# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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, ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...