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

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1answer
42 views

Find a function comparing two variables that is equal to 1 when they are equal and 0 when they are not

I am trying to find a function that compares two variables. When these two variables are equal, the function should equal 1. For any other value, the function should equal 0. Is there a way to do this ...
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0answers
10 views

Proof by induction that $\sum \limits _{i=1} ^n d^{-l_i} = 1$ sentence in a full tree [duplicate]

How do I prove by induction that $\sum \limits _{i=1} ^n d^{-l_i} = 1$ where: $d$ = the number of children of each node; $n$ = the number of leaves; $l$ = the depth of each leaf $l_1, \ldots, l_n$? ...
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3answers
79 views

Generating binary array

Programming side (C++) I have this task where I need to get 5 numbers in binary array, but i cant think of way to generate it. The building process is: Starts Width 10 then adds same number but 1 ...
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1answer
17 views

Convex or Quasiconvex Relaxed Binary Quadratic Optimization Problem

Let's say I have a quadratic problem with nonnegative triangular matrix Q and binary decision variables x. $$min_{x} f(x) = ...
2
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2answers
89 views

Binary number with base $-2$ (minus two) arithmetic algorithm

I have a number X represented as a sequence of $a_i \in {0,1}$ so $$X = \sum_{i=0}^{N-1} a_i(-2)^i$$ where $N \in \{ 1, \ldots, 100000 \}$. I need to find an algorithm to produce a number $Y = -X$ ...
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0answers
28 views

largest number machine number that this comp has, numerical analysis, mantissa

I am having trouble understanding the basics on how to calculate the m, s and c of the formula. I have looked in my textbook but not much of an explanation, also on the IEEE websites but nothing much. ...
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2answers
25 views

Formulating the bitwise OR operation

Considering the bitwise OR operation, wikipedia states, $$x\;\mathrm{OR}\;y = \sum_{n=0}^b ...
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0answers
8 views

How would I round the number $-(1.01101|01)_2$ down?

I'm trying to round $-(1.01101|01)_2$ in binary down. The $|01$ are the guard bits. The answer is $-(1.01110)_2$ My attempt is as follows: $-(1.01101|01)_2$ $-(0.00001|00)$ = $(1.01100)$ Not ...
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1answer
77 views

How many 10-bit binary sequences begin and end with the same bit?

Can you please teach me how to solve this? There are other similar questions that I have, but I don't understand how to solve them. Another question is: How many 10-bit binary sequences contain an odd ...
1
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1answer
43 views

Binary variable in a constraint

I have the following optimization problem: Model I: $$f(x,y) \\ s.t., \\ y\leq x+M(1-V)\\ y \leq MV \\ x \geq 0, y \geq 0$$ where x and y are continuous variables whereas V is a binary variable. M ...
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2answers
35 views

About 2-adic representation of integers

How would I express -3 in 2-adic representation? Is it just revercimal calculation of binary expression of -3? like: -3 = -11 in binary, so using revercimal, -11. in binary = 01. ?
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1answer
67 views

Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
3
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1answer
48 views

Proof that max value of $n$-bit binary number is $2^n - 1$

After reading this programming question , I wanted to prove the assertion. I'm wondering whether the below would be considered a complete and clear proof. Claim: $\sum_{i=0}^{n-1} 2^i = 2^n - 1.$ ...
5
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7answers
645 views

How to find the unique sums in the values 1,2,4,8,16, 32

I apologize but I'm not sure what you would even call this problem. I have some data that provide a numeric code for race as follows: hispanic(1) + american_indian_or_alaska_native(2) + asian(4) + ...
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1answer
35 views

What actually determines a 'borrow' in binary subtraction?

Given Byte1 and Byte2 in binary. How to determine whether Byte1 - Byte2 results in a borrow ...
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0answers
18 views

Is it possible to break a system into two BSCs to find the total capacity of a system?

I have been trying to solve this problem in manor of ways but I cannot seem to find a satisfactory solution. I have tried the basic way of calculating capacity through self information but I was ...
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0answers
15 views

How to perform binary transformations?

One of the steps in Binary Index Tree algorithm is to find a node's parent which is done by un-setting the rightmost SET bit. For example: if a node has index 1010 than it's parent is 1000. To apply ...
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1answer
128 views

How many binary numbers of length n do not contain the substring 000?

How many binary numbers of length $n$ do not contain the substring $000$? Denote this number by $Z_n$; find a relationship between $Z_n$, $Z_{n-1}$, and (something else not given) to form an ...
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0answers
34 views

How to find the number of subsets with a given length and XOR?

I have A (0<A<500000) elements (up to 10^6) in the set. I need to find in how many ways can I remove a subset, the size of ...
1
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1answer
46 views

Why not, first order logic to DNF conversion?

There seems to be huge amount of discussion about converting "first order logic to CNF". But don't see much about "first order logic to DNF" conversion. What is the reason?
2
votes
1answer
184 views

Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. ...
2
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0answers
118 views

Bounding the global intersection of a family of sets

Suppose that we have a decision tree of height $r + 1$ that describes how to increment an $n$-bit integer in the range $[0, 2^n -1]$. That is, the internal nodes are labelled with a bit position that ...
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2answers
48 views

Prove a map of binary expansion is continuous

Prove that the map: f: $\{0,1\}^\mathbb{N} \times \{0,1\}^\mathbb{N}$ $\to$ $[0,1] \times [0,1]$ is continuous. I know that the map can be written as ($m_1,m_2,m_3,m_4,...$) $\times$ ...
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0answers
42 views

Is there a way to approximate a polynomial as another, binary-coefficient polynomial?

Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$. Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n ...
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0answers
82 views

Is there any kind of known pattern to $\sqrt 2$ in base 2?

Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into ...
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2answers
28 views

Trouble understanding binary representation of numbers

"If $x\in[0,1]$, we will use a repeated bisection procedure to associate a sequence $(a_n)$ of $0$s and $1$s as follows. If $x\neq\frac{1}{2}$ belongs to the left subinterval $[0,\frac{1}{2}]$ we take ...
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0answers
15 views

How to add two negative binary numbers

Solve the following using two's complement binary numbers: $ (-111)_{10}-(110)_{10}=?$ $(-111)_{10}=(10010001)_2$ with 2s complement $-(110)_{10}=(10010010)_2$ with 2s complement But ...
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2answers
44 views

How to compute two's complement of a negative number?

I made the search on here and on google and couldn't find anything that answered the topic title. From my bit of understanding, two's complement can be used to make a decimal number, negative. Which ...
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0answers
28 views

Binary tree for sorting combinations

Let's imagine to have a binary n-array as input. All the possible combinations for this n-array will be 2^n. Right? Now, I want to put these combinations in a specific order. In particular, I want ...
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0answers
40 views

Finding a Hamiltonian cycle for $Q_4$

A hyper cube $Q_n$ is a graph that have the length-n binary sequences as its vertices. Two vertices are adjacent if they differ in one entry. I found a Hamilton cycle for $Q_3$ as follows $$000 \to ...
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1answer
63 views

Probability random binary number lies in an interval

Suppose we have a sequence $\{d_n\}$ where all the $d_n$ are either 1 or 0, with equal probability. Let $x=\sum_{n=1}^\infty d_n2^{-n}$. I need to show that $\mathbb{P}(x \in [a,b])=b-a$. I started ...
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2answers
49 views

How do you add 8-bit floating point with different signs?

Hi I have some trouble with how should I add two 8-bit floating points with different signs. The question is here, 1 100 1100 + 0 101 1011 = Thee 1st bit is the sign, next 3 bits are the exponent ...
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1answer
20 views

Binary symmetric channel

Binary symmetric matrix A sends $i,j$ and B gets $i,j$. Does it mean that $A$ != $B$? I would know how to solve this if A would be equal to B, but now I'm not sure how should I start, when A has 2 ...
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0answers
18 views

binary divison with remainder

I wanted to calculate this (binary): 10101.101/1.1. the result is: 1110.01101010101010101010101010... I succeed in calculating the integer part, but I didn't understand how I find the numbers that ...
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0answers
108 views

Reciprocal of a binary number

I'm looking for a way to compute the reciprocal of a binary number. The numbers are fractions, but that doesn't really matter. Therefore we can think of any number involved as $N \geq 0$. The idea is ...
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0answers
189 views

185 stored as a signed 8-bit number?

Several exercises in my textbook start with assumptions that confuse me. For example: Assume 185 and 122 are signed 8-bit decimal integers stored in sign-magnitude format. Assume 151 and ...
1
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1answer
70 views

What is the name for the binary sequence 0110100110010110?

This is a special sequence formed by: 0 0 + 1(its opposite) 01 + 10(its opposite) 0110 + 1001(its opposite) 01101001 + 10010110(its opposite) What is it called?
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2answers
31 views

What is the minimum $k$ such that $\sum_{i = 1}^{k}2^{a_i} = \sum_{j = 1}^{n}2^{w_j}$

Suppose you have a sequence of numbers $S = (w_1,...,w_n)$ and you want to know the minimum $k$ such that $2^{a_1}+...+2^{a_k} = 2^{w_1}+...+2^{w_n} = T$. I'm told that the minimum $k$ is equal to the ...
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2answers
45 views

Binary Sequence of Single Bit Transitions

First of all, I'll have to say that I believe this problem has no solution, but I'm unable to prove it. Here is the problem: I need an algorithm to generate a sequence of all possible transitions of ...
0
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1answer
14 views

binary relations defining an equivalence relation on S

Is this a true statement for binary relations defines an equivalence relation on S: S is the set of all n-digit binary sequences. We say that two binary sequences are in a relation if and only if ...
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2answers
24 views

Number of binary strings with sub-string constraint

How many binary strings of length $n$: $a_1a_2\dots a_n$ are there, such that for every sub-string of k consecutive numbers $a_ia_{i+1}\dots a_{i+k-1}$ and $\forall k, 1 \leq k \leq n$, the difference ...
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0answers
18 views

Canonical forms for matrices with binary elements.

Based on this answer to a combinatorics question I grew curious of results regarding similarities or canonical forms of matrices fulfilling these criteria: Elements of matrix are binary valued ...
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1answer
44 views

The class of $0-1$ matrices with row sums at least $2$, where distinct columns have dot product $1$

There is an $m\times n$ matrix of ones and zeros where the dot product of any two different columns is one and any row have at least two ones in it. My question is: Is this a popular matrix? Does it ...
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0answers
24 views

symbol for set of ones in binary vector

Is there a standard symbol for the set of elements that have value 1 in a binary vector f? So far, I was defining Pos(f), but obviously this is suboptimal as it gives the impression that the values ...
0
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1answer
29 views

permutations of binary sequences

What is the proof that there are $2^n$ distinct binary codes of length n I know this progression also applies to the decimal ($10^n$) and hex ($16^n$) systems but how can this be shown?
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0answers
28 views

partial states or partial probability?

I am trying to figure out an alternative way of representing a state probability space, to make certain ideas clearer (that I don't need to discuss here). Let's say I have a system of two elements, A ...
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1answer
49 views

Binary operations 1011 & (~0 << 2)

My thought process to solving this is that 1011 & (~0 << 2) = 1011 & (1 << 2) = 1011 & 0100 = 0000. But my book says the answer is 1000, ...
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2answers
26 views

finding binary relations between two Finite sets?

If there are two finite sets $A$ and $B$, then how to achieve the below How many binary relations between $A$ and $B$? How many functions from $A$ to $B$? $A$ and $B$ should be in terms of ...
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0answers
120 views

floating point subtraction for binary numbers

Consider that I want to do a binary operation on the following floating point numbers: 0.35-0.62 I can reach the end but I can not figure out how the sign bit is ...
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1answer
30 views

binary combinations/rule of sums

I'm currently working on this topic but I'm having a hard time. In an 8-bit string, there are 256 combinations (or is it permutations?). Either way, I know it is 2^8. If at least 3 elements must be ...