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

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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 ...
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15 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. ...
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28 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 ...
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31 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 ...
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25 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 ...
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46 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 ...
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32 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 ...
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17 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: ...
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49 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 ...
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57 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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61 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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107 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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33 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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18 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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28 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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107 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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354 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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95 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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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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317 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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81 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 ...
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383 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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18 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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3answers
46 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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17 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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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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28 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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35 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
10 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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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
36 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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39 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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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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48 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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381 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 ...
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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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31 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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47 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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78 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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390 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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36 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
54 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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159 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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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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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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322 views

$N$ perfect logicians wearing hats

I once came across the following riddle: (assume $N$ to be extremely large) There are $N$ perfect logicians arranged in a vertical row. They are allowed to strategize before the game, during the ...
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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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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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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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79 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 ...