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

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1answer
60 views

Probability random binary number lies in an interval

Suppose we have a sequence $\{d_n\}$ where all the $d_n$ are either 1 or 0, with equal probability. Let $x=\sum_{n=1}^\infty d_n2^{-n}$. I need to show that $\mathbb{P}(x \in [a,b])=b-a$. I started ...
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2answers
45 views

How do you add 8-bit floating point with different signs?

Hi I have some trouble with how should I add two 8-bit floating points with different signs. The question is here, 1 100 1100 + 0 101 1011 = Thee 1st bit is the sign, next 3 bits are the exponent ...
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1answer
19 views

Binary symmetric channel

Binary symmetric matrix A sends $i,j$ and B gets $i,j$. Does it mean that $A$ != $B$? I would know how to solve this if A would be equal to B, but now I'm not sure how should I start, when A has 2 ...
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0answers
18 views

binary divison with remainder

I wanted to calculate this (binary): 10101.101/1.1. the result is: 1110.01101010101010101010101010... I succeed in calculating the integer part, but I didn't understand how I find the numbers that ...
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0answers
93 views

Reciprocal of a binary number

I'm looking for a way to compute the reciprocal of a binary number. The numbers are fractions, but that doesn't really matter. Therefore we can think of any number involved as $N \geq 0$. The idea is ...
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0answers
176 views

185 stored as a signed 8-bit number?

Several exercises in my textbook start with assumptions that confuse me. For example: Assume 185 and 122 are signed 8-bit decimal integers stored in sign-magnitude format. Assume 151 and ...
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1answer
69 views

What is the name for the binary sequence 0110100110010110?

This is a special sequence formed by: 0 0 + 1(its opposite) 01 + 10(its opposite) 0110 + 1001(its opposite) 01101001 + 10010110(its opposite) What is it called?
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2answers
31 views

What is the minimum $k$ such that $\sum_{i = 1}^{k}2^{a_i} = \sum_{j = 1}^{n}2^{w_j}$

Suppose you have a sequence of numbers $S = (w_1,...,w_n)$ and you want to know the minimum $k$ such that $2^{a_1}+...+2^{a_k} = 2^{w_1}+...+2^{w_n} = T$. I'm told that the minimum $k$ is equal to the ...
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2answers
39 views

Binary Sequence of Single Bit Transitions

First of all, I'll have to say that I believe this problem has no solution, but I'm unable to prove it. Here is the problem: I need an algorithm to generate a sequence of all possible transitions of ...
0
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1answer
13 views

binary relations defining an equivalence relation on S

Is this a true statement for binary relations defines an equivalence relation on S: S is the set of all n-digit binary sequences. We say that two binary sequences are in a relation if and only if ...
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2answers
24 views

Number of binary strings with sub-string constraint

How many binary strings of length $n$: $a_1a_2\dots a_n$ are there, such that for every sub-string of k consecutive numbers $a_ia_{i+1}\dots a_{i+k-1}$ and $\forall k, 1 \leq k \leq n$, the difference ...
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0answers
17 views

Canonical forms for matrices with binary elements.

Based on this answer to a combinatorics question I grew curious of results regarding similarities or canonical forms of matrices fulfilling these criteria: Elements of matrix are binary valued ...
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1answer
44 views

The class of $0-1$ matrices with row sums at least $2$, where distinct columns have dot product $1$

There is an $m\times n$ matrix of ones and zeros where the dot product of any two different columns is one and any row have at least two ones in it. My question is: Is this a popular matrix? Does it ...
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0answers
22 views

symbol for set of ones in binary vector

Is there a standard symbol for the set of elements that have value 1 in a binary vector f? So far, I was defining Pos(f), but obviously this is suboptimal as it gives the impression that the values ...
0
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1answer
28 views

permutations of binary sequences

What is the proof that there are $2^n$ distinct binary codes of length n I know this progression also applies to the decimal ($10^n$) and hex ($16^n$) systems but how can this be shown?
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0answers
28 views

partial states or partial probability?

I am trying to figure out an alternative way of representing a state probability space, to make certain ideas clearer (that I don't need to discuss here). Let's say I have a system of two elements, A ...
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1answer
47 views

Binary operations 1011 & (~0 << 2)

My thought process to solving this is that 1011 & (~0 << 2) = 1011 & (1 << 2) = 1011 & 0100 = 0000. But my book says the answer is 1000, ...
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2answers
25 views

finding binary relations between two Finite sets?

If there are two finite sets $A$ and $B$, then how to achieve the below How many binary relations between $A$ and $B$? How many functions from $A$ to $B$? $A$ and $B$ should be in terms of ...
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0answers
110 views

floating point subtraction for binary numbers

Consider that I want to do a binary operation on the following floating point numbers: 0.35-0.62 I can reach the end but I can not figure out how the sign bit is ...
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1answer
30 views

binary combinations/rule of sums

I'm currently working on this topic but I'm having a hard time. In an 8-bit string, there are 256 combinations (or is it permutations?). Either way, I know it is 2^8. If at least 3 elements must be ...
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1answer
23 views

Binary subtraction with a lot of zeroes

I need to subtract $$\begin{array}{r} & 1000000000.0000\\ -& 0.0001\\ \hline \end{array}$$ The trick is that I need to borrow a $1$ for the first $0 - 1\ldots$ But there are zeroes ...
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1answer
62 views

Finding the mantissa from binary with floating point numbers?

Here is the example problem slide I am working with: I understand how to get the exponent, its just 2+128=130-127=3 I understand the first bit is the sign bit for positive or negative. I just ...
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0answers
24 views

Binary arithmatic. Overflow, carry, Zero flags

I am abit confused on how the binary arithmetic works. I though i got it right but after encountering this question i feel like i am wrong. ...
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1answer
82 views

Permutation and combination and binary numbers

what is the sum(in base 10) of all natural numbers less than 64 which have exactly three ones in their base 2 representation. answer given -630 I just need a starting point like from where should I ...
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1answer
37 views

$\lfloor\log_2(x)\rfloor + 1 \ \neq \lceil\log_2(x)\rceil?$

Is there any case where $$\lfloor\log_2(x)\rfloor + 1 \ \neq \lceil\log_2(x)\rceil ?$$ I'm in discrete mathematics, and my teacher stated the former formula to be finding how many bits are needed ...
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1answer
47 views

The limit points of a subset of binary space with ultra metric

let $Z$ be the subset of binary space $B$ consisting of all points that are eventually $0$. Then what is set of all limit points of $Z$. Binary Space $(B,d)$ is a metric space as follows. Let $BB$ be ...
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3answers
42 views

Determine if the solution to a division is odd or even

Is there a way to quickly determine if the solution to a division operation will be odd or even? I need a quick way determine that in a program. Doing a complete a complete division takes up too much ...
2
votes
1answer
26 views

Binary dividing with a remainder

How much is $\frac{101000}{1001}$ in binary? I checked in 3 sites, each displayed another result. The result I get is 100 with a reminder of 100. Can you please try and solve it and show how you did? ...
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0answers
24 views

Computing relative roundoff error of a correctly rounded binary number

This is related to a question that was asked and answered a moment ago. I need to answer the following: If $\displaystyle \frac{3}{5}$ is correctly rounded to the binary number ...
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0answers
74 views

Rounding up a binary number

I just converted the fraction $\frac{3}{5}$ to the following floating point binary number: $(1.001100110011001100110011\cdots)_{2}\times 2^{-1}$. Now, I am trying to find the two nearest machine ...
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0answers
13 views

solution of equations with inter-dependent variables.

Suppose I have few equations of the form a12*x+a13*y+a14*z+a15*xy+a16*xz+a17*yz=d1 In short I have n free variables and other (n choose 2) dependent variables. ...
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1answer
54 views

The set of all limit points of a binary space with ultra metric

Binary Space (B,d) is a metric space as follows. Let BB be the set of all infinite sequences of 0's and 1's. That is, points of B are of the form x=(x1,x2,x3,…), where xi=0 or 1 for every i=1,2,3,…. ...
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4answers
2k views

Why b%2==1 implies that rightmost digit of b in binary form is 1

How one can deduce that if b is any number and if b%2==1 then rightmost digit of b in binary form will be 1 without checking it manualy
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0answers
48 views

Ones and twos complement on hexadecimal numbers?

I have a question as below: Give the data representation of each of the following integers assuming 16 bits and each of the representations of sign and magnitude, one’s complement, and two’s ...
3
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3answers
79 views

How many numbers are in this set?

Let M a positive integer is called “Minos” if in its binary representation, any two consecutive appearance of digit 1 are separated by 2 or more 0. Example 36= 100100 (binary) is “Minos” number, but ...
3
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1answer
147 views

What are the three different ways negative numbers can be represented in Binary?

I am new to Binary and we are learning it for my computer hardware class. Since I am just learning, I am not very sure how to represent negative numbers in binary. I believe they are Signed ...
0
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1answer
88 views

A nice form of a given function

First let, $\oplus(a_1,a_2,\ldots,a_n)$ denote the bitwise xor of $a_1,a_2,\ldots,a_n$. Define the function $\Delta(a_1,a_2,\ldots,a_n)$ to be the maximum value of $a_i - ...
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0answers
14 views

Multiplying numbers in two's complement

I have 2 numbers, both in two's complement form. The problem is 101011 x 100111. I already sign extended the multiplicand to 2n bits, making it 111111101011. I arrived at my answer, 110100001101 which ...
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1answer
21 views

Integer division and modulo by $\mathbb{2^n-1}$

Dividing an integer number $k$ by $2^n$ (i.e., computing $\lfloor k/(2^n)\rfloor$) is easy: it is just equivalent to bitshifting $k$ by $n$ binary places, therefore removing the last $n$ binary digits ...
2
votes
1answer
35 views

Decimal Number to Octal

I have the number $9243$ In Decimal (Base 10), I'm trying to convert it to octal and then Binary. What I've done is (r IS THE REMAINDER) $4096 | 9243| 2$ $ R = 1051$ $512|1051| 2$ $r = 27$ ...
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0answers
22 views

Binary operations and proofs

$\text{binary}(a)$ = binary representation of a base 10 number $a$ Are the following statements correct? If yes, where can I find the proofs? (1) $\text{binary}(a\times b)=\text{binary}(a)\times ...
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1answer
28 views

Proof of function that gives 1's in binary representation

There is an older question here that asks about the function $f(n)$ that gives the number of 1's in the binary representation of $n$. Stated in an equivalent way, this function is: $f(n) = ...
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0answers
26 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
28 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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0answers
19 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
44 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 ...
2
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1answer
60 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
51 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 $21+11 = 32$. ...
2
votes
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
23 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 ...