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

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1answer
28 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
13 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 ...
3
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2answers
45 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
33 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
17 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 ...
0
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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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1answer
28 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. ...
0
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1answer
14 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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2answers
34 views

How many ones are there in the binary representation of a digit [closed]

in each case added to the previous value of count How many of 1, we need for n 0 = 0 -> 0 1 = 1 -> 0+1=1 2 = 10 -> 1+1=2 3 = 11 -> 2+2=4 4 = 100 -> 1+4=5 5 = 101 -> 2+5=7 6 = 110 ...
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1answer
17 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
23 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
123 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 ...
0
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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
123 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 ...
0
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1answer
21 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
66 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
votes
1answer
31 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 ...
0
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1answer
25 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.
2
votes
1answer
22 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 ...
0
votes
1answer
21 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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0answers
12 views

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
13 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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0answers
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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2answers
43 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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0answers
82 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. ...
1
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1answer
31 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
20 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 ...
3
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2answers
26 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} = ...
0
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1answer
18 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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0answers
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 + ...
0
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1answer
33 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 ...
0
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1answer
30 views

From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
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2answers
46 views

Determine whether a number has an even number of 1's or not in a binary base

Assume that I have an ordinary number in Decmical base, now what I want to know is determining whether it has an even number of 1's in a binary base or not.and yet again I should emphasize the fact ...
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2answers
26 views

Binary solving problem. 50 in binary number

The asnwers doesnt make sence for me. The binary number of 50 is 110010. Do I overlook something?
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1answer
15 views

Linearly independent rows in a binay matrix

I need the algorithm to finding only the linearly independent rows in a binary matrix using XOR function. Example 1: The result: Example 2: The result: R4 is not included because:
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3answers
3k views

Why are huge binary numbers about 3.3218 times longer than their decimal counterpart?

Why are huge binary nubers about $3.3218$ times longer than their decimal counterpart? I thought about this when I was writing this Python code: ...
0
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1answer
57 views

Is there a binary [10,6,4] code?

Using the sphere padding packing bound formula I can conclude that 1 + 12 + 66 $\ge$ $2^{6}$ which indicates that there MAY be a binary [10,6,4] code, however I cannot prove that there is. How can I ...
0
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1answer
28 views

Finding a standard generator matrix given a binary code

My question is how do I find the standard generator matrix of a binary [7,6,2] code? From what I understand a generator matrix for $C$ is any $ k \times n$ matrix $ G$ with entries in $ ...
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1answer
17 views

Binary Gaussian Elimination of a matrix

Can anyone help me find the algorithm for the Binary Gaussian elimination of a matrix. for example: The output:
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1answer
5 views

How can I determine the base of the following numbers for the operations to be correct?

Given: 24)A + 17)A = 40)A How can I find the base of the following number (A) so the operations are correct? NOTE: I am not sure what topic this would fall under. Hence sorry for any misplaced ...
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3answers
228 views

Convert the following hexadecimal representation of 2’s complement binary numbers to decimal numbers.

I need to convert the following hexadecimal representation of 2’s complement binary numbers to decimal numbers I am unsure if I am doing it correctly or am I missing a step? a. xF0 b. x7FF c. ...
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2answers
23 views

Prove that the binary representation of a number n will use floor(lg(n)) + 1 bits.

I'm taking Computer Algorithms class and one of my problems is from Skiena's Algorithm Design Manual, 2-41: Prove that the binary representation of $n \ge 1$ has $\lfloor \lg n \rfloor +1$ bits ...
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3answers
74 views

How many bits are needed to represent the integers 3^1000 and 2^1000?

I'm struggling with a math exercise here, and I would gladly appreciate some help. My problem is that I've encoutered some very big numbers such as $3^{1000}$ and $2^{1000}$ and I want to estimate ...
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vote
0answers
30 views

Leibniz Binary Representation of Squares

Leibniz claims to have found patterns in the square numbers and their binary representations. I cannot see any patterns at all. Here are the first ten squares and their binary representations, can ...
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0answers
11 views

Redundant Binary Representation

Is it possible to have a technique using Redundant Binary Representation so that repeated addition can be obtained with no carry propagation time?
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2answers
26 views

One output for input of $n$-tuples using AND, OR, NOT

Let $B$ be set of $\{0,1\}$ and $B_n$ be the set of all strings of length $n$. How many functions can be constructed from $B_n$ to $B$ using logical operators like AND, OR, NOT. Help $\rightarrow$ ...
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0answers
66 views

Proving that number of codes with even weight is the same as number of codes with odd weight for a specific code book

Consider the $[n,n]$ code-book $C_0=\{0,1\}^n$ with $n$ being odd and the codes $c_i \in C_0=[c_1,c_2,...,c_{2^n}]$ being sorted in the ascending order of hamming weight (from $0$ to $n$). Now let's ...