Questions related with (base 2) representation of numbers and their unique properties arising out of number representation.

learn more… | top users | synonyms

0
votes
0answers
15 views

A property of permutation codes

A permutation code $C$ is, for $k\ge2$ and $M\ge k+2$, is the rows of the matrix that one gets when put $M$-length columns where there are $k$ ones and $M-k$ zeros. In his paper ...
1
vote
4answers
654 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
3
votes
2answers
62 views

How to populate a $0-$line with $1$'s?

I have a line of $n$ $0$'s like this: Zeroth index -->$000...000$ I want to populate the line with $m$ $1$'s with the following rules: (1) They have to occur after the index ...
0
votes
0answers
13 views

Learning Booth's Algorithm, I Can't Find the Issue on Final Result

I am practicing using Booth's Algorithm to multiply a positive number and a negative number (specifically the problem is $-12 \times 4$). I have included my attempt, but I can't find the issue. If ...
0
votes
3answers
33 views

Efficient algorithm for taking powers of binary numbers?

Is there any efficient algorithm for taking powers of binary numbers? Or even just squaring one? Thanks ahead of time.
4
votes
1answer
57 views

Why do we divide or multiply by 2 when converting binary?

Trying to understand the fundamentals of binary rather than just following steps, I wanted to know why do we multiply by 2 to convert a decimal (0.5, 0.25) to a binary and why do we divide by 2 when ...
0
votes
1answer
19 views

Division binary using polynomials

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
1
vote
0answers
31 views

Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
2
votes
2answers
219 views

Longest sequence of 1s in binary representation of a number & Average Sequence length for number with N digits

I originally posted this to rec.puzzles in October 1992. I subsequently solved the problem (at least I think I did...) and published an article about it in early 1993. The article is nowhere to be ...
0
votes
1answer
19 views

Computing the Value of a minimax tree

I am asked to compute the value of a minimax tree, which each node labeled with its initial value. I am just unsure how to do it. I know that it is a minimax tree if: the root is a min node, the ...
0
votes
1answer
12 views

Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
0
votes
1answer
17 views

Probabilities of a Binary String

You generate a random $N$-bit binary string, and compute $X = \Sigma_1^N x_i$, where the $x_i$ are the $0$ and $1$ entries of the string. What is the probability that $X$ is odd, if $N$ is odd? What ...
0
votes
1answer
42 views

I'm trying to convert a number to binary. What am I doing wrong?

So I'm trying to convert a number to binary, but I keep getting the incorrect answer. If it matters I was roughly following this guide. Here is my work: $120 = 60 + 60$ $120 = 64 + 32 + 16 + 8$ ...
1
vote
1answer
55 views

the blood test riddle (number theory)

A microbiologist has been given a set of $100$ blood vials. Exact one of those $100$ vials is positive to a concrete disease X. The microbiologist desires to send only $7$ vials for analysis. He can ...
0
votes
1answer
31 views

Bitwise ops - The relationship between $a$, $b$, $a \wedge b$, $a \vee b$ and $a \oplus b$

In computer programming, the term bitwise operation is used to denote the use of boolean operators (and $\wedge$, or $\vee$, exclusive or $\oplus$) on corresponding bits of two numbers. Bits, in this ...
6
votes
1answer
156 views

Why is $2^{16}=65536$ the only power of $2$ less than $2^{31000}$ that doesn't contain the digits $1$, $2$, $4$ or $8$ in its decimal representation?

$65536$ is the only power of $2$ less than $2^{31000}$ that does not contain the digits $1$, $2$, $4$ or $8$ in its decimal representation. http://en.wikipedia.org/wiki/65536_%28number%29
0
votes
0answers
31 views

Is there an alternative encoding scheme to binary where similarity of pattern correlates with size of number?

If I compare binary for 7 111 and binary for 8 1000 there is no correlation between these two patterns that suggests that ...
-1
votes
1answer
47 views

Binary strings and recurrence relations [closed]

So the problem is: How many binary strings of length n contains 111? Give a recurrence relation Tn, where Tn is the number of binary strings of length n that contains 111. How could we possibly ...
2
votes
3answers
48 views

Determining if a relation is reflexive, symmetric, or transitive [closed]

Let $A = \{0,1,2,3\}$ Define a relation $T$ on $A$ as follows: $T = \{(0,1),(2,3)\}$ Is $T$ reflexive? symmetric? transitive?
0
votes
3answers
27 views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
0
votes
1answer
31 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
3
votes
0answers
44 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
1
vote
0answers
22 views

How to determine size and height balance of binary search tree?

I've been reading/ learning binary search trees and I've been stuck on the following question for a while now. I have the following tree, how do I determine the height and size balance of it? How do ...
0
votes
1answer
21 views

Algorithm for Converting Non-balanced Base-n to Balanced Base-n (for odd n)

Let $n \in 2 \mathbb{N} - 1$. I was wondering what sort of algorithms there are for converting (non-balanced) base-n to balanced base-n, where "balanced" is as is described in this article: ...
1
vote
2answers
76 views

How to check quickly $\frac{2}{3}=.101010… $ holds?

Every $x \in [0, 1]$ can be expressed in the form $\dfrac{a_1}{2}+\dfrac{a_2}{2^2}+\dots + \dfrac{a_m}{2^m}+\dots$ , where each $a_i$ equals either $0$ or $1$. For such $x$, we have the binary ...
0
votes
2answers
512 views

Quick logarithm calculation

In coming up with an algorithm for finding log (10) base 2, these are my thoughts. I wanted to know if this makes sense and how could I truly make it more efficient. The requirements are strictly not ...
1
vote
2answers
73 views

Recursively defining sets of strings discrete math

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set K? Can someone ...
1
vote
1answer
20 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
-2
votes
0answers
23 views

Recursively defining sets of strings [duplicate]

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set $K$? For (1) I got ...
0
votes
3answers
27 views

Binary remainder not equal to the decimal remainder

I am having a weird result. I am dividing the binary number $10101010100000$ by $10011$. In binary division. I get $R= 0100$ which is 4. However, If I consider the decimal representation of the ...
0
votes
2answers
1k views

binary representation of a fraction in two's complement

Could any one please explain what is a 16-bit two's-complement representation of -0.375 and the steps to calculate it? Also, what happens if I convert it back to decimal? Thanks
0
votes
1answer
26 views

Binary Representation of the Collatz Conjecture

What is the benefit of looking at the binary representation of the collatz conjecture. I know that it makes the computation easier because there is really one operation involved which is multiplying ...
0
votes
1answer
20 views

why borrow two 1s in binary substruction?

In binary subtraction i have seen some tutorial and some sites that during 0-1 process borrow value from next 1s. Let's say one of tutorial is this ...
2
votes
1answer
33 views

What is the minimum longest repeated substring of a binary string of size n?

The longest repeated substring of 0111011 is 011 for example. My question is given the size of a binary string, what is the shortest this longest repeated substring can be. I have computed values for ...
0
votes
1answer
12 views

Evaluating the decimal equivalent of binary numbers in; sign and magnitude, one's complement and two's complement

For example, i have this binary number : 1011 1101 Now i wish to evaluate the decimal equilant using sign and magnitude, one's complement and two's complement. Now for sign and magnitude, i know the ...
2
votes
2answers
189 views

Floating point binary arithmetic question

I'm doing a basic class on computer architecture and we dwell into Floating Point Arithmetic, I'm not looking for someone to solve my homework, I'm actually just going through old exams and I'm kinda ...
1
vote
1answer
15 views

Why Are The First and Last Entries of a Binary Reflected Gray Code Sequence Adjacent?

Some Information on Gray Codes If you're rusty or unfamiliar with Binary Reflected Gray Code, but want to try and help, here is a youtube tutorial explaining what they are: ...
0
votes
0answers
38 views

binary search worst case for a set database of entries some good some bad

A database has 10,000 sorted entries, 20% known to be good. When looking up a record in the database, the good entries account for 60%. Two design options are considered to store the data in the ...
0
votes
0answers
34 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
1
vote
2answers
7k views

Modulo 2 binary division (XOR not subtracting) method

I have attached an image showing a Modulo 2 binary division. I can roughly understand the working below which is using XOR calculation but I am not sure how the answer (in red) is being computed ...
3
votes
1answer
51 views

Proofs involving some general formulae for trees and binary trees.

So here I have 3 tree-related questions. 1) Let $n\geq2$ and let $d_1 ≤d_2 ≤···≤d_n$ be a sequence of integers. Show that there is a tree with degree sequence $d_1,d_2,...,d_n \Leftrightarrow \sum ...
4
votes
1answer
103 views

Disjoint subsets and Number of 1's in the binary representation

For a subset $S$ of $[n]$, let $\chi(S)$ denote the $n$ bit 'characterisitc vector' of $S$, i.e., $\chi(S)=(a_1, a_2, \ldots, a_n)$ where $a_i=1$ if $i \in S$ and $a_i=0 $ if $i \notin S$. Think of ...
-5
votes
4answers
6k views

Binary Subtraction of Two Unsigned Integers

For unsigned integers $ X = 00110101 $ and $ Y = 10110101 $, determine the following values: $ X + Y = (\text{My answer is}) ~ 11101010 $. $ X - Y = ~ ??? $ $ Y - X = ~ ??? $
6
votes
1answer
732 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
0
votes
2answers
28 views

Base-n arithmetic and multi-dimensional matrices

Interesting thing about binary numbers is to find their decimal value you can represent them as a multidimensional array, where each cell is indexed, starting from 0. For simplicity, let's start with ...
0
votes
0answers
12 views

Find minimum algorithm complexity

So, I have this task: Let us have square matrix $A \ size\ n\ \times\ n$ for which is true: $$A(k,l)<=A(m,p)\ if \ k <=m, l<=p$$ I need to find algorithm, which finds value X in such matrix, ...
0
votes
0answers
79 views

Looking for a best binary division tutorial

All I can found is such confusing "tutorials" like the below examples.. Even if i search it on Youtube, most videos are having dislikes and with short or unclear explanation. Is it a really hard ...
0
votes
0answers
21 views

How Leibniz invented the Binary System?

Do you know which reasoning and observations made Leibniz invent the Binary system ? Some say that he was inspired by Chinese mathematicians do we have any record of how he came with this idea ?
1
vote
3answers
65 views

Convert from binary to quinary

How to convert a number from binary to quinary system without using decimal system ? It is possible ? I want to write a program who does it.
10
votes
2answers
342 views

Sum of series with binary parity in the numerator

I'm now stuck with this question, and I don't even know where to start: Find sum of series$$\sum_1^\infty \frac{f(n)}{n(n+1)}$$, where f(n) - number of ones in binary representation of n. I wish I ...