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

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How is OEIS sequence A120933 'maximal leading nondecreasing subword ' to be understood?

For n=2 we only have these four binary words: 00 01 10 11 What is the procedure for calculating by hand T(2,1) and T(2,2)? I'm trying to understand the reasoning behind this sequence as I can't see ...
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13 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
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+50

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
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2answers
17 views

Find the solution of binary xor operator equation

I am working in binary xor operator $\mathbb Z_2$. I have to resolve my problem such as $$\begin {cases} x_1+x_2+x_3=1\\ x_1+x_2=0\\ x_1+x_3=1\\ \end {cases}$$ Could you suggest to me any method to ...
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1answer
12 views

Mathematical Terms for Binary Operations

I'm trying to represent binary operations on numbers in mathematically correct terminology. For example given two binary numbers: 42 : 101010 13 : 001101 I want ...
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36 views

Will the Russian Peasant work with anything other than base 2?

The Russian peasant method involves doubling and halving by 2. Therefore you will get exact remainders of either 1 or 0, which perfectly represents one of the multiplicands in binary form. I just ...
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1answer
14 views

Explanation of division/reduction in a binary Galois Field using bit-shifts

I've seen a lot of algorithms reducing the result of a multiplication in a Binary Field by using only bit-shifts and XOR. The number of positions to shift seems to be derived from the polynomial, but ...
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0answers
28 views

Counting binary strings that have atmost k consecutive 0's

I know how to count how many binary strings with length n and having exactly k 0's are there but i am not able to find a way to count the number of binary strings that have exactly x 0's and y 1's and ...
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68 views

Random walk on infinite binary tree (recurrence, transience)

Consider a random walk on the infinite binary tree with root $x$ which has the following transition probabilities. $$ p_{x,0}=p_{x,1}=\frac{1}{2},~~~p_{y,y0}=p,~~~p_{y,y1}=q,~~\text{and ...
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2answers
46 views

Solving the equation $a+e+2ae=a$ for $w$

Just need a quick answer of how my tutor got $e$ to $= 0$ from this equation. (I'm trying to find the identity of a binary operation) $$a+e+2ae=a$$ I feel like this is a very easy problem but I'm ...
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1answer
27 views

Every binary number coincides in more than half of bits

Let $n\in\mathbb{Z}^+$. We would like to pick some binary numbers of length $2n+2$ so that any binary number of length $2n+2$ coincides with one of the picked numbers in at least $n+2$ positions (that ...
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16 views

Convert a 6 bit binary to negative

"Explain how negative numbers are represented." My answer is that if we ie. have the number 3 and want to convert that to -3 i do as follows: 3 = 0011 in 4 bits. i switch the 1s and 0s => 1100 and ...
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1answer
28 views

Flipping coin(s) to decide who pays the bill

I'm not from a mathematical background. I found this video on YouTube rather confusing. I know basics of probability from school. My question: If a single coin is flipped among 3 people say A, B and C ...
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0answers
11 views

Forumlate Binary Optimization problem for maximizing total net profit for terminals installed and revenue between these terminals

I need help with formulating the problem, as I am not sure if I am doing it correctly and whether there is only 1 answer or multiple correct answers. Thanks
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1answer
36 views

Most efficient mental way to convert Decimal to Hexadecimal

My question is as follows: What is the most efficient mental way to convert Decimal to Hexadecimal? I've heard of many methods. Some people divide the decimal by 16 and find the remainder. Others ...
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1answer
17 views

Why do long division remainders give conversion from base 10?

I learned that you can convert base 10 numbers to other bases, like binary, with long division. I can do this, but I don't understand why this works. I can only understand that the first remainder of ...
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1answer
41 views

binary addition

Can any direct me to any resources online that teach how to approach binary addition such as this/ working with more complex binary arithmetic? I know the basics of binary addition and carrying the ...
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1answer
39 views

Greatest Common Divisor of two binary polynomials

How can I find the GCD of $x^4 + x^3 + x^2 + 1$ and $x^6 + x^5 + x^4 + x^3 + x^2 + 1$? I know that $x^4 + x^3 + x^2 + 1$ is an irreducible polynomial of degree $4$, and that it is not primitive, but ...
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1answer
17 views

Factoring binary polynomials

I need to factor two binary polynomials and present each as a product of powers of irreducible polynomials. a) x⁴ + 1 I have figured it out this far: x⁴ = (x²)² and 1 = 1² So I have something in ...
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2answers
67 views

How many binary sequences of length n are there that contain exactly m occurrences of the pattern 01?

I thought there were n-1 places between the first and last digit. In these places I hypothesized there are switches that change (from 0->1 or 1->0) For ...
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1answer
36 views

Is it really possible to make all possible numbers with an infinite binary table?

Suppose I have an imaginary computer, with an infinite binary table, like the one below: $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \cdots & 128s & 64s & 32s & 16s & 8s & 4s & ...
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1answer
41 views

Polynomial Arithmetic Modulo 2 (CRC Error Correcting Codes)

I'm trying to understand how to calculate CRC (Cyclic Redundancy Codes) of a message using polynomial division modulo 2. The textbook Computer Networks: A Systems Approach gives the following rules ...
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1answer
14 views

Determine whether or not a binary number is divisible by $3$

Let $K$ be a natural number with $n$ binary digits. Is there an $O(n)$ method for deciding whether or not $K$ is divisible by $3$? $3|K \iff d_1-d_2+d_3-d_4\dots\pm d_n=0$ works correctly up to ...
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29 views

Binary overflow

Which of the following hexadecimal numbers, representing signed 16-bit binary numbers, results in overflow when multiplied by 4? Here, a negative number is represented in 2's complement. ...
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1answer
23 views

Binary representation of the real numbers

I am solving the following exercise: for $n \in \mathbb{N}$ and $a_1,a_2, \ldots ,a_n \in \{0,1\}$ we define: $$ I(a_n, \ldots , a_n) := \left \lbrack \sum_{i=1}^n \frac{a_i}{2^i}, ...
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1answer
25 views

Convert from two's complement into unsigned number

There is an 8-bit numerical value, where a negative number is represented in two’s complement. When this value is represented in decimal, it becomes -100. When this value is regarded as an ...
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1answer
27 views

Multiple representations of ternary expansions of numbers

$x \in [0,1]$. If in binary expansions ie series $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{2^i}$ where each $x_i \in \{0,1\}$ we identify the sequences $\underline{x}$ and $\underline{x}'$ ...
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2answers
134 views

How many Binary numbers?

How many binary numbers of length $n$ can be generated where $n > 7$ and the number either start with $000$ or end with $111$? My questions is, can I choose an $n$ randomly? For example, let's say ...
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1answer
24 views

Finite binary representation of float number

How can I prove that float number $x$ has the finite binary representation if and only if it is written like that: $ x = m / 2^n $, where $m, n \in \mathbb N$. Should I consider something like 3 ...
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3answers
184 views

Number of binary palindromes in a range

I want to find the number of binary palindromes from $1$ to $N$. $0 \lt N \lt 2^{32}-1$. I observed a pattern that if we have an odd-length binary palindrome, it can generate only $1$ even-length ...
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1answer
33 views

Binary expansions of dyadic rationals in $[0,1]$

Completely stuck on this exercise! Hints and a starting point would be greatly appreciated. Nor do I see why is $x \in [ 0,1] \setminus D$ do not have $2$ binary expansions.
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2answers
70 views

Is it possible to not have irrational numbers?

(Math noob question): Is there a base that can be used like binary that produces no irrational numbers or numbers with an infinite amount of one number after the decimal (don't know the name)? I feel ...
2
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1answer
34 views

Getting the nth bit of a decimal number

I have a formula for decoding a 3-bit data object: $$T = 68 + 2 \sum_{i=0}^22^iTempA_i$$ where $TempA$ is the 3-bit object and $TempA_i$ is the $i$'th bit from the right. I am trying to rewrite this ...
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1answer
30 views

Where can I find a binary calculator that can do exponentiations, roots and logarithms?

I've searched on Google, but all I found was binary calculators that can do additions, subtractions, multiplications and divisions, nothing else.
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1answer
24 views

Express a binary operation in decimal

Is there a way to represent binary operation in decimal. What I mean with this is for example a set of decimal operators that would give the same result as a x>>n a ror(x), etc. So far the only thing ...
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1answer
69 views

Prove through structural induction that a binary tree has an odd number of nodes

A full binary tree is a binary tree where every node has either 0 or 2 children. Prove that every non-empty full binary tree has an odd number of nodes. I dont know how to get started with this ...
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Estimating how many searches will be needed to justify time spent on presorting an array.

Problem Estimate how many searches will be needed to justify time spent on presorting an array of $10^3$ elements if sorting is done by merge sort and searching is done by binary search. (You may ...
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1answer
14 views

If gen matrix has even weigth rows, do codewords have even weigth for non binary code?

Is that true that in a non binary code C every codeword has even weight if and only if every row of G has even weight?
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2answers
26 views

Trying to understand binary number equation

I'm reading a book called "The Elements of Computing Systems" by Noam Nisan/Shimon Schoken. There's an excerpt which includes some math that I'm struggling to understand (limited math background; I ...
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39 views

Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
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50 views

Equivalence relations for $\mathbb{N} \times \mathbb{N}$ question

On the set $\mathbb{N} \times \mathbb{N}$ define $(m, n) \sim (k, l)$ if $m + l = n + k$. Show that $\sim$ is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. Draw a sketch of $\mathbb{N} ...
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83 views

Why is $2^{16} = 65536$ the only power of $2$ less than $2^{31000}$ that doesn't contain the digits $1$, $2$, $4$ or $8$ in its decimal representation

$65536$ is the only power of $2$ less than $2^{31000}$ that does not contain the digits $1$, $2$, $4$ or $8$ in its decimal representation. http://en.wikipedia.org/wiki/65536_%28number%29
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1answer
22 views

Does decimal fraction has hex value?/can hex be fraction?

I was wondering if a decimal fraction could be converted into a hexadecimal fraction? I have seen it many times ? but I have been also told that decimal or binary fraction has no meaning in hex. ...
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1answer
39 views

How many binary string are there such that there are no k consecutive characters are the same?

Given number $n$ and $k$. Count the number of string with length $n$ such that there are no $k$ consecutive characters are the same. Example with $n = 3, k = 3$, the answer is $6$. ($110, 001, 101, ...
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1answer
23 views

Why do we add 6 in BDC addition.

When performing addition to BCD, if we get an invalid BCD, we remedy this by adding a binary $6$ to our sum. Example: $0101 + 0110 = 1011$ (Invalid in BCD) So, we add $6$ to fix this. $1011 + 0110 ...
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2answers
30 views

Why are these two conversion methods (base 10 to base 2) equivalent

I've come across two methods for converting a base 10 number into its base 2 equivalent. I want to know why they are equivalent. Method 1: We're given a number $N$ to convert into binary 1) Find the ...
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1answer
25 views

Can we transform given strings to get the same string?

There are 2 binaries string $A, B$ (string just contains $0$ or $1$) Input: $A_1, A_2,\dots,A_{50}$ and $B_1, B_2,\dots,B_{50}$ Note that: $A_{51} = B_{51} = A_{52} = B_{52} = \dots = A_{\inf} = ...
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1answer
19 views

Boolean Algebra - What is the meaning of f(x) = ∑(N)?

Suppose you had: f(x,y,z) = ∑(2,3,4,5) What does this represent or map? For each of the variables x,y,z?
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21 views

Converting From Different Number Systems

I've been taught to convert from base 2 to base 10 using the following process: 10110 = $0\times1 + 1\times2 + 1\times4 + 0\times8 + 1\times16 = 0\times2^0 + 1\times2^1 + 1\times2^2 + 0\times2^3 + ...
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1answer
38 views

Number of 2n-digit binary sequences

Find the number of 2n-digit binary sequences in which the number of 0's in the first n digits is equal to the number of 1's in the last n digits. I'm not sure how to approach the question. My ...