By Bryan H. Bunch
Publish yr note: First released in 1982
From old Greek arithmetic to 20th-century quantum concept, paradoxes, fallacies and different highbrow inconsistencies have lengthy questioned and intrigued the brain of guy. This stimulating, thought-provoking compilation collects and analyzes the main attention-grabbing paradoxes and fallacies from arithmetic, common sense, physics and language.
While focusing totally on mathematical problems with the 20 th century (notably Godel's theorem of 1931 and choice difficulties in general), the paintings takes a glance to boot on the mind-bending formulations of such very good males as Galileo, Leibniz, Georg Cantor and Lewis Carroll ― and describes them in comfortably obtainable element. Readers will locate themselves engrossed in pleasant elucidations of tools for false impression the true international via test (Aristotle's Circle paradox), being led off beam by way of algebra (De Morgan's paradox), failing to understand genuine occasions via good judgment (the Swedish Civil protection workout paradox), mistaking infinity (Euler's paradox), knowing how likelihood ceases to paintings within the actual international (the Petersburg paradox) and different complicated difficulties. a few highschool algebra and geometry is thought; the other math wanted is built within the textual content. interesting and mind-expanding, this quantity will entice somebody searching for not easy psychological routines.
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Additional resources for Mathematical Fallacies and Paradoxes
A brief description of the terminology that will be used in the remainder of this book is also necessary. The intent of this chapter is to introduce the mathematical concepts that underlie the Monte Carlo method and provide a basis for further development of selected topics. To obtain a broader and more rigorous development of the underlying mathematical concepts than is presented here the interested reader may consult any of a number of standard textbooks and references on statistics and probability.!
The sampling of events uses the following information: • The sample space from which a particular event or sample is to be selected, • The value of the random variables associated with every event in the sample space (this is required so that the Monte Carlo model can deal with real numbers), • The cumulative distribution and/or the pdf for the random variables involved in the problem, and • A method of obtaining a sequence of random numbers. These four elements are used in a Monte Carlo calculation in the following manner: • One or more random numbers are selected.
In order to calculate the statistical uncertainty in estimates of random variables, a few rules regarding the sums and products of variances and standard deviations are needed. 1 Stratified Sampling One effective technique for reducing the variance of a Monte Carlo estimate of a random variable is stratified sampling. With this technique the range over which the independent variable is sampled is divided into strata, and each stratum is sampled separately. 30. We wish to select subranges of the independent variable x to serve as strata for a demonstration of stratified sampling.
Mathematical Fallacies and Paradoxes by Bryan H. Bunch