Questions tagged [binary]

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

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18 views

How many maximum binary pairs are possible in a Poset?

Answer : $n(n+1)/2$ Maximum binary pairs is possible iff the poset is a toset. A toset is reflexive so I don't have control over self loops like $(1,1),(2,2).$ They have to be there. My approach to ...
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22 views

How do I convert a number in scientific notation to binary notation? [closed]

I have to convert the number $$-1.5845632 \cdot 10^{29} $$ to binary notation. Anyone can help me please?
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1answer
30 views

Two blocks binary transformation in generalized Josephus Problem

I've been struggling through the reasoning behind the generalization of Josephus problem and got confused on the very last step. Then I searched online but seems that this problem is not that well-...
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1answer
20 views

()Most significant bit; bit of the greatest value

A binary number is number expressed in the binary numeral system. Let $n$ be the binary number. Each digit in a number as a bit. And the definition of the most significant bit is the following: The ...
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1answer
46 views

Is this method of converting a integer to Gray code correct?

My method for converting an integer to Gray code (binary) uses successive divisions by powers of $2$ and looks at the parity of the rounded quotient. Example $29/2 = 14.5 \approx 15 \Rightarrow 1$ $29/...
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1answer
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Unfolding of recurrence in generalization of Josephus problem

I have been going through the bit-wise generalization of Josephus problem in Concrete Mathematics. And so the author came up with these relations (I do understand this part) $$f(1) = α ;$$ $$f(2n + j) ...
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1answer
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Determine the depth and number of node in perfect binary tree if index-number given

I need help in solving this problem, let's say we have given perfect binary tree, in perfect binary tree all nodes have the same distance starting from the root, and there are $n$ nodes such that $n+1$...
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1answer
50 views

Combining probability and binary system in a combinatorics problem

This question is not so difficult but I have been quite fascinated with the way it was solved. It is stated as follow: Given n students participating in a contest of m questions. At each stage, a ...
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1answer
42 views

What's this compression technique called?

Consider this string of 1's, 0's (spaces added for readability): 1010 1010 1010 1010 We can ...
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23 views

How to calculate the carry-on in multiplication circuit

I am trying to convert integer factorization to sat-3 by using a multiplication circuit. My first step is to convert the multiplication circuit into CNF. I best understand via examples so I started ...
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Deriving sets of basis vectors which can span a higher dimensional binary space

I'm wondering if there is a general approach to deriving all the possible sets of basis vectors that could fill the entire space of a binary $n\times m$ matrix, where $n \lt m$. That is, is there a ...
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1answer
25 views

How to obtain the following Chernoff Bound?

I'm reading a paper which states: "By the Chernoff Bound: $$ \Pr\big[\big||x|-\frac{n}{2}\big|>3\sqrt{n}\big]< 0.03 $$ Where: $x$ is a binary vector of length n (each index is either 0 or 1 ...
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there exist infinitely many pairs $i<j$ such that $S_2(a_j-a_i)=k$.

For a real number $\lambda > 100$, let $f(\lambda)$ denote the smallest positive integer $k$ satisfying the following property. For any integer sequence $0<a_1<a_2<...$, if $a_n\leq \...
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Is it possible to do XOR operation using 32-bit system on a 16 digit number?

I am trying to perform the bitwise XOR operation on a 32-bit system where both numbers are 16-digit decimals (base 10). However, the largest unsigned integer number enterable in the 32-bit system is ...
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25 views

Levi-Civita with binary multi-index

Consider the Levi-Civita symbol $\varepsilon_{ijkl}$ where $i,j,k,l \in \{0,1,2,3\}$. Consider now the binary representation of $i,j,k$ and $l$, such that $i \to (i_0 \, i_1)$, $j \to (j_0 \, j_1)$, $...
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6answers
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Is binary or hexadecimal better than decimal system

Human have been evolved to use decimal. But I have two views that say that either decimal or hexadecimal would be better Hexadecimal would be better as it would take less digits in hexadecimal ...
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What would be the notation for this bit string with these characteristics look like?

Every block of 0s of length greater or equal to 3 cannot be followed by a block of 1s greater than or equal to three....As for blocks of 0s of length 2 or shorter, it must be followed by 1 mod 4 ...
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1answer
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How to calculate expected code length?

How to calculate expected code length?
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Reference Request : The number of binary string with some special condition (open problem)

(Sorry for my poor english...) I wonder the reference of Simon Marais Mathematics competition 2019 problem B4. This problem is as follow. (They said this problem is open problem.) B4. A set $\...
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1answer
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Having problem with 10's complement subtraction

From what I've found, to find A - B using 10's complement; where A and B are decimals Let A = 215 , B = 155 Find 10's complement of B = (1000 - 155) = 845 Add 10’s complement of B to A If it ...
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2answers
35 views

Find a recursive relation for the case of a length $n$ binary string where there are no three consecutive $0$'s.

Find a recursive formula for the following example. The number of binary strings of length $n$, such that there are no $3$ consecutive $0's$. I started by considering the length $n$ as a block ($1$ ...
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1answer
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how to convert decimal 49.25 to hexadecimal?

please explain step by step procedure on how to convert 49.25 to hexadecimal. I don't understand how to convert the decimal part.
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3answers
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Infinite binary sequences countable set

I know that the set of all binary sequences is uncountable, and I'm asked to prove that the set of all binary sequences that are constant from a certain point ($n\in\mathbb{N}$) is countable, meaning ...
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1answer
37 views

Boolean expression for a problem

I want to express problems like this in boolean expression with say $XOR$, or operations etc. $HD$ = Hamming distance Say for $HD(2^4, 0000)\geq2\;$ the boolean expression is $$x1 (x2+x3+x4) + x2 (...
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4answers
217 views

How do you convert a number from base $10$ to binary?

I don't understand how you get the binary representation of a number. Say we have a number in base $10$, how do you change it into binary? I used the Google math converter as well. . All I know is ...
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arranging binary words in circle - induction

i need to prove that for set A which contains all of the binary words of length n can be arranged in a circle so each two adjacent words will be different only by one char. I tried solving it by ...
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Determining if $m$ can be written as a combination of distinct powers of $w$.

Given a weight $m$ and weights $w^0, w^1, w^2, \ldots, w^{100}$, determine if $m$ can be measured on a balance using these weights. In other words, is it possible to place a weight $m$ and some ...
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Binary representation unusual relation in theorem about immersed manifolds.

Question I'm reading Schuller's Lectures on the Geometric Anatomy of Theoretical Physics and he states the following theorem. I was surprised by this theorem and would like references for further ...
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1answer
28 views

Use structural induction to prove that $v(G) = e(G) + 1$

$G$ is an element of FBRT (full binary rooted trees), $v(G)$ = total vertices in $G$, and $e(G)$ = total edges in $G$. I know logically that this is true, but I'm not sure how to prove it using ...
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1answer
4k views

Signed binary subtraction using two's complement method

I'm sorry about this, but I'm at my wit's end. I'm trying do to an incredibly simple calculation. I want to calculate 4 - 7 using the two's complement method for signed binary numbers with a 4 bit ...
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1answer
2k views

How to convert decimal with scientefic notation, to binary?

I have number $-1e35$, and I'm supposed to convert it to binary. The answer is : $-1.10101001010110100101101...\text{e}–117.$ I can't figure out how to get this! and how we can calculate numbers ...
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1answer
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Converting base 2 to Base 10? [closed]

How is $0.11 \cdot 2^{1}$ converted to $1.5$ in base 10? I am unable to understand the theory behind this, since I believe $0.11 * 2^{1}$ corresponds to $1.1$ but I dont see the conversion process?
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1answer
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One's Complement to Decimal, Table Conversion Question

I have been supplied with the following table to indicate one's complement: This table suggests that -5 in one's complement signed binary equals 10000101. However, guidance found elsewhere suggests ...
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0answers
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How change in discrete bits is equivalent to sinusoidal function?

I am reading this blog about positional encoding. I came across an interesting comparison where the continuous representation of bit change is a sinusoidal function. See the following bit change: <...
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2answers
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8 bit binary number $01101110$ is on a computer using two’s complement representation. What should it be in decimal?

8 bit binary number $01101110$ is on a computer using two’s complement representation. What should it be in decimal? My solution is: reverse it to $10010001$ $10010001+1 = 10010010$ $10010010$ to ...
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1answer
12 views

Compute the minimum hamming distance

to compute the min hammning distance of the 4 codewords: 1110001110010111 1001011010001110 0010111101101111 1100000000011111 Do I have to compute the hamming distance between every two codewords? ...
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12 views

Number of combinations of a list with a given parameter of max distance between two values

Given a list $${1,1,0,0,0,0,0,0}$$ On can solve can find all combinations given the equation $$n!/(r!(n-r)!)$$ which for the above example is $$(8!/(2!(8-2)!) = 28$$ What would be the equation if ...
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1answer
12 views

The FlipBits tree

Take a number, say $n=99_{10}$. Express it in binary, $11000111$. Now flip the bits, $0011100$ and discard leading $0$'s: $11100 = 28_{10}$. Continue until $0$ is reached: $(99,28,3,0)$. Here's a ...
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1answer
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Greedy approach to find maximum product $P = (A-k_{1})*(B-k_{2}).$

Suppose you are given two integers $A$ and $B$; $A \leq B$. I have to calculate the maximum possible value of the expression $P = (A-k_{1})*(B-k_{2}).$ Here $k_{1}$ and $k_{2}$ both are variables but ...
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1answer
30 views

How use probability distribution to generate a binary array with k number 1s acording a range of lower and upper bounds

I need to generate a binary array of size N with K numbers 1 and (n-k) numbers 0 on it,does not matter in which position. The amount of K numbers 1s belongs to an interval Min <= K <= Max , ...
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8 views

Is there any stablished concept of a numeric base that increases linearly the alphabet according to the digit position?

I have seen something about multi-radix systems like the factorial one, but I have a more specific question. Suppose base-10: a 3-digit number can have all possible digits in the alphabet $[0,1,...9]$....
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the height(h) of a Btree, with n keys, where every node that is not a leaf has exactly d sons, keeps this trait $h\le log_d((d-1)n+1)-1$

how can I prove that the height(h) of a Btree, with n keys, where every node that is not a leaf has exactly d sons, keeps this trait? $$h\le log_d((d-1)n+1)-1$$ I tried this with nodes, but i am ...
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1answer
15 views

Ternary random generator from coinflip

Provided a coinflip generator (a function that can be called to provide an output of either 1 or 0 with equal probability of $1/...
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Minimum number of nodes perfect binary tree

definition of perfect binary is "which all internal nodes have two children and all leaves are at same level" perfect binary tree i know max. node number: 2^h - 1 (h>=1) but i cant figure out, ...
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About binary subtraction: how does this work?

So there's a course I'm watching online and it operates binary subtraction this way: asking y-x, where: y=0111 (or decimal 7) x=0010 (or decimal 2) And instead of using a 2's complement to change ...
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1answer
3k views

negative binary subtraction using 2's complement (and 5 bit representation)

I'm wanting to carry out the calculation of 8 - 11 (assuming that 5 bits represents a number and also using 2s complement representation), however, I can't seem to get the correct answer. This is what ...
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2answers
46 views

Binary multiplication 1111 x 1111?

I am confuse that how carry works while multiplying two binary numbers 1111 x 1111 = 011100001 but how to handle carries in calculation? Can anyone please explain? I know how to handle partial ...
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1answer
186 views

3.1 x 10^-9 in 4-bit sign-magnititude representation problem possible?

I'm puzzled with this math question below and would like the help of others, the only answer I can imagine to this question is that the above number will need more than 4 bits to be expressed in ...
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Does this identity over {0,1} have any relation to logarithms?

I was working with Bernoulli Mixture models for a Machine Learning class, and I stumbled across a (to me, surprising) identity. Question The identity in question is $$\forall x \in \{0,1\}, a,b \in ...
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1answer
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Check if in each case borrow and overflow are generated?

Note: Just tell me if my reasoning or logic is wrong. So I have two numbers 01001 and 1110. And the question asks to subtract them as both signed and unsigned system. Unsigned System For this I ...

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