IRRATIONALITY OF π AND e 1. Introduction Numerical estimates for

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IRRATIONALITY OF π AND e
KEITH CONRAD
1. Introduction
Numerical estimates for π have been found in records of several ancient civilizations.
These estimates were all based on inscribing and circumscribing regular polygons around a
circle to get upper and lower bounds on the area (and thus upper and lower bounds on π
after dividing the area by the square of the radius). Such estimates are accurate to a few
decimal places. Around 1600, Ludolph van Ceulen gave an estimate for π to 35 decimal
places. He spent many years of his life on this calculation, using a polygon with 262 sides!
With the advent of calculus in the 17-th century, a new approach to the calculation of π
became available: infinite series. For instance, if we integrate
1
= 1 − t2 + t4 − t6 + t8 − t10 + · · · , |t| < 1
1 + t2
from t = 0 to t = x when |x| < 1, we find
x3 x5 x7 x9 x11
+
−
+
−
+ ··· .
3
5
7
9
11
Actually, this is also correct at the boundary point x = 1. Since arctan 1 = π/4, (1.1)
specializes to the formula
π
1 1 1 1
1
(1.2)
=1− + − + −
+ ··· ,
4
3 5 7 9 11
which is due to Leibniz. It expresses π in terms of an alternating sum of reciprocals of
odd numbers. However, the series in (1.2) converges too slowly to be of any numerical
use. For example, truncating the series after 1000 terms and multiplying by 4 gives the
approximation π ≈ 3.1405, which is only good to two places after the decimal point.
There are other formulas for π in terms of arctan values, such as
π
1
1
= arctan
+ arctan
4
2
3
1
1
= 2 arctan
+ arctan
3
7
1
1
= 4 arctan
− arctan
.
5
239
Since the series for arctan x is more rapidly convergent when x is less than 1, these other
series are more useful than (1.2) to get good numerical approximations to π. The last such
calculation before the use of computers was by Shanks in 1873. He claimed to have found
π to 707 places. In the 1940s, the first computer estimate for π revealed that Shanks made
a mistake in the 528-th digit, so all his further calculations were in error!
Our interest here is not to ponder ever more elaborate methods of estimating π, but
to prove something about the structure of this number: it is irrational. That is, π is not
(1.1)
arctan x = x −
1
2
KEITH CONRAD
a ratio of integers. The idea of the proof is to argue
√ by contradiction. This is also the
principle behind the simpler proof that √
the number 2 is irrational. However, there is an
essential difference
between proofs that 2 is irrational and proofs that π is irrational. One
√
can prove 2 is√irrational using only algebraic manipulations with a hypothetical rational
expression for 2 to reach a contradiction. But all known proofs of the irrationality of π
are based on techniques from calculus, which can be used to prove irrationality of other
numbers, such as e and rational powers of e (aside from e0 = 1).
The remaining sections are organized as follows. In Section 2, we prove π is irrational
with definite integrals. Irrationality of e is proved by infinite series in Section 3. A general
discussion about irrationality proofs is in Section 4, and we apply those ideas to prove
irrationality of nonzero rational powers of e in Section 5. In Section 6 we introduce complex
numbers into a proof from Section 5 in order to obtain another proof that π is irrational.
2. Irrationality of π
The first proof that π is irrational is due to Lambert in 1761. His proof involves an
analytic device that is not covered in calculus courses: continued fractions. (A discussion of
this work is in [3, pp. 68–78].) The irrationality proof for π we give here is due to Niven [5]
and uses integrals instead of continued fractions.
Theorem 2.1. The number π is irrational.
Proof. For any nice function f (x), a double integration by parts shows
Z
Z
0
f (x) sin x dx = −f (x) cos x + f (x) sin x − f 00 (x) sin x dx.
Therefore (using sin(0) = 0, cos(0) = 1, sin(π) = 0, and cos(π) = −1),
Z π
Z π
f (x) sin x dx = (f (0) + f (π)) −
f 00 (x) sin x dx.
0
0
In particular, if f (x) is a polynomial of even degree, say 2n, then repeating this calculation
n times gives
Z π
(2.1)
f (x) sin x dx = F (0) + F (π),
0
f 00 (x)
where F (x) = f (x) −
+ f (4) (x) − · · · + (−1)n f (2n) (x).
To prove π is irrational, we argue by contradiction. Assume π = p/q for positive integers
p and q (since π > 0). We are going to apply (2.1) to a carefully (and mysteriously!) chosen
polynomial f (x) and wind up with an integer that lies between 0 and 1. No such integer
exists, so we have a contradiction and therefore π is irrational.
For any positive integer n, set
xn (p − qx)n
xn (π − x)n
=
.
n!
n!
This polynomial of degree 2n depends on n (and on π!). We are going to apply (2.1) to this
polynomial and find a contradiction when n becomes large.
Before working out the consequences of (2.1) for f (x) = fn (x), we note the polynomial
fn (x) has two important properties:
• for 0 < x < π, fn (x) is positive and (when n is large) very small,
• all the derivatives of fn (x) at x = 0 and x = π are integers.
(2.2)
fn (x) = q n
IRRATIONALITY OF π AND e
3
To show the first property is true, the positivity of fn (x) for 0 < x < π is immediate from
its defining formula. To bound fn (x) from above when 0 < x < π, note that 0 < π − x < π,
so 0 < x(π − x) < π 2 . Therefore
2n (qπ 2 )n
n π
=
(2.3)
0 < fn (x) ≤ q
.
n!
n!
The upper bound tends to 0 as n → ∞, so the upper bound is less than 1 for large n.
To show the second property is true, first take x = 0. The coefficient of xj in fn (x) is
(j)
fn (0)/j!. Since fn (x) = xn (p − qx)n /n! and p and q are integers, the binomial theorem
tells us the coefficient of xj can also be written as cj /n! for an integer cj . Therefore
j!
cj .
n!
Since fn (x) has its lowest degree nonvanishing term in degree n, cj = 0 for j < n, so
(j)
(j)
fn (0) = 0 for j < n. For j ≥ n, j!/n! is an integer, so fn (0) is an integer by (2.4).
To see the derivatives of fn (x) at x = π are also integers, we use the identity fn (π − x) =
(j)
(j)
fn (x). Differentiate both sides j times and set x = 0 to get (−1)j fn (π) = fn (0) for all j.
Therefore, since the right side is an integer, the left side is an integer too. This concludes
the proof of the two important properties of fn (x).
Now we look at (2.1) when f = fn . Since all derivatives of fn at 0 and π are integers,
the
R π right side of (2.1) is an integer when f = fn (look at the definition of F (x)). Therefore
0 fn (x) sin x dx is an integer for every n. Since fn (x) and sin x are positive on (0, π), this
integral is a positive integer. However, when n isRlarge, |fn (x) sin x| ≤ |fn (x)| ≤ (qπ 2 )n /n!
π
by (2.3). As n → ∞, (qπ 2 )n /n! → 0. Therefore 0 fn (x) sin x dx is a positive integer less
than 1 when n is very large. This is absurd, so we have reached a contradiction. Thus π is
irrational.
fn(j) (0) =
(2.4)
This proof is quite puzzling. How did Niven choose the polynomials fn (x) or know to
compute the integral (2.1)? Here is a reworking of Niven’s proof in terms of recursions, due
to Markov and Zhou [8].
Proof. Set
π
(x(π − x))n
sin x dx.
n!
0
The integrand is continuous and positive on (0, π), so In > 0. By explicit calculation, I0 = 2
and I1 = 4:
π
Z π
I0 =
sin x dx = − cos x = −(−1) + 1 = 2
Z
In =
0
0
and
Z
π
(πx − x2 ) sin x dx
Z π
Z π
= π
x sin x dx −
x2 sin x dx
0
π 0
π
2
= π(sin x − x cos x) − (2x sin x − (x − 2) cos x) I1 =
0
0
= π(π − 0) − ((π 2 − 2) − 2)
= 4.
0
4
KEITH CONRAD
Using integration by parts twice, for n ≥ 2 we have In = (4n − 2)In−1 − π 2 In−2 , so by
induction In is a polynomial in π n of degree at most n with integral coefficients:
In = cn,n π n + cn−1,n π n−1 + · · · + c1,n π + c0,n .
(2.5)
The table below gives the first few explicit values of In to illustrate this formula, although
we don’t need to know them except for I0 and I1 .
n
In
0
2
1
4
−2π 2 + 24
2
3
−24π 2 + 240
4
2π 4 − 360π 2 + 3360
5
60π 4 − 6720π 2 + 60480
6
6 −2π + 1680π 4 − 151200π 2 + 1330560
Assume
is rational, so P
we can write π = p/q for positive integers p and q. By (2.5),
Pπ
n
n
n−k pk ∈ Z. Since I > 0, we have q n I ∈ Z+ .
nπk =
c
q
q n In =
n
n
k=0 ck,n q
k=0 k,n
At Rthe same time, since |x(π − x)| ≤ π 2 for 0 ≤ x ≤ π we have the bound |q n In | ≤
π
q n 0 ((π 2 )n /n!) dx = π((qπ 2 )n /n!), so In → 0 as n → ∞. This contradicts q n In being a
positive integer, so we have a contradiction.
3. Irrationality of e
We turn now to a proof that e is irrational. This was first established by Euler in 1737
using continued fractions. We will prove the irrationality in a more direct manner, using
infinite series, by an argument of Fourier (1815).
Theorem 3.1. The number e is irrational.
Proof. Write
e=1+
1
1
+ + ··· .
2! 3!
For any n,
1
1
1 + + + ··· +
2! 3!
1
1
=
1 + + + ··· +
2! 3!
e =
1
1
1
+
+
+ ···
n!
(n + 1)! (n + 2)!
1
1
1
1
+
+
+ ··· .
n!
n! n + 1 (n + 2)(n + 1)
The second term in parentheses is positive and bounded above by the geometric series
1
1
1
1
+
+
+ ··· = .
2
3
n + 1 (n + 1)
(n + 1)
n
Therefore
1
1
1
1
0 < e − 1 + + + ··· +
≤
.
2! 3!
n!
n · n!
Write the sum 1 + 1/2! + · · · + 1/n! as a fraction with common denominator n!: it is pn /n!
with pn ∈ Z. Clear the denominator n! to get
1
(3.1)
0 < n!e − pn ≤ .
n
IRRATIONALITY OF π AND e
5
So far everything we have done involves no unproved assumptions. Now we introduce the
rationality assumption. If e is rational, then n!e is an integer when n is large (since any
integer is a factor by n! for large n). But that makes n!e − pn an integer located in the open
interval (0, 1/n), which is absurd. We have a contradiction, so e is irrational.
4. General Ideas
It’s time to think more systematically. A basic principle we need to understand is that
numbers can be proved to be irrational if they can be approximated “too well” by rationals.
Of course each number can be approximated arbitrarily closely by
√ rational numbers: use a
truncated decimal expansion. For instance, we can approximate 2 = 1.41421356... by
14142
1414213
(4.1)
= 1.4142,
= 1.414213.
10000
1000000
With truncated decimals, we achieve close estimates at the expense of rather large denominators. To see what this is all about, compare the above approximations with
1393
99
= 1.41428571...,
= 1.41421319...,
(4.2)
70
985
where we have achieved just as close an approximation with much smaller denominators
(e.g., the second one is accurate to√6 decimal places with a denominator of only 3 digits).
These rational approximations to 2 are, in the sense of denominators, much better than
the ones we find from decimal truncation.
To measure the “quality” of an approximation of a real number α by a rational number
p/q, we should think not about the difference |α − p/q| being small in an absolute sense,
but about the difference being substantially smaller than 1/q (thus tying the error with the
size of the denominator in the approximation). In other words, we want
|α − p/q|
p
= q α − = |qα − p|
1/q
q
to be small in an absolute sense.
Measuring the approximation of α by p/q using |qα − p| rather than |α − p/q| admittedly
takes some getting √used to, if you are new to the idea. Consider what it says about our
approximations to 2. For example, from (4.1) we have
√
√
|10000 2 − 14142| = .135623, |1000000 2 − 1414213| = .562373,
and these are not small when measured against √
1/10000 = .0001 or 1/1000000 = .000001.
On the other hand, from the approximations to 2 in (4.2) we have
√
√
|70 2 − 99| = .005050, |985 2 − 1393| = .000358,
which are small when measured against 1/70 = .014285 and 1/985 = .001015. We see
vividly
approximations
√ that 99/70 and 1393/985 really should be judged as “good” rational √
to 2 while the decimal truncations are “bad” rational approximations to 2.
The importance of this point of view is that it gives us a general strategy for proving
numbers are irrational, as follows.
Theorem 4.1. Let α ∈ R. If there is a sequence of integers pn , qn such that qn α − pn 6= 0
and |qn α − pn | → 0 as n → ∞, then α is irrational.
In other words, if α admits a “very good” sequence of rational approximations, then α
must be irrational.
6
KEITH CONRAD
Proof. Since 0 < |qn α − pn | < 1 for large n, by hypothesis, we must have qn 6= 0 for large
n. Therefore, since only large n is what matters, we may change terms at the start and
assume qn 6= 0 for all n.
To prove α is irrational, suppose it is rational: α = a/b, where a and b are integers (with
b 6= 0). Then
α − pn = a − pn = qn a − pn b .
qn
b
qn
bqn
Clearing the denominator qn ,
qn a − p n b .
|qn α − pn | = b
Since this is not zero, the integer qn a − pn b is nonzero. Therefore |qn a − pn b| ≥ 1, so
1
|qn α − pn | ≥ .
b
This lower bound contradicts |qn α − pn | tending to 0.
It turns out the condition in Theorem 4.1 is not just sufficient to prove irrationality, but
it is also necessary: if α is irrational then there is a sequence of integers pn , qn such that
|qn α − pn | is not zero and tends to 0 as n → ∞. We will not have any need for this necessity
(except maybe for its psychological boost) and therefore omit the proof. See [4, p. 277].
Of course, to use Theorem 4.1 to prove irrationality of a number α we need to find the
integers pn and qn . For the number e, these integers can be found directly from truncations
to the infinite series for e, as we saw in (3.1). In other words, rather than saying e is
irrational because the proof of Theorem 3.1 shows in the end that rationality of e leads
to an integer between 0 and 1, we can say e is irrational because the proof of Theorem
3.1 exhibits a sequence of good rational approximations to e. In other words, the proof of
Theorem 3.1 can stop at (3.1) and then appeal to Theorem 4.1.
While other powers of e are also irrational, it is not feasible to prove their irrationality
by adapting the proof of Theorem 3.1. For instance, what happens
try to prove e2
P if we
k
2
is irrational from taking truncations of the infinite series e = k≥0 2 /k!? Writing the
P
truncated sum nk=0 2k /k! in reduced form as, say, an /bn , numerical data suggest bn e2 − an
does not tend to 0, (For example, the value of bn e2 − an at n = 22, 23, and 24 is roughly
.0026, 1.4488, and .3465. Since the corresponding values of bn have 12, 16, and 17 decimal
digits, these differences are not small by comparison with 1/bn , so the approximations an /bn
to e2 are not that good.) Thus, these rational approximations to e2 probably won’t fit the
conditions of Theorem 4.1 to let us prove the irrationality of e2 . (However, a well-chosen
subsequence of the partial sums does work. See the appendix.)
5. Irrationality of rational powers of e
To find good rational approximations to positive integral powers of e (good enough, that
is, to establish irrationality of those powers), we will not use a series expansion, but rather
use the interaction between ex and integration. Some of the mysterious ideas from Niven’s
proof of the irrationality of π will show up in this context.
We will use Theorem 4.1 to prove the following generalization of the irrationality of e.
Theorem 5.1. For every positive integer a, ea is irrational.
Before we prove Theorem 5.1, we note two immediate corollaries.
IRRATIONALITY OF π AND e
7
Corollary 5.2. When r is a nonzero rational number, er is irrational.
Proof. Since e−r = 1/er , it suffices to take r > 0. Then r = a/b for positive integers a and b.
If er is rational, so is (er )b = ea , but this contradicts Theorem 5.1. Thus er is irrational. Corollary 5.3. For any positive rational number r 6= 1, ln r is irrational.
Proof. The number ln r is nonzero. If ln r is rational, then Corollary 5.2 tells us eln r is
irrational. But eln r = r is rational. We have a contradiction, so ln r is irrational.
The proof of Theorem 5.1 will use the following lemma, which tells us how to integrate
e−x f (x) when f (x) is any polynomial.
Lemma 5.4 (Hermite). Let f (x) be a polynomial of degree m ≥ 0. For any number a,
Z a
m
m
X
X
e−x f (x) dx =
f (j) (0) − e−a
f (j) (a).
0
j=0
j=0
R
Proof. We compute e−x f (x) dx by integration by parts, taking u = f (x) and dv = e−x dx.
Then du = f 0 (x)dx and v = −e−x , so
Z
Z
−x
−x
e f (x) dx = −e f (x) + e−x f 0 (x) dx.
Repeating this process on the new indefinite integral, we eventually obtain
Z
m
X
e−x f (x) dx = −e−x
f (j) (x).
j=0
Now evaluate the right side at x = a and x = 0 and subtract.
Remark 5.5. It is interesting to make a special case of this lemma explicit: for f (x) = xn ,
Z a
n
1 X
n(n − 1) · · · (n − j + 1)an−j .
e−x xn dx = n! − a
e
0
j=0
Letting a → ∞ (n is fixed), the second term on the right tends to 0, so
This integral formula for n! is due to Euler.
R∞
0
e−x xn dx = n!.
Now we prove Theorem 5.1.
Proof. Rewrite Hermite’s lemma (Lemma 5.4) by multiplying through by ea :
Z a
m
m
X
X
a
−x
a
(j)
(5.1)
e
e f (x) dx = e
f (0) −
f (j) (a).
0
j=0
j=0
Equation (5.1) is valid for any positive number a and any polynomial f (x). Let a be a
positive integer at which ea is assumed to be rational. We want to use for f (x) a polynomial
(actually, a sequence of polynomials fn (x)) with two properties:
• the left side of (5.1) is positive and (when n is large) very small,
• all the derivatives of the polynomial at x = 0 and x = a are integers.
8
KEITH CONRAD
Then the right side of (5.1) will have the properties of the differences qn α − pn in Theorem
4.1, with α = ea and the two sums on the right side of (5.1) being pn and qn .
Our choice of f (x) is
(5.2)
fn (x) =
xn (x − a)n
,
n!
where n ≥ 1 is to be determined. (Note the similarity with (2.2) in the proof of the
irrationality of π!) In other words, we consider the equation
(5.3)
ea
Z
a
e−x fn (x) dx = ea
0
2n
X
fn(j) (0) −
j=0
2n
X
fn(j) (a).
j=0
We can see (5.3) is positive by looking at the left side. The number a is positive and the
integrand e−x fn (x) = e−x xn (x − a)n /n! on the interval (0, a) is positive, so the integral is
positive. Now we estimate the size of (5.3) by estimating the integral on the left side. By
the change of variables x = ay on the left side of (5.3),
Z a
Z 1
y n (y − 1)n
e−x fn (x) dx = a2n+1
e−ay
dy,
n!
0
0
so we can bound the left side of (5.3) from above by
Z a
Z
ea a2n+1 1 −ay
0 < ea
e−x fn (x) dx ≤
e
dy.
n!
0
0
As a function of n, this upper bound is a constant times (a2 )n /n!. As n → ∞, this bound
tends to 0.
(j)
(j)
To see that, for any n ≥ 1, the derivatives fn (0) and fn (a) are integers for every j ≥ 0,
first note that the equation fn (a − x) = fn (x) tells us after repeated differentiation that
(j)
(j)
(−1)j fn (a) = fn (0). Therefore it suffices to show all the derivatives of fn (x) at x = 0
(j)
are integers. The proof that all fn (0) are integers is just like that in the proof of Theorem
2.1, so the details are left to the reader to check. (The general principle is this: for any
polynomial g(x) that has integer coefficients and is divisible by xn , all derivatives of g(x)/n!
at x = 0 are integers.)
P
(j)
The first property of the fn ’s tells us that qn ea − pn is positive, where pn = 2n
j=0 fn (a)
P
(j)
a
and qn = 2n
j=0 fn (a), and qn e − pn tends to 0 as n → ∞. The second property of the
fn ’s tells us that pn and qn are integers. Therefore the hypotheses of Theorem 4.1 are met,
so ea is irrational.
What really happened in this proof? We actually wrote down some very good rational
approximations to ea . They came from values of the polynomial
Fn (x) =
2n
X
fn(j) (x).
j=0
Indeed, Theorem 5.1 tells us Fn (a)/Fn (0) is a “good” rational approximation to ea when n
is large. (The dependence of Fn (x) on a is hidden in the formula for fn (x).) The following
table illustrates this for a = 2, where the entry at n = 1 is pretty bad since F1 (0) = 0.
IRRATIONALITY OF π AND e
9
n |Fn (0)e2 − Fn (2)|
1
4
2
1.5562
3
.43775
4
.09631
.01739
5
6
.00266
7
.00035
8
.00004
a
If we take a = 1, the rational
Pnapproximations we get for e = e by this method are
different from the partial sums k=0 1/k!.
Although the proofs of Theorems 2.1 and 5.1 are similar in the sense that both used
estimates on integrals, the proof of Theorem 2.1 did not show π is irrational by exhibiting a
sequence of good rational approximations to π. The proof of Theorem 2.1 was an “integer
between 0 and 1” proof by contradiction. No good rational approximations to π were
produced in that proof. It is simply harder to get our grips on π than it is on powers of e.
6. Returning to irrationality of π
The proof of Theroem 5.1 gives a broader context for the proof in Theorem 2.1 that π
is irrational. In fact, by thinking about Theorem 5.1 in the complex plane, we are led to a
slightly different proof of Theorem 2.1.
Proof. We are going to use Hermite’s lemma for complex a, where integrals from 0 to a
are obtained using any path of integration between 0 and
as the straightline path).
Pa (such
n
a
The definition of e for complex a is the infinite series n≥0 a /n!. In particular, for real
t, breaking up the series for eit into its real and imaginary parts yields
eit = cos t + i sin t.
Thus |eit | = 1 and eiπ = −1. Consider Hermite’s lemma at a = iπ:
Z iπ
m
m
X
X
(6.1)
e−x f (x) dx =
f (j) (0) +
f (j) (iπ).
0
j=0
j=0
Extending (5.2) to the complex plane, we will use in (6.1) f (x) = fn (x) = xn (x − iπ)n /n!
for some large n to be determined later (so m = 2n as before). To put (6.1) in a more
appealing form, we apply the change of variables x = iπy. The left side of (6.1) becomes
Z iπ
Z 1
y n (y − 1)n
−x
n 2n+1
(6.2)
e fn (x) dx = i(−1) π
e−iπy
dy.
n!
0
0
Let gn (y) = y n (y − 1)n /n! (which does not involve i or π), so fn (iπy) = (iπ)2n gn (y) and
differentiating j times gives
(iπ)j fn(j) (iπy) = (iπ)2n gn(j) (y).
(6.3)
Feeding (6.2) and (6.3) into (6.1), with f = fn ,
Z 1
2n
2n
X
X
n 2n+1
(6.4)
i(−1) π
e−iπy gn (y) dy =
(iπ)2n−j gn(j) (0) +
(iπ)2n−j gn(j) (1).
0
j=0
j=0
10
KEITH CONRAD
This is the key equation in the proof. It serves the role for us now that (5.3) did in the
(j)
(j)
proof of irrationality of powers of e. Notice the numbers gn (0) and gn (1) are all integers.
(j)
Suppose (at last) that π is rational, say with positive denominator q. For j < n, gn (x)
vanishes at x = 0 and x = 1, so the sums in (6.4) really only need to start at j = n. That
means the largest power of π in any (nonzero) term on the right side of (6.4) is π n , so the
largest denominator on the right side of (6.4) is q n . Multiply both sides by q n :
Z 1
2n
2n
X
X
n 2n+1
e−iπy gn (y) dy =
(6.5)
i(−q) π
(iπ)2n−j q n gn(j) (0) +
(iπ)2n−j q n gn(j) (1).
0
j=0
j=0
We estimate the left side of (6.5). Since |eiπy | = 1, an estimate of the left side is
Z
n 2n+1 1 −iπy
q n π 2n+1
(qπ 2 )n
q π
≤
=
π
.
e
g
(y)
dy
n
n!
n!
0
As n → ∞, this bound tends to 0.
On the other hand, since the nonzero terms in the sums on the right side of (6.5) only
start showing up at the j = n term, and π 2n−j q n ∈ Z for n ≤ j ≤ 2n, the right side of (6.5)
is in the integral lattice Z + Zi. It is nonzero, as we see by looking at the real part of the
left side of (6.5), which is
Z 1
Z 1
y n (y − 1)n
sin(πy)
sin(πy)gn (y) dy = (−q)n π 2n+1
(−q)n π 2n+1
dy.
n!
0
0
The integrand here has constant sign on (0, 1), so (6.5) is in Z + Zi and doesn’t vanish for
any n ≥ 1 since the real part is not 0. A sequence of nonzero elements of Z + Zi can’t tend
to 0. We have a contradiction, so π is irrational.
We used complex numbers in the above proof to stress the close connection to the proof
of Theorem 5.1. If you take the real part of every equation in the above proof (especially
starting with (6.4), then you will find a proof of the irrationality of π that avoids complex
numbers. (For instance, we showed (6.5) is nonzero by looking only at the real part, so
taking real parts everywhere should not damage the logic of the proof.) By taking real
parts, the goal of the proof changes slightly. Instead of showing the rationality of π leads
to a nonzero element of Z + Zi with absolute value less than 1, showing the rationality of π
leads to a nonzero integer with absolute value less than 1. Is such a “real” proof basically
the same as the first proof we gave that π is rational? As a check, try to adapt the first
proof of Theorem 2.1 to get the following.
Corollary 6.1. The number π 2 is irrational.
Proof. Run through the previous proof, starting at (6.4), but now assume π 2 is rational
with denominator q ∈ Z. While it is no longer true that π 2n−j q n ∈ Z for n ≤ j ≤ 2n, we
instead have π 2n−j q n ∈ Z for j even and π 2n−j q n ∈ (1/π)Z for j odd. Since i2n−j is real
when j is even and imaginary when j is odd, we now have (6.5) lying in the set Z + Z(i/π),
whose nonzero elements are not arbitrarily small.
If you write up all the details in this proof and take the real part of every equation, you
will have the proof of the irrationality of π in [7, Chap. 16].
The numbers π and e are not just irrational, but transcendental. That is, neither number
√
is the root of a nonzero polynomial with rational coefficients. (For comparison, 2 is
irrational but it is a solution of x2 − 2 = 0, so it is in some sense linked to the rational
IRRATIONALITY OF π AND e
11
numbers through this equation.) Proofs of their transcendence can be found in [1, Chap. 1],
[2, Chap. II], and [6, Chap. 2]. The proof that e is transcendental is actually not much more
complicated than the our proof of Theorem 5.1, taking perhaps two pages rather than one.
The idea is to use Hermite’s lemma and a construction of a sequence of rational functions
whose values give good rational approximations simultaneously to several integral powers
of e, not only to one power like ea . The construction of such rational functions generalizes
the fn (x)’s in Theorem 5.1, which gave good rational approximations to ea . By comparison
to this, the proof that π is transcendental is much more involved than the proof of its
irrationality.
Historically, progress on π always trailed that of e. Euler proved e is irrational in 1737,and
Lambert proved irrationality of non-zero rational powers of e and irrationality of π in 1761.
Their proofs used continued fractions, not integrals. (Lambert’s proof for π was actually a
result about the tangent function. When r is a nonzero rational where the tangent function
is defined, Lambert proved tan r is irrational. Then, since tan(π/4) = 1 is rational, π must
be irrational in order to avoid a contradiction.) Transcendence proofs for e and π came 100
years later, in √
the work of Hermite (1873) and Lindemann (1882). In addition to e and√π,
transcendental. The status of 2e , π e , π 2 ,
the numbers 2 2 , log 2, and eπ are known to beP
e + π, e log 2, and Euler’s constant γ = limn→∞ nk=1 1/k − log n is still open. Surely these
numbers are all transcendental, but it is not yet proved that any of them is even irrational.
Appendix A. Irrationality of e2
In Section
4, we saw that the partial sums of the Taylor series for ex at x = 2, namely the
Pn
k
sums k=0 2 /k!, do not seem to be a sequence of rational approximations to e2 that allow
us to prove e2 is irrational via Theorem 4.1. This was circumvented in Theorem 5.1, where
the nonzero integral powers of e (not just e2 ) were proved to be irrational using rational
approximations coming from something
than the Taylor series for ex . What we will
Pn other
k
show here is that the partial sums k=0 2 /k! can, after all, be used to prove irrationality
of e2 by focusing on a certain subsequence of the partial sums and exploiting a peculiar
property P
of 2. The argument we give is due to Benoit Cloitre.
Write nk=0 2k /k! in reduced form as an /bn . Then e2 > an /bn , so bn e2 − an > 0. While
numerical data suggest bn e2 − an does not go to 0 as n → ∞, it turns out that these nonzero
differences tend to 0 as n runs through the powers of 2. The table below has some limited
evidence in this direction.
n bn e2 − an
2
2.389
4
.3890
8
.5526
16
.0881
32
.0006
64
.0211
128
.0005
256
.0001
If these differences do tend to 0, then this proves e2 is irrational by Theorem 4.1. To
prove this phenomenon is P
real, we use Lagrange’s form of the remainder to estimate the
difference between e2 and nk=0 2k /k! before setting n to be a power of 2. Lagrange’s form
12
KEITH CONRAD
of the remainder says: for an infinitely differentiable function f and integer n ≥ 0,
f (x) =
n
X
f (k) (0)
k=0
k!
xk +
f (n+1) (c) n
x ,
(n + 1)!
where c is between 0 and x. Taking f to be the exponential function and x = 2,
e2 =
n
X
2k
k=0
k!
+
ec 2n
,
(n + 1)!
where 0 < c < 2. Bring the sum to the left side, multiply by n!/2n , and take absolute
values:
n
n!
k
X
n!
e2
2
.
(A.1)
≤
n e2 − n
2
2
k! n + 1
k=0
Set
n
n! X 2k
cn = n
,
2
k!
k=0
Pn
dn =
n!
,
2n
2k /k!
= an /bn and (A.1) says |dn e2 −cn | ≤ e2 /(n+1). Thus |dn e2 −cn | → 0
so cn /dn = k=0
as n → ∞ and dn e2 −cn 6= 0. Be careful: the numbers cn and dn are not themselves integers
(as we will see), so the expression |dn e2 − cn | is not quite in a form suitable for immediate
application of Theorem 4.1. But we can get good control on the denominators of cn and
dn .
P
Writing cn = (1/2n ) nk=0 (n!/k!)2k , since n!/k! is an integer cn is an integer divided by
2n , so cn has a 2-power denominator in reduced form. Since dn is an integer divided by 2n ,
its reduced form denominator is also a power of 2. What are the powers of 2 in the reduced
form for cn and dn ?
For a nonzero rational number r, write ord2 (r) for the power of 2 appearing in r,
e.g., ord2 (40) = 3 and ord2 (21/20) = −2. By unique prime factorization, ord2 (rr0 ) =
ord2 (r)+ord2 (r0 ) for nonzero rationals r and r0 . Another important formula is ord2 (r+r0 ) =
min(ord2 (r), ord2 (r0 )) when ord2 (r) 6= ord2 (r0 ) and r + r0 6= 0. (These properties both resemble the degree on polynomials and rational functions, except the degree has a max where
ord2 has a min on sums.)
We want to compute ord2 (cn ) and ord2 (dn ). As dn = n!/2n is simpler than cn , we look at
it first. Since ord2 (dn ) = ord2 (n!/2n ) = ord2 (n!) − n, we bring in a formula for the highest
power of 2 in a factorial, due to Legendre: ord2 (n!) = n − s2 (n), where s2 (n) is the sum of
the base 2 digits of n. For example, 6! = 24 · 32 · 5 and 6 = 2 + 22 , so 6 − s2 (6) = 6 − 2 = 4
matches ord2 (6!). With this formula of Legendre,
ord2 (dn ) = ord2 (n!) − n = (n − s2 (n)) − n = −s2 (n).
For n ≥ 1 this is negative (there is at least one nonzero base 2 digit in n, so s2 (n) ≥ 1),
which proves dn is not an integer.
What about ord2 (cn )? Writing
cn =
n
X
n!2k
k=0
2n k!
=
n!
n! · 4
n! · 23
n! · 2
+
+
+
+ · · · + 1,
2n 2n · 1 2n · 2
2n · 6
IRRATIONALITY OF π AND e
13
each term has at worst a 2-power denominator since n!/k! ∈ Z when 0 ≤ k ≤ n. To figure
out the power of 2 in the denominator we will compute ord2 (n!2k /2n k!). For k = 0 this is
ord2 (n!/2n ) = −s2 (n). For k ≥ 1 this is
ord2 (n!) + k − n − ord2 (k!) = −s2 (n) + s2 (k) > −s2 (n)
since s2 (k) ≥ 1. Therefore every term in the sum for cn beyond the k = 0 term has
a larger 2-power divisibility than the k = 0 term, which means ord2 (cn ) is the same as
ord2 (n!/2n ) = −s2 (n). In other words, cn and dn have the same denominator: 2s2 (n) .
If we let n be a power of 2 then cn and dn have denominator 21 = 2, so 2cn and 2dn are
integers. Then the estimate
0 < |2dn e2 − 2cn | = 2|dn e2 − cn | ≤
2e2
→0
n+1
proves e2 is irrational by Theorem 4.1.
p
It is natural to ask if the same argument yields a proof that
Pne isk irrational for prime
x
n
p > 2 using the Taylor series for e at x = p. Let cn = (n!/p ) k=0 p /k! and dn = n!/pn ,
so |dn ep − cn | ≤ ep /(n + 1) as before. Legendre’s formula for ordp (n!), the highest power
of p in n!, is (n − sp (n))/(p − 1), where sp (n) is the sum of the base p digits of n. The
fractions cn and dn have the same p-power denominator, with exponent ordp (n!) − n =
n(1 − 1/(p − 1)) + sp (n)/(p − 1). Alas, for p > 2 this exponent of p in the denominator of
cn and dn blows up with n because 1 − 1/(p − 1) > 0 for p > 2. When p = 2 this exponent
in the denominator is s2 (n), which can stay bounded by restricting n to the powers of 2.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
A. Baker, “Transcendental Number Theory,” Cambridge Univ. Press, Cambridge, 1975.
A. O. Gelfond, “Transcendental and Algebraic Numbers,” Dover, New York, 1960.
E. Hairer and G. Wanner, “Analysis by its History,” Springer-Verlag, New York, 1996.
K. Ireland and M. Rosen, “A Classical Introduction to Modern Number Theory,” 2nd ed., SpringerVerlag, New York, 1990.
I. Niven, A simple proof that π is irrational, Bull. Amer. Math. Soc. 53 (1947), 509.
A. Shidlovski, “Transcendental Numbers,” de Gruyter, Berlin 1989.
M. Spivak, “Calculus,” 2nd ed., Publish or Perish, Wilmington, Del., 1967.
L. Zhou and L. Markov, Recurrent Proofs of the Irrationality of Certain Trigonometric Values, Amer.
Math. Monthly 117 (2010), 360–362.
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