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

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27 views

is the Root of a binary Tree counted as a node

I am working on this Homework questions and there's one thing I can't seem to understand. We are trying to proof using structural induction that some elements in T hold for the following statement :...
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1answer
30 views

Converting from Binary to Decimal help?

Can anyone help me solve this? Converting to decimal from binary that is signed...so also using twos-compliment. ...
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20 views

Questions about this type of problem

Problem Consider the general chip-and-be-conquered recurrence relation: $T(n) = b_1T(n - 1) + b_2T(n - 2) + ... + b_kT(n - k) + f(n)$; for $n >= k$ for some constant $k >= 2$. The ...
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1answer
45 views

How would one go about solving these types of problems?

I'm totally lost. All I know is it has to do with binary trees and may need to be solved using induction. Show that every 2-tree with $n$ internal nodes has $n + 1$ external nodes. Show that the ...
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1answer
63 views

What was the original purpose for the binary system?

Obviously computers weren't around when binary was first created... was there a particular use for binary back then or was it just developed as another number system?
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1answer
55 views

Is this sum of Binary Numbers an Overflow?

I have a question of the likes of 21 + 11 I converted each number to binary getting: 010101 + 001011 I got a result of : $100000$ which is $32$ in decimal Thus it is correct that $21+11 = 32$. ...
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1answer
40 views

Decimal - Binary

Let N = 1 000 … 000 1 (n zeros), and M = 111 … 1110 (n ones). What is the decimal value of M and N assuming: a) Unsigned Binary Representation b) Two's compliment Representation c) Signed/Magnitude ...
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2answers
25 views

How many binary numbers can be represented with X number of number places?

How do we find out that in the binary number system, how many different numbers can be represented with a certain number of number-places? For example, suppose we have 8 number places, i.e. a 1's ...
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1answer
50 views

Binary expansion and bijection

Using the idea of binary expansion ($1011$ is $1\cdot2^0 + 1\cdot2^1+0\cdot2^2+1\cdot2^3 = 11$) to find a bijection that (a) maps $\mathbb{N}$ to the set of all finite subsets of $\mathbb{N}$ (b) ...
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1answer
54 views

Decoding BCH codes with permuted parity-check matrix

A normal t-error correcting $BCH(n,k)$ code over $GF(2^m)$ would be constructed using a generator polynomial g(x), which is the LCM of the minimal polynomials of $a,a^2,..a^{2t}$, with $a$ being a ...
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0answers
30 views

Operations on binary numbers, that have equal numbers of two available digits, that preserve this property?

I'm looking for operations on binary numbers that have the same number of the available digits, $0$ and $1$, such as $100011$, $100101$ and $10$. These could be integers as the examples given are, or ...
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1answer
52 views

Expected number of evaluated nodes

I have an array of zeroes with a length of $1024$. Suppose $n$ random elements are changed to $1$, the array now has $(1024-n)$ zeroes and $n$ ones. I want to find the position of all $n$ elements ...
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2answers
119 views

Switch flipping sequence with no repeats

Okay, you have N switches. They are all off. You may flip one switch at a time. You must visit each possible state of switches being flipped without repeating any state. At the end, you must be ...
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1answer
96 views

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T .

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T . I have no idea what this question is even asking me. What ...
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1answer
176 views

Converting a negative decimal with fractions to binary and hex

I am asking the question even though it has been answered before because I am looking at 3 different pages with 3 different answers right now. I am wondering how to convert a negative decimal with a ...
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1answer
256 views

orthogonal binary sequences

How to show that two binary sequences are orthogonal? For an example verify whether [0110001] and [0011101] are orthogonal
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1answer
134 views

Combinatorics - Counting the number of binary strings with specified length and sum, with substring constraints

Suppose I have a string of bits of length R. The sum of the bits must be equal to S, so there are S ones and R-S zeros. The longest string of ones cannot exceed X in length. Also the number of places ...
6
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1answer
81 views

Tzaloa 2015 game problem (piles with $1,2,4 \dots 2^{19}$ coins each)

We have $20$ piles with $1,2,4,8\dots 2^{19}$ coins repectively and two players. In each turn a player must select five piles that have at least one coin and remove exactly one coin from each. Player $...
2
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1answer
115 views

Whether a real number is a dyadic rational iff its binary expansion terminates?

In self-studying a textbook on computability theory, I found that many of the exercises depend on the following factlet: A dyadic rational is a rational number whose denominator is a power of two, ...
0
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1answer
29 views

inference of causality for binary variables

Let's say that a data set has N random binary variables Xi and we want to infer which of these variables have a causal relationship with X1. The following table would describe the data, where each ...
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2answers
513 views

How would I convert a long decimal to binary without using a calculator?

I am aware that you can keep dividing a decimal number by two, finding the remainders in the process, in order to convert the number to binary. However, when I am working with a long decimal number ...
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1answer
27 views

Could you explain this signed fixed point number equation?

How do I interpret this equation? Decimal value of signed fixed point number: $$V=(-1)^{b_{N-1}}\times 2^{N-P-1}+\sum_{i=0}^{N-2}(b_i\times 2^i)\times 2^{-P}$$ Original picture $N$ is ...
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1answer
226 views

signed 2's complement of 4-bit binary numbers

In signed 2's complement representation of 4-bit positive binary numbers are: 0000 -> +0 0001 -> +1 0010 -> +2 .... 0111 -> +7 Negative binary numbers(from -1 to -7) are obtained by taking ...
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0answers
67 views

minimum number of leaves in a perfect binary tree

I'm trying to prove that the number of leaves in a perfect binary tree is at least H+1 where H is the height of the tree. This is what I've done up til now: No of leaves at height $H = 2^H$ Base ...
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2answers
61 views

A function for decimal to binary conversion

I want to convert a decimal (base 10) number to its binary (base 10) equivalent. The binary string has to be of infinite length. Is any of the following functions correct for non-negative integers $x$:...
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2answers
160 views

When decoding a block code, how do you know which error a syndrome corresponds to?

I'm working with forward error correcting block codes such as Hamming(7,4) and Golay(23,12). I'm quite new to this field, so there are some things that I don't yet understand. I chose these codes ...
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1answer
251 views

Converting decimal fractions to binary

I know that if we multiply the fraction by 2 repetitively and take out the integer part every time, we will get the binary form. But why does this method work? Why should we multiply by 2 for the ...
0
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1answer
52 views

Binary tree node value by level

How can I calculate the value of given node level, for example: (let's use this image I found on Google Images and invert the level: starting at bottom 0..1..2..3..4) Knowing that each node pays ...
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2answers
218 views

Binary expansion, finding the greatest power of $2$ less than a given number

I'm looking to better understand binary for a CS50 problem set. I'm not understanding transferring decimal notation to binary. For example, use 237. How to find the largest power of $2$ less than $...
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3answers
133 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
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5answers
950 views

Why are sums of powers of 2 able to give all numbers?

It is known that If we sum up a combination of numbers that are positive powers of 2(starting from 0 to infinity), we can get any number possible. (Correct me if this is wrong). Can anyone ...
0
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1answer
107 views

Why is 2's complement called this way?

So, I'm preparing for a test and one of the preparation questions is as follows: I can tell easily that 'a' and 'e' are just wrong and therefore these answers are irrelevant, but taking a look at ...
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1answer
56 views

How many bits to represent this integer?

If $$x = \left(\frac{n+1}{4}\right)^{(n+1)/2},$$ then how many bits do we need to represent $x$ in binary?
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1answer
62 views

Binary Integer Programming

I need to form teams. There are 8 projects and 60 students. Each project has different requirements. For example, out of 5 total requirements, project 1 has 2 requirements: must have a programmer and ...
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3answers
292 views

Why does the division algorithm work for converting between number bases?

I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this ...
0
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1answer
47 views

Decompose negative power of ten in finite series

Suppose we have numbers $10^{-1}, 10^{-2}, 10^{-3}, ..., 10^{-n}$. We need to represent each number by its closest representation that is less than original by means of finite sum of two in negative ...
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1answer
27 views

Number of ways to choose rows with inclusion condition

I have a large collection of lists consisting of 1's and 0's, each list the same length. I call each list a row. I want to know the number of ways to select rows such that their cumulative OR results ...
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1answer
84 views

External operation: binary and unary perhaps???

Consider the following examples from which some definitions are derived: Let us take an element from the set R of real numbers (say, the number 8) and another from the set L of lengths (say, 4m). ...
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1answer
109 views

Given a binary number, how do we get the last decimal digit?

Given a binary representation of 25 i.e 11001, if I am interested only in the last decimal digit, how do I get it?
2
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1answer
22 views

Binary addition preserving Hamming weights

Let $x,y$ be two $n$-bit strings, with Hamming weights (number of $1$ bits) equal to $w_{1},w_{2}$, respectively. Let $z$ be the binary representation of the sum $x+y$, where we interpret $x$ and $y$ ...
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3answers
247 views

Exponential generating function for the number of binary strings of length $n$

I know that the generating function of the sequence counting the number of binary strings of length $n$ is $e^{2x}$. But my book doesn't explain why this is the case. Could you give me a little more ...
2
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1answer
63 views

Does a bijection from the reals to the any binary form?

It is fairly simple to store all rational numbers in a binary format (not base 2) (a language composed of only 1s and 0s, no . marking) by simply storing one integer, a seperator, and another integer. ...
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3answers
73 views

binary representation of integers congruent 1 and 3 modulo 4

Let $k=b_nb_{n-1}\ldots b_3b_2b_1b_0$ be the binary representation of an odd positive integer. Prove: If $k\equiv 1 \mod 4$ then $b_1=0$. If $k\equiv 3 \mod 4$ then $b_1=1$. I think that to prove ...
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2answers
73 views

How many bit-flips are required to achieve random distribution?

If I have a binary number W bits wide, initially all set to zero, and I repeatedly pick a random bit and toggle it from zero to one or vice versa, how many times would I need to do this to achieve ...
0
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1answer
86 views

Signed Number's Binary Addition

I'm going to discuss about Signed Number's Binary addition, I searched about it and even read books. Now I make little changes in it's logic and start my own logic to solve it. Let me show 4 bit ...
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1answer
62 views

Looking for expectation of the number of substrings

The question is formulated as follows: if we are given $n$ random binary strings of length $n$, what is the expectation of the number of substrings they have in common? Sounds pretty simple, but if ...
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5answers
224 views

Simplifying sum equation. (Solving max integer encoded by n bits)

Probably a lack of understanding of basic algebra. I can't get my head around why this sum to N equation simplifies to this much simpler form. $$ \sum_{i=0}^{n-2} 2^{-i+n-2} + 2^i = 2^n - 2 $$ ...
2
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1answer
236 views

Multiplication and binary xor

I have to prove one thing that combines logical xor and arithmetical sum of binary representation of some numbers. Could you direct me what can I read on this topic? Specifically, I need to prove ...
2
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2answers
56 views

What is the minimum length of maximal palindrome of a binary word of length $n$?

For example if we have $n=4$ then the minimum length of maximal palindrome is 2. Here are all four digit possible binary words along with their maximal palindromes: ...
3
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3answers
760 views

What is the last digit? [closed]

Consider all 100 digit numbers, i.e., those between $0$ and $10^{100} - 1$ (inclusive). For each number, take the product of non-zero digits (treat the product of digits of $0$ as $1$) , and sum ...