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Posted: October 29th, 2022

Powers of Two and Their Decimal Digits

A MOST SIGNIFICANT NUMBER: Powers of Two and Their Decimal Digits
Powers of Two and Their Decimal Digits
Background and Inquiry. Enumerated the first few powers of the number 2:
1, 2, 4, 8, 16, 32, 64, . . .
and ponder: which digits can occur as the most significant digits of these exponential expressions? (Clearly, 1, 2, 3, 4, 6 and 8 come up, but what about the rest?). Proceeded with a purely-human investigation and reasoning, reached a conclusion and provided sufficient rationale to constitute a proof.

(i) Define the concept of most significant digit in a way aimed to be accessible to a reader with pre-college mathematics background.

(ii) Describe the “Most significant Question”, for powers of 2, posed above using language and terms accessible to a student with background similar to a University-level knowledge, but without classroom experience.

(iii) Provide an answer to the question and prove, using purely human terms (read: no computer involvement) that your answer is correct. Use lemmas, theorems, etc. in this part, but do so in a way that is deemed accessible to any student in the same class. Be sure to five references and acknowledge any sources or tools used that do not constitute your own work.

(iv) Discuss the value of trying to answer the question in purely human terms in today’s world, where powerful software and hardware tools are available.

(v) Imagine that someone solved the problem many years ago, when no computers were available, and just remembered the problem again now. What additional insights might they gain with help from a computer?

Some Explanations:
An explanation of digit significance: the study of power towers was focused on the convergence of a power tower of infinite order. Precisely the study of the last digit and the digital root of a power tower of a positive integer, and find a pattern when the order increases. As a number is expressed as a sequence of digits based on a base-10 system written from left to right. The digits on the left contribute higher value to the number and are more significant than the ones to their right. Being that any number can be expanded to a sum of powers of 10 with coefficients less than 10. For example, 1234 has four digits; the expanded form is k1 ∗ (10^3)l + k2 ∗ (10^2)) + k2 ∗ (10^1)) + 4. Therefore, the last coefficient in the expanded form in descending order is the last digit of the number.

With a base of 2 where the exponents range from 2^1 to 2^1000. We notice that the most significant digit that occurs is as follows {(1, 301), (2, 176), (3, 125), (4, 97) (5, 79), (6, 69) (7, 56), (8, 52), (9, 45).}. Buy a base 10 number system we see that the higher the most significant number is the life at least amount of times it occurs. For example 9 only show up 45 times as the most significant digit in the range of 2^1 to 2^1000; in comparison to 1 that occur 301 times as the most significant digit.

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