Mathematics Database Programming Web Design Price List     Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise  # Bernoulli's Sophism Complex Analysis. FreeTutorial

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

Ln(-z) = Ln(z) for any .

 "Proof:" 1. Ln[(-z)2] = Ln(z2); 2. Ln(-z) + Ln(-z) = Ln(z) + Ln(z); 3. 2Ln(-z) = 2Ln(z); 4. Ln(-z) = Ln(z). Where is the mistake?

An explanation:

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

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

Ln(-z) = ln(|z|) + i[arg(z) +(2k+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+A 2·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) Top  Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers. Free Lessons Contact us