Block Conversion from Binary to Octal
The following is a proof that this procedure works as advertised. Suppose ???? is a nonnegative integer and its binary expansion is given by: (Proof Attached)
Questions marked [P] require a rigorous proof. Questions marked [V] ?only? require a clear, succinct verbal explanation. [VA] requires such a verbal explanation, backed up by algebra, but omitting the formalities of a rigorous proof, such as strict, explicit attention to quantification, assumptions and conclusions.
1. What is the meaning of ?without loss of generality?? What is the purpose here of that device? How can the number of terms in the sum, an arbitrary integer, always be a multiple of 3? [V]
2. What does the quantity q represent? [V]
3. Why is the single sigma sum equal to the double sigma sum? [VA] 4. What algebraic laws are used to simplify the sigma sum? Why could the 23???? be factored out of the inner sigma sum, but not the outer one? [V]
5. Fill in a detail that the given proof omits, namely why the ???????? satisfy the inequality 0 ? ???????? ? 7. [VA]
6. Rewrite the given proof to demonstrate how and why block conversion from octal to binary works. [P]
7. Write a similar proof to demonstrate how and why block conversion from binary to hexadecimal works. [P]
8. You can?t block convert from binary to decimal. At which specific point(s) would a proof that follows the lines of the proof given here fail? [V]
9. For which numbers ???? can binary numbers be block converted to base-b? [Answer only, no explanation necessary.]
10. Write a proof that block conversion always works for those numbers. [P]
This question was answered on: Sep 18, 2020
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