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

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Explanation of carry in carry out borrow in and borrow out for binary addition and subtraction with examples

Hi I am having a hard time understand what carry in, carry out, borrow in and borrow out mean can anyone help me out and show me some examples thanks
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System of equations with binary variables

Is there any way to simplify solving system of equations, assuming that all variables are binary? All equations, however, are seen as equations with real (or, to make things simple, integer) ...
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125 views

Can every periodic binary sequence be expressed as the output of a Linear or Non-Linear Feedback Shift Register?

Quick background: The output of a Linear Feedback Shift Register (LFSR) with $n$ taps is a binary sequence which is periodic of length dividing $2^n-1$. From a mathematical point of view, such a ...
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42 views

Multiplication over $F_{2^{31}-1}$ by power of $2$

I'm reading the source code of a stream cypher (zuc): I cannot understand properly why they define the multiplication by power of 2 in this way: ...
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477 views

Simplifying Boolean Expression

I am asked to simply the following expression $$F(a,b,c) = c’ab + c’b’ + aba + b’cb + abc + c’b$$ using the Boolean identities and finding $F'(a, b, c)$ using DeMorgan’s law I have been trying for ...
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Binary notation in Magma

As a part of programming an Elliptic Curve Method with montgomery coordinates in Magma, I need to have an algorithm to convert a number from decimal notation to binary notation. Since there is no ...
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200 views

A question dealing with residual codes.

I've been reading about residual codes, and have come across several statements that I am having trouble showing. (1.) By using the residual code, one can show that a $[16, 5, 8]$ binary code ...
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3answers
318 views

Binary Sequences

Let $B_n$ = $\mathcal{P}(\{1, 2, \dots, n\})$. The set $\{0,1\}^n = \{a_1, a_2, ... , a_n : a_i \in \{0,1\}\}$ is called the length of binary sequences of length $n$. I want to verify and work on ...
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131 views

Binary Sequence Block

More of an informatics question, rather than applied mathematics - Source - Zonal Informatics Olympiad 2011 Question Paper Although, I've tried a few brute methods, I haven't really understood ...
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1answer
61 views

How can I quantify the amount of space required to store all possible 128kilobit mp3s?

Somone has suggested that Within, say, a collection of every possible 30 second long MP3 file encoded at 128kbps, I'd probably be infringing on a few thousand copyrighted works. 128kilobits per ...
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583 views

How can I solve simple equations involving binary operators?

I have some simple equations like: A = (X AND 1779038349) XOR ((X AND 3144134329) XOR 7047511487) Where A is some constant and X is unknown (all numbers are 32 ...
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197 views

What are the last four digits in the binary expansion $1234^{5555} + 4321^{5555}$?

I'm having a lot of trouble figuring out this discrete math question: What are the last four bits in the binary expansion of 1234^5555 + 4321 ^5555? I need to ...
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35 views

Element in a 4 by 4 matrix mult-group over $\mathbb{F}_2$ such that it has a stabilizer subgroup of order 64

I want to show that there exists an column or row vector with four entries in $\mathbb{F}_2$ such that there are 64 4 by 4 binary matrices $M$ where $Mv =v$, ie $M$ leaves $v$ fixed. ie, the ...
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189 views

An Upper Bound for an $[n,k,d]$ Linear Binary Code.

I've been reading about the various upper bounds for different types of codes. Recently, I came across a statement that is similar to the Singleton Upper Bound that I am having trouble proving. The ...
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10answers
582 views

Function that sends $1,2,3,4$ to $0,1,1,0$ respectively

I already got tired trying to think of a function $f:\{1,2,3,4\}\rightarrow \{0,1,1,0\}$ in other words: $$f(1)=0\\f(2)=1\\f(3)=1\\f(4)=0$$ Don't suggest division in integers; it will not pass for ...
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2answers
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How to perform the addition of 2 (base 16) numbers?

for example 0101011(base 16) + 0111011 (base 16) =? another ex: 7FE + 3AB = ?
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2answers
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It it possible to “compress” a list of large numbers using their prime factors?

On a computer I can have integers on arbitrary size thanks to GMP, so it's represented in base 2 in memory. I'm wondering if it's possible in theory to use less memory if I store only prime factors ...
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1answer
353 views

Calculating CRC code

I think I may be under a misconception. When calculating the CRC code, how many bits do you append to the original message? Is it the degree of the generator polynomial (e.g. x^3+1 you append three ...
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73 views

interpret reduced cost of all binary variables

I have prepared and run a Linear Programming model in SAS. I have some questions about the output that I can’t find answers to, and am hoping that someone can help. My model contains the decision ...
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114 views

Maximum number of truths in an optimized truth table.

I have a math-related question: I have a set of predicates that need to be evaluated. These predicates can have two kinds of operators; AND/OR. When such an expression is constructed my code builds ...
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2answers
359 views

Binary Logistic Regression Model Processing

Thanks for showing interest and wanting to help out. My aim is to develop a model that - as accurately as possible - predicts how entities in a population will either cooperate or defect, as a % of ...
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1answer
83 views

Partitions of binary numbers into binary numbers with fixed digits?

If we are to have (two, for example) binary numbers, such that their sum is $100111010_2$, and given that the first number has 5 ones, and the second number has 3 ones, can I find the numbers that ...
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1answer
74 views

How to convert these numbers?

I am trying to understand number systems: (binary and decimal) How can I convert the following numbers using the least amount of digits: $(47)\text{base}-10$ to signed binary. $(-27)\text{base}-10$ ...
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2answers
183 views

A similarity measure for binary sequences from a partition

I'm onto a problem about binary sequence similarity for which I have not found any existing solution. I want to share it and the approaches I have taken, although none of them convince me. Consider a ...
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517 views

Analogy of Binary subtraction to decimal subtraction [duplicate]

Possible Duplicate: How to do +, -, *, / with number in a base b? Subtraction of numbers with arbitrary bases This is a very basic question. But I come from non computer science ...
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3answers
132 views

Mathematical alternative to bit manipulation to set all bits to $0$ except the two most significant (highest order) set bits.

The following describes a function which I want to solve mathematically rather than resorting to binary bit manipulation is possible: $y = f(x)$ where $x$ is an arbitrary integer equal to or ...
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115 views

What function $f$ such that $a_1 \oplus\, \cdots\,\oplus a_n = 0$ implies $f(a_1) \oplus\, \cdots\,\oplus f(a_n) \neq 0$

For a certain algorithm, I need a function $f$ on integers such that $a_1 \oplus a_2 \oplus \, \cdots\,\oplus a_n = 0 \implies f(a_1) \oplus f(a_2) \oplus \, \cdots\,\oplus f(a_n) \neq 0$ (where the ...
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254 views

Explanation of why the height of a binary tree $\theta({lg}(n))$.

From Heap Sort chapter of Introduction to algorithms : Since a heap of n elements is based on a complete binary tree , its height is $\theta({lg}(n))$. I know this is correct but how can this ...
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72 views

Proving that $A_2(13,7) = 8$

It is not too difficult to find a binary code consisting of $8$ words, each $13$ bits long, keeping the distance between every pair of words at least $7$. I know it is not possible to find $9$ words ...
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1answer
157 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 ...
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4answers
3k views

binary representation of a real number

In Baby Rudin,I find reference to the fact that the binary representation of a real number implies the uncountablity of the set of real numbers.(page 30)But I have two questions: Does every real ...
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2answers
280 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 ...
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2answers
706 views

Count the number of n-bit strings with an even number of zeros.

I am currently self-studying introductory combinatorics by reading Introduction to combinatorial mathematics. I am currently in the first chapter, and I have a question regarding one of the examples. ...
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131 views

1s surpassing 0s in binary strings of odd length

Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in ...
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414 views

The sum of powers of two and two's complement – is there a deeper meaning behind this?

Probably everyone has once come across the following "theorem" with corresponding "proof": $$\sum_{n=0}^\infty 2^n = -1$$ Proof: $\sum_{n=0}^\infty q^n = 1/(1-q)$. Insert $q=2$ to get the result. Of ...
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1answer
62 views

q-ary code/Latin squares

For any value of $q$ the largest number of elements in any q-ary code $C$ of length $4$, distance $3$ is $q^2$. How can we prove that this is attainable iff there are a pair of mutually orthogonal ...
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2answers
337 views

Period of a finite binary sequence

Let $G:N\to\{0,1\}$, and let $L$ be some period of $G$, so that $G(i+kL)=G(i)$. What's the best a good way to find the smallest period of $G$? I mean an algorithm that takes ($G$,$L$) and outputs the ...
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2answers
156 views

Number 1s in a binary grid

Consider a binary grid of size 4*4, each of cell can either have 0 or 1. Among all possible 2^16 arrangement how many arrangement of such grid exist in which each row and column contains even number ...
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1answer
121 views

first 1 in a bitmask using log2

I am trying to get the last 1 in a bitmask. More mathematically speaking, I have a number k, that can be written in its binary form as a sequence of 1 and 0. I want the "weight" or "index" of the last ...
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2k views

How many bits needed to store a number

How many bits needed to store a number $55^{2002}$ ? My answer is $2002\;\log_2(55)$, is it correct?
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257 views

Finding the number of distinct sub-strings in a binary string.

Whilst solving a question, I have come across a problem regarding the maximal number of possible distinct $k$-length binary sub-strings in an $n$-length binary string. My thought process was that if ...
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400 views

Where does binary arithmetic/manipulation enter the mathematics/engineering curriculum?

Binary arithmetic is both an educational basis for elementary logic and an pervasive tool for practical mechanics in managing computer systems (at a very particular level). That is the state of ...
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1answer
2k views

Binary division - find a remainder

I have difficulty in finding the remainder of the next expression: 111010011/1101. What I did: ...
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150 views

Recurrence relation for the digits of the integer square root in binary

I was investigating a question on the Electrical Engineering Stack Exchange site, available here: ...
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437 views

Finding Expressively Adequate truth Functions

I was wondering if someone could help me count the total number of Truth Functions of 3 variables, that can generate all the possible truth functions.. I got 56 but I'm not sure of the answer. EDIT: ...
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Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
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Terminating decimal number which is not terminating in binary

I know that when converting a decimal number from base 10 to base 2, the result might be not terminating, even though the number is terminating in base 10. For instance, 0.2 -> 0.0011 0011 0011 ... ...
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Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
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Calculating CRC by long division: How to decide the top number of long division?

Ok, I know how to use long division by using regular numbers, but when comes to binary numbers I'm getting confused. In following calculation I can see the equation solved but I don't understand ...
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3answers
246 views

Representing a binary number

Suppose you wanted to write the number 100000. If you type it in ASCII, this would take 6 characters (which is 6 bytes). However, if you represent it as unsigned binary, you can write it out using ...