Mathematical Error
Author
Lester Telser
Title
Mathematical Error
Description
Irrational numbers are unavoidable sources of error
Category
Essays, Posts & Presentations
Keywords
irrational numbers, Pi
URL
http://www.notebookarchive.org/2021-02-472zs79/
DOI
https://notebookarchive.org/2021-02-472zs79
Date Added
2021-02-09
Date Last Modified
2021-02-09
File Size
11.3 kilobytes
Supplements
Rights
Redistribution rights reserved
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Mathematical Error Is Unavoidable
L. G. Telser
L. G. Telser
Everybody knows there is measurement error in observations but few know measurement error is inherent in pure mathematics. Irrational numbers are its source. Pythagoras implied this more than two thousand years ago when he discovered that is not a rational number. Now more than ever the error inherent in irrational numbers demands attention especially in models that use Fourier series. Two illustrations are next.
2
Then-orderFourierseriesoff(t)isbydefaultdefinedtobewith=f(t)tNextarecommonsettings
th
n
∑
k=-n
c
k
kt
c
k
1
2π
π
∫
-π
-kt
c k 1 2π π ∫ -π -kt | n ∑ k=-n c k kt |
Two transcendental numbers enter this expression, π and . They are written as if they are exact numbers, no different in this respect than integers or rationals. One who uses this expression implicitly assumes π and satisfy the axioms of arithmetic such as the Peano axioms. However, because these irrational numbers are limits of convergent series, they belong to an open set defined by the accuracy of the approximation given by the infinite series that make them. One can say the set contains different limits arbitrarily close to each other but never equal. This is the simplest case.
The name of the set for a given transcendental number, say π, is not a number. A member of the set is a number. The infinite series that compute π produce sums as members of the set perhaps identified by the number of their terms and their estimated accuracy. The set named π has many members. The symbol π itself is not a number. The famous formula given by Euler. =-1, is exact but not a counter example to my analysis. It follows from the definition = Cos[x] + i Sin[y]. Set x = 0, y = π and get -1.
π
x+y
This analysis applies to one transcendental, but a Fourier series involves many operations, even as simple as addition and multiplication. The result is a more complicated open set. It is a delusion to assume this has no effect on applications of these series to science and reality. The Monte Carlo method invented by Ulam has the great virtue of dispelling delusions about accuracy when interpreting the results. That Ulam was applying complicated theories to understand the quantum world was probably an important motive for his invention as well as the source of its widespread use.
Quantum computing in IBM as described in a recent article in Fortune supports this interpretation of error.
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Cite this as: Lester Telser, "Mathematical Error" from the Notebook Archive (2021), https://notebookarchive.org/2021-02-472zs79
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