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o6report normalnumbers.pdf - Adobe Reader File Edit View Document Tools Window Help 06report-normal ® Find Normal Numbers are Normal By Davar Khoshnevisar Department of Mathematics University of Utah ABSTRACT. A number is normal in base b if every sequence of k symbols in the letters 0, 1,... .b -1 occurs in the base-b expansion of the given number with the expected frequency b *. From an informal point of view, we can think of numbers normal in base 2 as those produced by flipping a fair coin, recording 1 for heads and 0 for tails. Normal numbers are those which are normal in every base In this expository article, we recall Borel's result that almost all numbers are normal. Despite the abundance of such numbers, it is exceedingly difficult to find specific exemplars. While it is known that the Champernowne number 0.123456789101112131415. is normal in base 10, it is (for example) unknown whether v2 is normal in any base. We sketch a bit of what is known and what is not known of this peculiar class of numbers, and we discuss connections with areas such as computability theory 1. Introduction Let r be a real number between zero and one. We can write it, in binary form, as r = 0.r1T2 . .. where each r, takes the values zero and one. We are interested first of all in "balanced" numbers- numbers r such that half of their binary digits are zeros and the remaining half are ones. More

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o6report normalnumbers.pdf - Adobe Reader File Edit View Document Tools Window Help 06report-normal ® Find Normal Numbers are Normal By Davar Khoshnevisar Department of Mathematics University of Utah ABSTRACT. A number is normal in base b if every sequence of k symbols in the letters 0, 1,... .b -1 occurs in the base-b expansion of the given number with the expected frequency b *. From an informal point of view, we can think of numbers normal in base 2 as those produced by flipping a fair coin, recording 1 for heads and 0 for tails. Normal numbers are those which are normal in every base In this expository article, we recall Borel's result that almost all numbers are normal. Despite the abundance of such numbers, it is exceedingly difficult to find specific exemplars. While it is known that the Champernowne number 0.123456789101112131415. is normal in base 10, it is (for example) unknown whether v2 is normal in any base. We sketch a bit of what is known and what is not known of this peculiar class of numbers, and we discuss connections with areas such as computability theory 1. Introduction Let r be a real number between zero and one. We can write it, in binary form, as r = 0.r1T2 . .. where each r, takes the values zero and one. We are interested first of all in "balanced" numbers- numbers r such that half of their binary digits are zeros and the remaining half are ones. More

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Uploaded Mar 09, 2010 at 02:30AM EST

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X Y is X (Redundant Adjectives are Redundant)

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