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

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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 ...
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0answers
39 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 ...
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2answers
8k 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 ...
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1answer
52 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 ...
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1answer
105 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 ...
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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 = ~ ??? $
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1answer
746 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) ...
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2answers
29 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 ...
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0answers
13 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, ...
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0answers
80 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 ...
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0answers
22 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 ?
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3answers
75 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.
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2answers
346 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 ...
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2answers
198 views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...
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0answers
19 views

Determining whether two trees are isomorphic

Is there a (probably recursive) algorithm that can be used to determine whether two not necessarily binary ordered (sub)trees are isomorphic or not?
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1answer
361 views

Probability of a binary communication system

A binary communication system is used to send one of two messages: message A is a sequence of 0s with probability 2/3, message B is a sequence of 1s with probability 2/3, The ith received bit is ...
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3answers
38 views

modulus to find 10-bit samples in 8-bit memory

For one of my electronics projects I am using an EEPROM chip that holds up to 512k bits of information, arranged in 8-bit (1 Byte) address registers. Multiply 512 x 1024 and divide by 8 and you'll get ...
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0answers
14 views

Distribution of binary sequences?

I have two binary sequence and wanted to test if there is a similar patterns between the two. This is a completely new topic for me. Are there ways to define distribution of a binary sequence ...
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0answers
12 views

why use Unpacked BCD

I wonder why someone should ever use unpacked BCD, as packed BCD is a lot more compact, and you can also form each number. Is there a good reason to use it or not?
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0answers
24 views

What are the possible relations?

If there are two sets A and B. Set A has two elements so does Set B. How many different possible relations can we have from A to B. For example. Set A has (David, Max) and Set B has (x,y). Would the ...
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1answer
26 views

Why is the enumeration of binary strings misaligned?

If you enumerate the set of binary strings in ascending order, you get: ...
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0answers
34 views

Performing arithmetical operations (with binary numbers) using propositional logic

Clarifying some terms. By arithmetical operations I mean the four basic operations of addition, subtraction, multiplication and division. By binary numbers I mean numbers in the binary system. By ...
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1answer
8 views

haw to find the values of binary numbers in octal?

what is the value of 101111.0111 of base 2 (binary) in octal(base 8) we know that in order to translate from binary to octal we need to group the bits in groups of three and the value is going to be ...
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0answers
38 views

What are the possible dimensions of a binary cyclic code of length 37?

What about the possible dimensions of a 16-ary cyclic code of length 37? [Edit]: The cyclic codes of length $37$ are in a bijective correspondence with factors of $x^{37}-1$. Furthermore, the ...
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3answers
31 views

Binary subtraction A0 - E4

I have two numbers. The negative one is translated using 2's complement. $$ a = 160_{10} = A0_{16} = 1010 0000_{2} \\ b = -28_{10} = E4_{16} = 1110 0100_{2} $$ And I have to perform the following ...
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1answer
34 views

What is a signed number?

When is the meaning of "signed number"? A signed number simply means a number that is negative, correct? Sorry if this seems like a stupid question but I'm just starting to get deep into ...
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1answer
51 views

How many numbers from $1$ to $2^n$ will have $``11"$ as substring in binary representation?

For example say, $n = 2$. So our set is $\{1, 2, 3, 4\}$ in base $10$ and $\{1, 10, 11, 100\}$ in base $2$. So Output $1$, because only one number i.e. $3$ is there such that it has $``11"$ in it. ...
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2answers
10k views

Subtract Binary Numbers with 1`s Complement

I'm trying to figure out how to subtract two binary numbers with a complementary one or two, When I need to address carrier and when not? Do I need to solve the problem in decimal numbers? And ...
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2answers
47 views

What am I doing wrong when multiplying binary numbers together?

This is from Discrete Mathematics and its applications I was able to get sum pretty easy. I am trying to follow this example in the book to get the product of the two binary numbers Here's my ...
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1answer
281 views

Convert hexadecimal numbers to floating-point format using single-precision IEEE 754 format

I need help with converting hexadecimal numbers to floating-point format using single-precision IEEE 754 format. Hex numbers such as 312A. how do I go about converting it? I know how to convert from ...
0
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1answer
38 views

Converting from twos complement to decimal?

I am currently reading a textbook and I can't seem to understand what the examples in the book did. I do believe it is an error with the book, but if not can someone explain? How come there is no ...
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0answers
30 views

How to convert a decimal to a $2$'s complement [duplicate]

On this link How can I convert 2's complement to decimal? is a good description. I need to know how to convert from a negative say $-26$ to a two's complement binary and from a decimal with a decimal ...
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1answer
37 views

Simplifying Simple Boolean XOR Expression (!AB + A!B)

I am trying to simplify the 5 gate XOR from a A!B + !AB expression to a (A + B)!(A + B) implementation. How can I convert ...
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1answer
243 views

How to convert from floating point binary to decimal in half precision(16 bits)?

I'm trying to convert a 16 bit precision binary number to decimal format however I am completely failing to do so. The binary I'm trying to convert is $0101011101010000$ My current method is: ...
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2answers
74 views

Possible distinct binary tree structures at depth d

I'm trying to figure out a recursive formula for the number of possible distinct binary trees at any depth d. I haven't been able to find any sort of sources on this. basically, at depth 0, the only ...
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0answers
32 views

Binary to Decimal Floating Point Number

I seem to be having difficulties trying to figure this out: I have a Binary 0101011101010000 and I'm trying to compute the decimal floating point number, in the IEEE-754 format Can somebody help? ...
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1answer
134 views

Grade School Multiplication Algorithm for Binary Numbers explanation

I under stand the shifting but not why it will always give the right answer? For Example: ...
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1answer
45 views

Converting from Octal to Decimal.

I have the value $(3738)_8$ and I want to convert it to decimal. The answer i believe is $$(3 \times 8^3) + (7 \times 8^2)+ (3 \times 8^1) + (8 \times 8^0) = 2016$$. My question is that on some ...
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0answers
63 views

Generating function from a set of binary strings

So this question is in my textbook and there's no solution, so I'm seeing if I can get a confirmation? Q: Let $S$ be the set of all binary strings of length 4, where for each string $a\in S$, the ...
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0answers
42 views

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation?

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation? $2$ is the only prime with $1$ one, the Fermat ...
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3answers
114 views

Defining a bijection with binary strings

Let $n\in\mathbb N$. Then a binary string of length $n$ has the form $a_1a_2...a_n$ such that each $a_i$ is either 0 or 1. Define $E_n$ to be the set of all binary strings of length $n$ with even ...
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1answer
46 views

Understanding binary combinatorial problem

It has been quite some time since I've done permutations and combinations, and I'm attempting to remember the proper way to go about solving this issue (not a homework assignment, more of a thought ...
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1answer
67 views

Using XOR operation repeatedly

There are $n$ binary digits, from $A_0$ to $A_{n-1}$. Each operation consists of the following 2 steps: Each digit is replaced by the XOR addition of itself with the next digit. ...
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2answers
60 views

Mathematical Relations in Computing - Unary

I have this question that's bugging my mind: "Discuss by giving suitable examples the role of mathematical relations (Unary, binary and ternary) in computing." I'm sure it's a very simple question, ...
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0answers
22 views

Binary Overflow Detection

I am trying to solve several problems, which are binary and encoded using the 2's complement system. One problem has stuck out to me: 0111 + 0001 Both are positive, with 0111 being 1+2+4 or 7, and ...
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1answer
77 views

Find number between $A$ and $B$ with maximum set bits?

Given two integers $A,B$. Find number $N$ which has maximum number of set bits in its binary form and lies between $A$ and $B$ inclusive. Is there any approach for this question. Also if there are ...
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2answers
76 views

Converting Decimal to Hexadecimal

MathExchange, I am trying to learn more about computers, and one thing I have opted to teach myself is decimal to binary, and decimal to hex conversion. From the web, I have found tutorials on ...
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0answers
51 views

An expression for the number of n-bit binary strings with at most k ones (without summations)

Say we need to find an expression for the number of binary strings of length $n$, which have at most $k$ ones. My solution was to split the problem into $k+1$ cases, where the number of ones, ...
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2answers
36 views

Correlation between Binary and N-dimensional simplexes

I found an interesting correlation between binary numbers and $n$-dimensional simplexes and I'm trying to find where I can find more information on the subject. I noticed that binary representations ...
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2answers
33 views

How many numbers with given amount of ones in their binary form?

I was practicing for a programming competition and I got the following problem, which I was unable to solve: It is given a number N. Find the amount of x, y values, where x > N, y < N and the ...