Eilenberg–Mazur swindle (nonfiction)

In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums.

In geometric topology it was introduced by Mazur (1959, 1961) and is often called the Mazur swindle.

In algebra it was introduced by Samuel Eilenberg and is known as the Eilenberg swindle or Eilenberg telescope (see telescoping sum).

The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0:

1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0 This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense, to show that if A + B = 0 then A = B = 0.

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics

Nonfiction cross-reference

• Samuel Eilenberg (nonfiction)
• Geometric topology (nonfiction) - the study of manifolds and maps between them, particularly embeddings of one manifold into another.
• Grandi's series (nonfiction) - the infinite series 1 − 1 + 1 − 1 + ⋯, named after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2.
• Mathematician (nonfiction)
• Mathematics (nonfiction)
• Barry Mazur (nonfiction)
• Telescoping series (nonfiction) - a series whose partial sums eventually only have a fixed number of terms after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.

External links:

• Eilenberg–Mazur swindle @ Wikipedia