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

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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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54 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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69 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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34 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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402 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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48 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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732 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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29 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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44 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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52 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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115 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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38 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
162 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
31 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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2answers
393 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
67 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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83 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 ...
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1answer
88 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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34 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
40 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
239 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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1answer
16 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
56 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
72 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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157 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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34 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
259 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
42 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
93 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
26 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
22 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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51 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 ...
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1answer
38 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
109 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
65 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
38 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 binary representations of huge numbers about $3.3218$ times as long as their decimal representations?

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: ...
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75 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 ...
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1answer
189 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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211 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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9 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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5k 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
51 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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197 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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0answers
62 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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43 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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29 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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144 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 ...
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1answer
171 views

Computing $ 001001_2 - 110101_2$ (base $2$), and representing the result in signed magnitude format

I've been asked (for homework) to do $ 001001_2 - 110101_2$ (base 2) and to represent the answer in a signed magnitude format. EDIT: I'm specifically asked: Perform subtraction on the given ...
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1answer
69 views

Converting 16bit float to Base10 and vice versa

Hi! I have some difficulties understanding how I'm supposed to calculate this 16bit float to base10. This is something that is coming up on a test so I would be pleased to learn how this is supposed ...