1
vote
3answers
85 views

Product of “reversed” numbers

Consider any 2 binary numbers, e.g.: 10101011 ; 11111101 and their product, say P. "Reverse" (mirror image) all the digits of the 2 numbers, e.g.: ...
2
votes
1answer
68 views

Calculating 6 decimal digits of $3^{\sqrt2}$ using a calculator.

How can we calculate $3^{\sqrt2}$ to 6 decimal digits, using only a semi-basic calculator (Which has the square root too) and a pen and paper? I asked this question from my teacher and he ...
0
votes
0answers
62 views

What is the number theory behind this?

I am given $3^{1000}$ and asked to find, in base $2$, now many digits it takes to represent this number. According to Wolfram, it is $1585$, but I don't know why. I understand that $2^n$ would be ...
0
votes
2answers
119 views

fill-in-the-blank induction proof

I'm stuck at homework task I'm working on. I would really appreciate some help. Here is the task: Let $f$ be a function on binary numbers defined recursively as follows. $f(0) = 1$ and ...
8
votes
2answers
79 views

Are the high-order bits of $n^2$ as likely to be zeroes as ones?

Let $B_i(n)$ be the $i$th bit in the binary expansion of $n$, so that $n=\sum B_i(n)2^i$. Now let $n$ be randomly and uniformly chosen from some large range, and let $E(j)$ be the expected value of ...
1
vote
1answer
41 views

Multiples of $5$ in base $2$

As a follow-up question, what can be said about multiples of $5$ in base 2? I think I must look at the last two digits, but I am not sure what the whole idea might be.
1
vote
1answer
91 views

Even numbers in base 2

We all know even numbers are the ones that end in even digits. How do we analyze even numbers in base 2?
5
votes
2answers
310 views

Why does the $2$'s and $1$'s complement subtraction works?

The algorithm for $2$'s complement and $1$'s complement subtraction is tad simple: $1.$ Find the $1$'s or $2$'s complement of the subtrahend. $2.$ Add it with minuend. $3.$ If there is ...