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

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2answers
13 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, which is the probability that the two numbers will differ by exactly one bit? From my understanding, the ...
0
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5answers
108 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 ...
1
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1answer
42 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
39 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 ...
2
votes
2answers
331 views

Counting binary strings that have atmost k consecutive 0's

I know how to count how many binary strings with length n and having exactly k 0's are there but i am not able to find a way to count the number of binary strings that have exactly x 0's and y 1's and ...
1
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1answer
26 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
0
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1answer
40 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
44 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 ...
2
votes
3answers
51 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
29 views

Algorithm for Converting Non-balanced Base-n to Balanced Base-n (for odd n)

Let $n \in 2 \mathbb{N} - 1$. I was wondering what sort of algorithms there are for converting (non-balanced) base-n to balanced base-n, where "balanced" is as is described in this article: ...
0
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0answers
17 views

Closest finite series representation of original number < 1

I have another question related to this Decompose negative power of ten in finite series. Suppose we have number $0.1$ decomposed in finite sum of two in negative power: $10^{-1} = 2^{-4} + 2^{-5} + ...
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2answers
2k views

binary representation of a fraction in two's complement

Could any one please explain what is a 16-bit two's-complement representation of -0.375 and the steps to calculate it? Also, what happens if I convert it back to decimal? Thanks
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1answer
28 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 ...
4
votes
1answer
21 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 ...
1
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3answers
86 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 ...
0
votes
1answer
431 views

Convert hexadecimal numbers to floating-point format using single-precision IEEE 754 format

I need help with converting hexadecimal numbers to floating-point format using single-precision IEEE 754 format. Hex numbers such as 312A. how do I go about converting it? I know how to convert from ...
1
vote
1answer
39 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
votes
1answer
14 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 $y$ ...
2
votes
1answer
51 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. ...
0
votes
3answers
27 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 ...
1
vote
1answer
362 views

The maximum number of digits in binary multiplication

Given 2 binary numbers of n digits. (n can be from 1 to something large) What is the maximum number of digits of their product? I think it's 1 in 1-digit*1-digit, and 2n otherwise. But I want some ...
0
votes
0answers
60 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
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2answers
64 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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0answers
9 views

Using the algebraic expression ((x-2)+3) / ((2-(3+y))*(w-8)) show the results of performing a preorder, an inorder, and a postorder search.

Using the algebraic expression ((x-2)+3) / ((2-(3+y))*(w-8)) show the results of performing a preorder, an inorder, and a postorder search. Preorder is root, left, right. Inorder is left, root, ...
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0answers
21 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 ...
3
votes
5answers
144 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. $$ \sum_{i=0}^{n-2} 2^{-i+n-2} + 2^i = 2^n - 2 $$ ...
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vote
3answers
619 views

Quick logarithm calculation

In coming up with an algorithm for finding log (10) base 2, these are my thoughts. I wanted to know if this makes sense and how could I truly make it more efficient. The requirements are strictly not ...
1
vote
1answer
36 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 ...
1
vote
1answer
23 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
votes
1answer
18 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: ...
1
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1answer
184 views

Can every periodic binary sequence be expressed as the output of a Linear or Non-Linear Feedback Shift Register?

Quick background: The output of a Linear Feedback Shift Register (LFSR) with length $n$ is a binary sequence which is periodic of length dividing $2^n-1$. From a mathematical point of view, such a ...
2
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3answers
149 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 ...
0
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2answers
30 views

How many bytes with this configuration given in bits?

If the maximum number of bits of certain field is set to be 10 bits max. How many bytes can be set within this limitation? The solution of such a problem suggests number of bytes ranged is 1023. Can ...
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votes
1answer
44 views

Is binary more ideal than decimal? [closed]

We only chose the decimal system because we have 10 fingers. Binary is the most basic positional numbering system, so would it make sense to say that it would be the most ideal system? Is it better ...
0
votes
1answer
39 views

converting decimal to hexadecimal using division method

Okay so I know the basic procedure of converting a decimal number to any base-r is to divide by r and keep up with reminders until you reach zero. The reminders form the new base that is equivalent to ...
2
votes
1answer
46 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
0
votes
0answers
28 views

Bitwise Operations and the Naming Convention of their Operators

So I just recently came across a bitwise operation on StackOverflow which shifts the bits in a binary number to the right while zero-filling from the left. The left side zero-filling overwrites the ...
6
votes
3answers
446 views

What is the next number having the same number of bit 1s? [duplicate]

You are given a number, $A$, and you have to determine a number, $B$, such that $B>A$ and the number of $1's$ in the binary representation of $A =$ number of $1's$ in the binary representation of ...
2
votes
1answer
70 views

IEEE-754 Format Conversion

Represent 11.0011 x 2^10 using the IEEE-754 standard for 32-bit floating point representation. 0-Sign Bit 10001010-Exponent 1001100000000000000000-Mantissa Is this answer is correct ? I am bit ...
1
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0answers
73 views

Generator polynomial creates a 127 bit sequence

I have been reading a paper that states that a generator polynomial $$ G(D)= 1+ D^4+D^7$$ creates a 127 bit sequence which is as follows 00001110 11110010 11001001 00000010 00100110 00101110 ...
16
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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: ...
0
votes
1answer
34 views

Generating binary palindromes

I have the following Number Systems problem. "Given the numbers from 2-2047 (inclusive) represented in binary, how many are palindromes? The leading digit can't be zero." How would I go about solving ...
6
votes
4answers
221 views

Floor function to the base 2

I'm not a math guy, so I'm kinda confused about this. I have a program that needs to calculate the floor base $2$ number of a float. Let say a number $4$, that base $2$ floor would be $4$. Other ...
4
votes
0answers
69 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
0
votes
3answers
93 views

converting to octal, hexadecimal and binary

Revising for an exam - could someone explain to me how you can convert an ordinary numbers to octal, hexadecimal and binary would be appreciated
8
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2answers
327 views

$N$ perfect logicians wearing hats

I once came across the following riddle: (assume $N$ to be extremely large) There are $N$ perfect logicians arranged in a vertical row. They are allowed to strategize before the game, during the ...
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1answer
22 views

calculation of binary power like $a^b$ where a,b are binary numbers

Is it possible to calculate power of binary number like $a ^ b$where a,b both are binary numbers.
1
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2answers
56 views

Calculating all possible sums of the numbers $2^0, 2^1, \ldots, 2^{(n-1)}$

Using the simple equation $2^{n-1}$ you get answers such as: $1,2,4,8,16,32,64,128,256,$ etc. How can I find all possible number combinations within this range? For example, then numbers $1,2,4,8$ ...
5
votes
5answers
120 views

How many bit strings of length $12$ have a substring $01$?

My question is should it be $11C2$ or should it be $11C1$? Since $01$ are connected together, I take them as a single unit and there are $11$ different positions where they can be placed. Is the ...
0
votes
0answers
20 views

Binary block code

I don't fully understand how i would construct binary block codes? As far as I know we have a code, say, $(5,4,3)-$code, then $5$ would represent the length, $4$ would represent the number of codes ...