**Topic: Representation of Real Numbers in Decimal Form**Define a digit to be an element of the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.Show that any real number $x \in [0, 1]$ can be represented as $x = 0.a_1a_2a_3 \ldots$ for some sequence of digits $(a_n)_{n=1}^\infty$.

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Answered: Define a digit to be an element of the…

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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Consider numbers of the form a = a₁ª2a3….. an, with n digits, each digit from the set {1, 2, 3, 4, 5, 6, 7, 8}. For example, if n = 4, the number a = a₁ª2ª3a4 will have 4 digits from the set {1,2, 3, 4, 5, 6, 7, 8}. Let n = 7. a. How many such numbers are there? b. How many c. How many such numbers are there for which the sum of the digits is even? such numbers contain more even digits than odd digits?
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Define a digit to be an element of the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Show that any real number x & [0, 1] can be represented as x = 0.a₂a₂a.....for some sequence of digits {a} n=1