From Vladimir Zorich Analysis I:
A number $x$ is represented on a computer as $$x=\pm q^{p}\sum_{n=1}^{k}\frac{a_n}{q_n}$$
where $p$ is the order of $x$ and $M=\sum_{n=1}^{k}\frac{a_n}{q_n}$ is the mantissa of the number $x$ ($\frac{1}{q} \leq M < 1)$.
Now a computer works only with a certain range of numbers: for $q=2$ usually $\left| p \right| \leq 64$ and $k =35$. Evaluate this range in the decimal system.
Definition:
My textbook by the principle of Archimedes and simple algebraic inequality manipulations shows (in a less formal manner) that there exists rational approximation $r_n$ of any $x \in \mathbb{R}^{+}$, such that:
$$r_n = {a_p}{q^p}+...+a_{p-n}q^{p-n}$$
where $q^{p-m}$ for $m \in \{0, 1, ...,n\}$, $q \in \mathbb{R}^{+}/\{0\}$ and $a_{p-m}$ for $m \in \{0, 1, ...,n\}$. Usually, $p$ is called an order of $x$ in the base $q$.
For example, when $q=10$,
$$123.45=1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 + 4 \times 10^{-1} + 5 \times 10^{-5}$$
The term "mantissa" mentioned above is slightly misleading (as often quoted by Donald Knuth), but in this case it obviously represents a significand.
My Attempt:
I'm unable to understand explicitly the purpose of the question, but I've tried simplifying the question by initially omitting significand part (or mantissa part):
Let $r_n = {a_p}{q^p}+...+a_{p-n}q^{p-n}$ be an approximation for any integer $x \in \mathbb{Z}$, let $A=\{{a_p},...,a_{p-n}\}$ and $O = \{\sigma_1, ... , \sigma_n\}$ be a set of all permutations of $A$, such that $\sigma_{i} : A \rightarrow A$. Usually on the computers, if $q=2$, the "maximum" order of $O$ for integral part is $q^{64}$, what is the "maximum" order of $O$ if $q=10$?
The answer to the question I've proposed is $10^{\log_{10}{(2^{64})}} \approx 10^{19.3}$, and perhaps the same logic could be applied to the significand part of $x$ if it was a rational number (instead of integer, as assumed in my question). I think this seems reasonable, since I've evaluated the ranges in different bases by simply using the logarithmic equation.
Did I understand the task properly? My answer seems too straightforward to be valid, do I have to use lower-order logic to prove relation between different ranges of different basis?
Thank you!
I don't think you are meant to set $q=10.$
I think the intention rather is to say, given that $$x=\pm q^p\sum_{n=1}^{k}\frac{a_n}{q_n}$$ where $q=2$, $\lvert p \rvert \leq 64$, and $k =35$, what is the largest possible value of $x$? The answer should be evaluated exactly as written above (in particular, $q=2\neq 10$), but use decimal notation when showing the result.
It might also be desired for you to write the smallest possible non-zero value of $x,$ again evaluating it exactly as written in the formula but showing the answer in decimal notation.
That said, I wonder what kind of computers Prof. Zorich works with on which $\lvert p \rvert \leq 64$ and $k =35$. Those parameters seem to imply a $42$-bit word.