This sophism was constructed
by Swiss mathematician John Bernoulli
(1667 - 1748), who was one of the eight
outstanding mathematicians in the Bernoulli
family.

Find a mistake in the
following chain of arguments, pretending
to prove that

**An
explanation:**

The conclusion that
Ln(-z) = Ln(z)
is false, because

Ln(*z*)
= ln(|*z*|)
+* i*[arg(*z*)
+2*k*],
*k*
= 0, ±1, ±2, ... ,

Ln(*-z*)
= ln(|*z*|)
+* i*[arg(*z*)
+(2*k*+1)],
*k*
= 0, ±1, ±2, ... ,

and none of the numbers representing
the value of Ln(*z*)
is the same as any of the numbers
representing Ln(*-z*).

The error occurs in
going from line 2 to line 3 because

Ln(-z) + Ln(-z)
2Ln(-z),

Ln(z) + Ln(z)
2Ln(z).

The following example elucidates
the situation:

Let **A** be the set
of two numbers **3**
and **4**.

A set **B=A+A** is the
set of numbers **6**;**
7**;** 8** because
**3+3=6**, **3+4=7**
and **4+4 =8**.

A set **C=2·A**
is the set of numbers **9**;**
12**;** 16 **because
**3·3=9**; **3·4=12**
and **4·4=16**.

So, a set **A+A2·A**

Ln(-z) + Ln(-z)
2Ln(-z),

Ln(z) + Ln(z)
2Ln(z).

by
Tetyana Butler

**Reference**

A.I. Markushevich, "Theory
of functions of a complex variable"
, 1–2 , Chelsea (1977) (Translated
from Russian)