prove π is irrational ??
Answers
Answer:
In the 1760s, Johann Heinrich Lambert proved that the number π (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus.
Answer:
Suppose
π
=
a
/
b
. Define
f
(
x
)
=
x
n
(
a
−
b
x
)
n
n
!
and
F
(
x
)
=
f
(
x
)
−
f
(
2
)
(
x
)
+
f
(
4
)
(
x
)
−
.
.
.
+
(
−
1
)
n
f
(
2
n
)
(
x
)
for every positive integer
n
.
First note that
f
(
x
)
and its derivatives
f
(
i
)
(
x
)
have integral values for
x
=
0
, and also for
x
=
π
=
a
/
b
since
f
(
x
)
=
f
(
a
/
b
−
x
)
.
We have
d
d
x
(
F
′
(
x
)
s
i
n
x
−
F
(
x
)
c
o
s
x
)
=
F
′′
(
x
)
s
i
n
x
+
F
(
x
)
s
i
n
x
=
f
(
x
)
s
i
n
x
whence
∫
π
0
f
(
x
)
s
i
n
x
d
x
=
[
F
′
(
x
)
s
i
n
x
−
F
(
x
)
c
o
s
x
]
π
0
=
F
(
π
)
+
F
(
0
)
∈
Z
But for
0
<
x
<
π
, we have
0
<
f
(
x
)
s
i
n
x
<
π
n
a
n
n
!
which means we have an integer that is positive but tends to zero as
n
approaches infinity, which is a contradiction.
Step-by-step explanation:
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