The decimal representation of a positive integer a is given by a = an-1an-2a₁a0 where a = an-110n-¹ + An−210¹−² + + a₁10 + ao and the digits an-1ªn-2a₁ª₁ are in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with an-1 ‡ 0. In this case we say that the integer a is an n digit number or that a is n digits long. For example, the number 756 = 7.10² + 5∙10 + 6 and is 3 digits long. Let a E N. If a is an n digit number, then from the decimal representation given above we know 10n-1 < a < 10". Use this inequality to prove that n = [log10 a] + 1.
The decimal representation of a positive integer a is given by a = an-1an-2a₁a0 where a = an-110n-¹ + An−210¹−² + + a₁10 + ao and the digits an-1ªn-2a₁ª₁ are in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with an-1 ‡ 0. In this case we say that the integer a is an n digit number or that a is n digits long. For example, the number 756 = 7.10² + 5∙10 + 6 and is 3 digits long. Let a E N. If a is an n digit number, then from the decimal representation given above we know 10n-1 < a < 10". Use this inequality to prove that n = [log10 a] + 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The decimal representation of a positive integer a is given by a = an-1 ªn-2 ··· A₁ ao where
a = an-110¹-¹+ an-210¹-² + + a₁10 + ao
and the digits an-1an-2 ··· A₁ ao are in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with an-1 # 0. In
this case we say that the integer a is an n digit number or that a n digits long. For
example, the number 756 = 7.10² + 5∙10 + 6 and is 3 digits long.
Let a E N. If a is an n digit number, then from the decimal representation given above we
know 10¹-1< a < 10". Use this inequality to prove that n [log₁0 a] + 1.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed14a3ea-da26-4be7-a143-8b845df95e91%2F6029036c-36bd-413e-8354-4ede9e11f96f%2Fzlqrumd_processed.png&w=3840&q=75)
Transcribed Image Text:The decimal representation of a positive integer a is given by a = an-1 ªn-2 ··· A₁ ao where
a = an-110¹-¹+ an-210¹-² + + a₁10 + ao
and the digits an-1an-2 ··· A₁ ao are in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with an-1 # 0. In
this case we say that the integer a is an n digit number or that a n digits long. For
example, the number 756 = 7.10² + 5∙10 + 6 and is 3 digits long.
Let a E N. If a is an n digit number, then from the decimal representation given above we
know 10¹-1< a < 10". Use this inequality to prove that n [log₁0 a] + 1.
=
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