Mathematical Fallacies And Paradoxes Pdf
A
DDIS
A
BABA UNIVERSITY
Mathematical Fallacies and Paradoxes
Miliyon T.
March 10, 2015 Abstract:
Mathematicalfallacies
are errors, typically committed with an in- tent to deceive, that occur in a mathematical proof or argument. A fallacy in an argument doesn't necessarily mean that the conclusion is necessarily incor- rect, only that the argument itself is wrong. However, fallacious arguments can have surprising conclusions. Apart from a mathematical fallacy a
paradox
is a statement that goes against our intuition but may be true, or a statement that is or appears to be self-contradictory. Mathematical paradoxes result from either counter-intuitive properties of infinity, or self-reference.
F
IRST
F
ALLACY
AfallacyduetoJohnBernoulli,maybestatedasfollows. Wehave(
−
1)
2
=
1.Takelogarithms, 2log(
−
1)
=
log1
=
0. log(
−
1)
=
0
−
1
=
e
0
−
1
=
1 The same argument may be expressed thus. Let
x
be a quantity which satisfies the equa- tion
e
x
=−
1 Square both sides
e
2
x
=
1
⇒
2
x
=
0
⇒
x
=
0
⇒
e
x
=
e
0
But
e
x
=−
1 and
e
0
=
1,
−
1
=
1 1
S
ECOND
F
ALLACY
Suppose that
a
=
b
, then
ab
=
a
2
ab
−
b
2
=
a
2
−
b
2
b
(
a
−
b
)
=
(
a
+
b
)(
a
−
b
)
b
=
a
+
b b
=
2
b
1
=
2
T
HIRD
F
ALLACY
Let
a
and
b
be two unequal numbers, and let
c
be their arithmetic mean, hence
a
+
b
=
2
c
(
a
+
b
)(
a
−
b
)
=
2
c
(
a
−
b
)
a
2
−
2
ac
=
b
2
−
2
bc a
2
−
2
ac
+
c
2
=
b
2
−
2
bc
+
c
2
(
a
−
c
)
2
=
(
b
−
c
)
2
a
=
b
F
OURTH
F
ALLACY
From Taylor's expansion, we know that log(1
+
x
)
=
x
−
1 2
x
2
+
1 3
x
3
−···
. If
x
=
1, the resulting series is convergent; hence we have log2
=
1
−
1 2
+
1 3
−
1 4
+
1 5
−
1 6
+
1 7
−
1 8
+
1 9
−···
. 2log2
=
2
−
1
+
2 3
−
1 2
+
2 5
−
1 3
+
2 7
−
1 4
+
2 9
−···
. Taking those terms together which have a common denominator, we obtain 2log2
=
1
+
1 3
−
1 2
+
1 5
+
1 7
−
1 4
+
1 9
−··· =
1
−
1 2
+
1 3
−
1 4
+
1 5
−··· =
log2 2
=
1 2
F
IFTH
F
ALLACY
This fallacy is very similar to that last given. We have log2
=
1
−
1 2
+
1 3
−
1 4
+
1 5
−
1 6
+··· =
1
+
1 3
+
1 5
+···
−
1 2
+
1 4
+
1 6
+···
=
1
+
1 3
+
1 5
+···
+
1 2
+
1 4
+
1 6
+···
−
2
1 2
+
1 4
+
1 6
+···
=
1
+
1 2
+
1 3
+···
−
1
+
1 2
+
1 3
+···
=
0 The error in each of the foregoing examples is obvious, but the fallacies in the next exam- ples are concealed somewhat better.
S
IXTH
F
ALLACY
We can write the identity
−
1
= −
1 in the form
−
1 1
=
1
−
1
−
1
1
=
1
−
1 (
−
1)
2
=
(
1)
2
−
1
=
1
S
EVENTH
F
ALLACY
Again, we have
a
·
b
=
ab
−
1
· −
1
=
(
−
1)(
−
1) (
−
1)
2
=
1
−
1
=
1
E
IGHTH
F
ALLACY
The following demonstration depends on the fact that an algebraical identity is true what- ever be the symbols used in it, and it will appeal only to those who are familiar with this fact. We have, as an identity,
x
−
y
=
i
y
−
x
(1) where
i
stands either for
+ −
1 or for
− −
1. Now an
identity
in
x
and
y
is necessarily true whatever numbers
x
and
y
may represent. First put
x
=
a
and
y
=
b
,
a
−
b
=
i
b
−
a
(2) 3
Mathematical Fallacies And Paradoxes Pdf
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