and measure

From the point of view of measure theory, the set of all Liouville numbers L is small. More precisely, its Lebesgue measure, λ(L), is zero. The proof given follows some ideas by John C. Oxtoby.[1]^:8

For positive integers n > 2 and q ≥ 2 set:

${\displaystyle V_{n,q}=\bigcup \limits _{p=-\infty }^{\infty }\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right)}$

we have

${\displaystyle L\subseteq \bigcup _{q=2}^{\infty }V_{n,q}.}$

Observe that for each positive integer n ≥ 2 and m ≥ 1, we also have

${\displaystyle L\cap (-m,m)\subseteq \bigcup \limits _{q=2}^{\infty }V_{n,q}\cap (-m,m)\subseteq \bigcup \limits _{q=2}^{\infty }\bigcup \limits _{p=-mq}^{mq}\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right).}$

Since

${\displaystyle \left|\left({\frac {p}{q}}+{\frac {1}{q^{n}}}\right)-\left({\frac {p}{q}}-{\frac {1}{q^{n}}}\right)\right|={\frac {2}{q^{n}}}}$

and n > 2 we have

${\displaystyle {\begin{aligned}\mu (L\cap (-m,\,m))&\leq \sum _{q=2}^{\infty }\sum _{p=-mq}^{mq}{\frac {2}{q^{n}}}=\sum _{q=2}^{\infty }{\frac {2(2mq+1)}{q^{n}}}\\[6pt]&\leq (4m+1)\sum _{q=2}^{\infty }{\frac {1}{q^{n-1}}}\leq (4m+1)\int _{1}^{\infty }{\frac {dq}{q^{n-1}}}\leq {\frac {4m+1}{n-2}}.\end{aligned}}}$

Now

${\displaystyle \lim _{n\to \infty }{\frac {4m+1}{n-2}}=0}$

and it follows that for each positive integer m, L ∩ (−m, m) has Lebesgue measure zero. Consequently, so has L.

In contrast, the Lebesgue measure of the set of all real transcendental numbers is infinite (since the set of algebraic numbers is a null set).

Structure of the set of Liouville numbers

For each positive integer n, set

${\displaystyle U_{n}=\bigcup \limits _{q=2}^{\infty }\bigcup \limits _{p=-\infty }^{\infty }\left\{x\in \mathbf {R} :0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}\right\}=\bigcup \limits _{q=2}^{\infty }\bigcup \limits _{p=-\infty }^{\infty }\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right)\setminus \left\{{\frac {p}{q}}\right\}}$

The set of all Liouville numbers can thus be written as

${\displaystyle L=\bigcap \limits _{n=1}^{\infty }U_{n}=\bigcap \limits _{n\in \mathbb {Z} ^{+}}\bigcup \limits _{q\geqslant 2}\bigcup \limits _{p\in \mathbb {Z} }\left(\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right)\setminus \left\{{\frac {p}{q}}\right\}\right).}$

Each U_n is an open set; as its closure contains all rationals (the ${\displaystyle \displaystyle p/q}$ from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense G_δ set.

Irrationality measure

The Liouville-Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number x is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any n in the power of q, we find the largest possible value for μ such that ${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}}$ is satisfied by an infinite number of integer pairs (p, q) with q > 0. This maximum value of μ is defined to be the irrationality measure of x.[2]^:246 For any value μ less than this upper bound, the infinite set of all rationals p/q satisfying the above inequality yield an approximation of x. Conversely, if μ is greater than the upper bound, then there are at most finitely many (p, q) with q > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of q. In other words, given the irrationality measure μ of a real number x, whenever a rational approximation x ≅ p/q, p,q ∈ N yields n + 1 exact decimal digits, we have

${\displaystyle {\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu +\varepsilon }}}}$

for any ε>0, except for at most a finite number of "lucky" pairs (p, q).

For a rational number α the irrationality measure is μ(α) = 1.[2]^:246 The Thue–Siegel–Roth theorem states that if α is an algebraic number, real but not rational, then μ(α) = 2.[2]^:248

Almost all numbers have an irrationality measure equal to 2.[2]^:246

Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number e has μ(e) = 2.[2]^:185 The irrationality measures of π, log 2, and log 3 are at most 7.103205334137,[3] 3.57455391,[4] and 5.125,[5] respectively.[6]

It has been proven that if the series ${\displaystyle \displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}}$ (where n is in radians) converges, then ${\displaystyle \pi }$'s irrationality measure is at most 2.5.[7][8]

The Liouville numbers are precisely those numbers having infinite irrationality measure.[2]^:248

Irrationality base

The irrationality base is a weaker measure of irrationality introduced by J. Sondow,[9] and is regarded as an irrationality measure for Liouville numbers. It is defined as follows:

Let ${\displaystyle \alpha }$ be an irrational number. If there exists a real number ${\displaystyle \beta \geq 1}$ with the property that for any ${\displaystyle \varepsilon >0}$, there is a positive integer ${\displaystyle q(\varepsilon )}$ such that

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}{\text{ for all integers }}p,q{\text{ with }}q\geq q(\varepsilon )}$,

then ${\displaystyle \beta }$ is called the irrationality base of ${\displaystyle \alpha }$ and is represented as ${\displaystyle \beta (\alpha )}$

If no such ${\displaystyle \beta }$ exists, then ${\displaystyle \alpha }$ is called a super Liouville number.

Example: The series ${\displaystyle \varepsilon _{2e}=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots }$ is a super Liouville number, while the series ${\displaystyle \tau _{2}=\sum _{n=1}^{\infty }{\frac {1}{^{n}2}}={\frac {1}{2}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{2^{2}}}}+{\frac {1}{2^{2^{2^{2}}}}}+{\frac {1}{2^{2^{2^{2^{2}}}}}}+\ldots }$ is a Liouville number with irrationality base 2. (${\displaystyle {^{b}a}}$ represents tetration.)

Liouville numbers and transcendence

Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental. However, not every transcendental number is a Liouville number. The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that π is another such example.[10]

The proof proceeds by first establishing a property of irrational algebraic numbers. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers, where the condition for "well approximated" becomes more stringent for larger denominators. A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. The following lemma is usually known as Liouville's theorem (on diophantine approximation), there being several results known as Liouville's theorem.

Below, we will show that no Liouville number can be algebraic.

Lemma: If α is an irrational number which is the root of a polynomial f of degree n > 0 with integer coefficients, then there exists a real number A > 0 such that, for all integers p, q, with q > 0,

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>{\frac {A}{q^{n}}}}$

Proof of Lemma: Let M be the maximum value of |f ′(x)| (the absolute value of the derivative of f) over the interval [α − 1, α + 1]. Let α₁, α₂, ..., α_m be the distinct roots of f which differ from α. Select some value A > 0 satisfying

${\displaystyle A<\min \left(1,{\frac {1}{M}},\left|\alpha -\alpha _{1}\right|,\left|\alpha -\alpha _{2}\right|,\ldots ,\left|\alpha -\alpha _{m}\right|\right)}$

Now assume that there exist some integers p, q contradicting the lemma. Then

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|\leq {\frac {A}{q^{n}}}\leq A<\min \left(1,{\frac {1}{M}},\left|\alpha -\alpha _{1}\right|,\left|\alpha -\alpha _{2}\right|,\ldots ,\left|\alpha -\alpha _{m}\right|\right)}$

Then p/q is in the interval [α − 1, α + 1]; and p/q is not in {α₁, α₂, ..., α_m}, so p/q is not a root of f; and there is no root of f between α and p/q.

By the mean value theorem, there exists an x₀ between p/q and α such that

${\displaystyle f(\alpha )-f({\tfrac {p}{q}})=(\alpha -{\frac {p}{q}})\cdot f'(x_{0})}$

Since α is a root of f but p/q is not, we see that |f ′(x₀)| > 0 and we can rearrange:

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|={\frac {\left|f(\alpha )-f({\tfrac {p}{q}})\right|}{|f'(x_{0})|}}=\left|{\frac {f({\tfrac {p}{q}})}{f'(x_{0})}}\right|}$

Now, f is of the form ${\displaystyle \sum _{i=0}^{n}}$ c_i xⁱ where each c_i is an integer; so we can express |f(p/q)| as

${\displaystyle \left|f\left({\frac {p}{q}}\right)\right|=\left|\sum _{i=0}^{n}c_{i}p^{i}q^{-i}\right|={\frac {1}{q^{n}}}\left|\sum _{i=0}^{n}c_{i}p^{i}q^{n-i}\right|\geq {\frac {1}{q^{n}}}}$

the last inequality holding because p/q is not a root of f and the c_i are integers.

Thus we have that |f(p/q)| ≥ 1/qⁿ. Since |f ′(x₀)| ≤ M by the definition of M, and 1/M > A by the definition of A, we have that

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|=\left|{\frac {f({\tfrac {p}{q}})}{f'(x_{0})}}\right|\geq {\frac {1}{Mq^{n}}}>{\frac {A}{q^{n}}}\geq \left|\alpha -{\frac {p}{q}}\right|}$

which is a contradiction; therefore, no such p, q exist; proving the lemma.

Proof of assertion: As a consequence of this lemma, let x be a Liouville number; as noted in the article text, x is then irrational. If x is algebraic, then by the lemma, there exists some integer n and some positive real A such that for all p, q

${\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {A}{q^{n}}}}$

Let r be a positive integer such that 1/(2^r) ≤ A. If we let m = r + n, and since x is a Liouville number, then there exist integers a, b where b > 1 such that

${\displaystyle \left|x-{\frac {a}{b}}\right|<{\frac {1}{b^{m}}}={\frac {1}{b^{r+n}}}={\frac {1}{b^{r}b^{n}}}\leq {\frac {1}{2^{r}}}{\frac {1}{b^{n}}}\leq {\frac {A}{b^{n}}}}$

which contradicts the lemma. Hence, if a Liouville number exists, it cannot be algebraic, and therefore must be transcendental.

See also

• Brjuno number
• Diophantine approximation

References

1. Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. 2 (Second ed.). New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4684-9339-9. ISBN 0-387-90508-1. MR 0584443.
2. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. 193. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139017732. ISBN 978-0-521-11169-0. MR 2953186. Zbl 1260.11001.
3. Zeilberger, Doron; Zudilin, Wadim (2019). "Irrationality measure of pi". arXiv:1912.06345 [math.NT].
4. Marcovecchio, Raffaele. "The Rhin–Viola method for log 2" (PDF).
5. Salikhov, V. Kh. (2007-12-01). "On the irrationality measure of ln3". Doklady Mathematics. 76 (3): 955–957. doi:10.1134/S1064562407060361. ISSN 1531-8362.
6. Zudilin, Wadim (2004). "An essay on the irrationality measure of π and other logarithms". Chebyshevskii Sbornik (in Russian). 5 (2(10)): 49–65. arXiv:math/0404523. Bibcode:2004math......4523Z. MR 2140069. Zbl 1140.11036.
7. Alekseyev, Max A. (2011). "On convergence of the Flint Hills series". arXiv:1104.5100.
8. Weisstein, Eric W. "Flint Hills Series". MathWorld.
9. Sondow, Jonathan (2004). "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik". arXiv:math/0406300. Bibcode:2004math......6300S. Cite journal requires |journal= (help)
10. The irrationality measure of π does not exceed 7.6304, according to Weisstein, Eric W. "Irrationality Measure". MathWorld.

External links

• The Beginning of Transcendental Numbers