# Questions tagged [binary]

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

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### How to find the number of Hamming weight 4 bit vectors with Hamming distance at least 4 for length N?

Given a set of bit vectors of length N with the restrictions that each vector should have a Hamming weight of 4 (4 of the bits are 1, the rest are 0) and each vector is at least a Hamming distance of ...
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### Hamming distance between consecutive primes

The Hamming distance between strings of equal length is the number of mismatches between the strings MYSTERY MASTERS ^ ^ This distance can be defined for ...
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### Find a sum of all numbers in the list which have zero in the high-order digit.

Let's fix a positive integer number $p$. Let $x$ be a non-negative integer that can be expressed using binary system in such a way: $x=\overline{x_{p-1},...,x_0}$, where $x_0,...,x_{p-1}\in \{0,1\}$. ...
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### How to calculate the right-most digits of $\lim_{n\to\infty}\sum_{k=0}^n k!$

Consider the the sequence $S_n=\sum_{k=0}^n k!$. Note that for any base $b$, as $k$ increases, the base-$b$ expansion of $k!$ will have a monotonically increasing number of right-most zeroes. Hence, ...
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### Negative binary minus positive binary? [closed]

I'm subtracting binary numbers (like the binary of 10-5) and I understand that if do: (positive) binary A - (positive) binary B then I have to add the two's compliment of binary B to binary A. But ...
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### Is there a rule or equation to convert decimal to binary without recursion

Every solution I’ve seen is always some method you have to repeat or iterate. But is there some equation I could just plug any number into and get the binary value. For context, I’m trying to do this ...
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### Is a binary sequence generated by repetition and subsequent digit flipping periodic?

Consider a binary sequence (i.e. consisting of 1s and 0s), $a_{0}=0$. The members of the sequence are generated by repetition and flipping : $a_{n}$ are all the digits in $a_{n-1}$ followed by all the ...
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### Amazing math question please find the logic. [duplicate]

Please note that my English is not the best, I appologize in advance. My question: Like in the binary system each of us have 2 parents your parents have total 4 parents (your grandparents) your ...
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### How many squarings are needed to exceed 2?

Given the base 2 representation of $x\in\mathbb Q$ ($1<x<2$ and has a finite number of digits in base 2), find a number $k\in\mathbb N$ such that $2<x^{2^k}<4$. The final answer/algorithm ...
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### What is the size of the largest subset with a pairwise hamming distance of 3

Consider all binary strings of length $n$. Is there any known bounds on the size of the maximum subset such that the pairwise hamming distance between any two elements is at least 3.
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### A puzzle problem of Digital VLSI design?

My thoughts: Here price of chocolate is increasing in Geometric progression with common ratio 2 and first term 1, so on nth day price of chocolate will be $T_{n}=2^{n-1}$ irrespective of wheather ...
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### Binary System with Sigma Summation

Let 100110101 be a number from the binary system with the digits being $$b_i \in \left\{0,1\right\}$$ If we calculate this number into the decimal system, the result is 309. What formula can be ...
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### Equation to get the reversed binary number

I am wondering if there is a mathematical formula that can convert one number to another number that is equal to the reversal of its bits for the number's binary representation by some length of bit ...
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### Binary addition using Linear algebra

Is it mathematically possible to do binary addition using linear algebra? To be more precise, when the binary numbers are represented by vectors, each element containing respectively the ones, tens, ...
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### Binary String Combinations

For this question, imagine finite binary strings consisting of $0$ and $1$. For instance, $00111$ is a string. Order does not matter, so the string $1100$ is considered equivalent to $0011$. We will ...
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### Pattern for all the binary chains divisible by 5

For instance, $x = 101$ is divisible by $5$ because it is the integer 5. Same thing for $x=1111$ is also divisible by 5 as it is the integer 15. However, $x=1100$ is not divisible by $5$ as it is the ...
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### Dimension and basis of set of linear codes of even weight

I am working on questions about coding theory. The set C is the set of all words in binary code that have an even weight and are of length n. I have already proven that this is a linear code. Now I ...
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### Can this binary number be represented as a fraction $a/b$? [closed]

Can the binary number $0.1010\ldots$ be represented as a fraction $a/b$ with $a$ and $b$ as integer numbers?
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### Determine the following sum

Let $S(n)$ be the sum of the digits of n in its binary representation. For example the binary notation for 19 is $10011$ and $S(19)=3$. Determine $\sum_{n=1}^{\infty} \frac{S(n)}{n(n+1)}$? I've made ...
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### Convert Hexadecimal $8D(16)$ to binary in signed magnitude

I'm supposed to covert hexadecimal value, $8D(16)$ into $8$-bit binary if signed magnitude representation is used. $8D(16)$ $\to$ $1000$ $1101(2)$ For signed magnitude, the left most bit is used to ...
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### Binary representation of a number $n$ with exactly one leading 0 can be turned into a binary representation of $n+1$

Prove that for $\forall n \in \mathbb{N}$, the binary representation of $n$ with exactly one leading 0 can be turned into a binary representation of $n + 1$ by flipping exactly one bit from 0 to 1, ...
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### What does this pic mean?(guess it's a binary number) [closed]

This number was presented in yesterday class and explained by teacher that they were lined up according to a certain rule. I think we'll use nibbles to solve this. I also think that the circle means ...