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

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1answer
54 views

Probability of Binary Word Matching

Thanks for reading. Would be lovely if somebody could help me out on this (but not just post the answer) but also how you got there. I'm a programmer and I've ran across this problem which I can't ...
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0answers
13 views

Can $1.0*2^{-1}$ and $1.1*2^0$ both equal -0.5 in twos complement?

I am happy that if you write -0.5 in binary as -0.1 and then convert to twos-complement you get 1.1. You can read that as $-1+1/2=-1/2$. However if you use standard form and write $0.5_{10}$ as ...
18
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4answers
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: ...
3
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1answer
45 views

Proof that max value of $n$-bit binary number is $2^n - 1$

After reading this programming question , I wanted to prove the assertion. I'm wondering whether the below would be considered a complete and clear proof. Claim: $\sum_{i=0}^{n-1} 2^i = 2^n - 1.$ ...
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0answers
57 views

Name of this formula or more explantation of the proof?

I have found this formula which is a combinatorial identity for counting binary words. I'd like more information on it, or the name of the proof. I am also not totally clear on the step between the ...
0
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0answers
18 views

Floating point binary to half precision floating point

I am trying to convert to $16$ bit half precision floating point however I ran into a possible error and am unsure if a negative exponent is ok. I am trying to convert $0010011100010000$ I separate ...
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1answer
38 views

Show that every number $a$ can be shown in the following form: $a=\sum_{i=1}^{k}2^{x_i}\cdot 3^{y_i}$

Show that every integer $a>0$ can be shown in the form: $$a=\sum_{i=1}^{k}2^{x_i}\cdot 3^{y_i}$$ where $0\le x_1< x_2< \dots < x_k$ and $0\le y_k < y_{k-1} < \dots < y_1$ are ...
0
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2answers
47 views

With 8 bits is it possible to obtain an integer in more than one way? [duplicate]

This is just a curiosity that just came to my mind while thinking at IP addresses. A byte is composed of 8 bits. A bit can either be $0$ or $1$. IPv4 addresses are composed of a group of 4 bytes. ...
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3answers
58 views

why $10$ in any base number system written as $10?$

I am a student trying to write an article in number system can same one give me an idea why $10$ in any base number system written as $10$ $?$
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1answer
21 views

Binary to Decimal Right of The Binary Point

I need to convert 10000000110.011 to decimal. The first part I fully understand, 2^10 + 2^2 + 2^1 = 1030. Wolfram Alpha confirmed that 1030 is correct. But for the right of the binary point, WA gives ...
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1answer
32 views

How many combinations have the same amount of ones and zeros?

In an even n digit binary number, how do you calculate the number of combinations that have the same number of ones and zeros.
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2answers
92 views

Powers of 10 written as binary numbers

Consider powers of 10 written as binary numbers. 10 = 1010 , 100 = 1100100 , 1000 = 1111101000 How can I find a formula for the number of zeroes between the last and the second last 1?
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1answer
17 views

decimal to binary conversion of recurring decimal numbers in matlab

I have made a function convert fractional part of a decimal number to binary, the code is given below: ...
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2answers
48 views

Subtracting Binary Numbers with one and two's complement

An example: $01001 - 11010 = -10001 = -17$ one's complement: $00101$ two's complement: $00110$ so the above statement should evaluate the same as: $01001 + 00101 = 01110$ (inverting it) $= 10001$ ...
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2answers
24 views

why shifting left 1 bit is the same as multiply the number by 2

I have recently faced a problem .The problem is here.We know that if we represent a decimal number in binary and move left all the bits by one. The left most bit is lost! and at the rightmost, a zero ...
0
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1answer
18 views

Retrieve position from binary number ordered by number of ones

I have binary numbers of length s. They are ordered by numbers of ones, and they can have at most j zeros. That is: first are ordered all numbers containing (s; 0) possible subsets of s numbers, next ...
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1answer
24 views

Prove by induction the number of edges in a tree given the leaves.

Define a cs130A tree to be a single leaf node or an internal node (the root) connected to two disjoint subtrees, which are themselves cs130A trees. Prove by induction that for all cs130A trees the ...
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1answer
18 views

Find what row a given number exists in within a binary heap tree.

If I have a binary heap tree like so: 1 2, 3 4, 5, 6, 7 8, 9, 10, 11, 12, 13, 14, 15 16 ... 31 32 ... 63 64 ... 127 I ...
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0answers
49 views

Probability of having n more 0's than 1's in a subsequence

We have a number K>2 and a sequence of m binary values $$b_1, ... b_m $$ Given a subsequence $$b_i, ... b_j, i>j $$ we define: W(i,j) is the number of 1's in the sequence L(i,j) is the number ...
0
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1answer
47 views

Binary subtraction with borrowing vs. 2's complement

Consider the following two binary numbers which are using sign-magnitude representation. Therefore, both are positive numbers. ...
0
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1answer
22 views

Determining if the result of adding (subtracting) two binary numbers is correct as NBC and 2's

My task is to convert two numbers to binary, add them and determine if the result is correct as a NBC and 2's. Then I have to subtract B from A by converting by using 2's for calculating -B [that is ...
4
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2answers
102 views

Integers with consecutive nonzero digits in base 2

Note that $2016=11111100000$. How many integers $\le n$ have their nonzero bits consecutive in their binary (base $2$) expansion? The case $n=2^m-1$ is easy. The question is about a closed formula for ...
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0answers
52 views

Two's complement involving binary numbers [Single-precision floating point number]

Given the following bit pattern: 0010 0100 1001 0010 0100 1001 0010 0100 What decimal number does it represent, assuming it is a single-precision floating point number?
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1answer
71 views

Two's complement involving binary numbers [closed]

$$0010\ 0100\ 1001\ 0010\ 0100\ 1001\ 0010\ 0100$$ What decimal number does it represent, assuming it's a two’s complement integer?
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1answer
60 views

why borrow two 1s in binary substruction?

In binary subtraction i have seen some tutorial and some sites that during 0-1 process borrow value from next 1s. Let's say one ...
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1answer
476 views

Binary Gaussian Elimination of a matrix

Can anyone help me find the algorithm for the Binary Gaussian elimination of a matrix. for example: The output:
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0answers
74 views

Designing a Turing machine for Binary Division

I'm trying to design a TM to binary divide 2 numbers. The best approach I've found is the method: division as repeated subtraction, but I don't know if there's a fastest way or an easier way. ...
0
votes
1answer
17 views

Convex or Quasiconvex Relaxed Binary Quadratic Optimization Problem

Let's say I have a quadratic problem with nonnegative triangular matrix Q and binary decision variables x. $$min_{x} f(x) = ...
0
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1answer
42 views

Find a function comparing two variables that is equal to 1 when they are equal and 0 when they are not

I am trying to find a function that compares two variables. When these two variables are equal, the function should equal 1. For any other value, the function should equal 0. Is there a way to do this ...
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0answers
10 views

Proof by induction that $\sum \limits _{i=1} ^n d^{-l_i} = 1$ sentence in a full tree [duplicate]

How do I prove by induction that $\sum \limits _{i=1} ^n d^{-l_i} = 1$ where: $d$ = the number of children of each node; $n$ = the number of leaves; $l$ = the depth of each leaf $l_1, \ldots, l_n$? ...
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3answers
77 views

Generating binary array

Programming side (C++) I have this task where I need to get 5 numbers in binary array, but i cant think of way to generate it. The building process is: Starts Width 10 then adds same number but 1 ...
2
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2answers
82 views

Binary number with base $-2$ (minus two) arithmetic algorithm

I have a number X represented as a sequence of $a_i \in {0,1}$ so $$X = \sum_{i=0}^{N-1} a_i(-2)^i$$ where $N \in \{ 1, \ldots, 100000 \}$. I need to find an algorithm to produce a number $Y = -X$ ...
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0answers
26 views

largest number machine number that this comp has, numerical analysis, mantissa

I am having trouble understanding the basics on how to calculate the m, s and c of the formula. I have looked in my textbook but not much of an explanation, also on the IEEE websites but nothing much. ...
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2answers
25 views

Formulating the bitwise OR operation

Considering the bitwise OR operation, wikipedia states, $$x\;\mathrm{OR}\;y = \sum_{n=0}^b ...
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0answers
8 views

How would I round the number $-(1.01101|01)_2$ down?

I'm trying to round $-(1.01101|01)_2$ in binary down. The $|01$ are the guard bits. The answer is $-(1.01110)_2$ My attempt is as follows: $-(1.01101|01)_2$ $-(0.00001|00)$ = $(1.01100)$ Not ...
2
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2answers
2k views

How do I calculate the binary form of a polynomial?

The 4 bytes used in CDROM sectors is just a usual 32 bit CRC. It uses the polynomial P(x) = (x^16 + x^15 + x^2 + 1).(x^16 + x^2 + x + 1) which expands to ...
1
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1answer
677 views

XOR of Binary Numbers to Reach a Given Number

Given a set S = { s1, s2, s3, ... sn} of Binary Numbers , I need to find if a given Binary Number X with only 1 bit position set as 1 (..00001000...), can be ...
1
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1answer
42 views

Binary variable in a constraint

I have the following optimization problem: Model I: $$f(x,y) \\ s.t., \\ y\leq x+M(1-V)\\ y \leq MV \\ x \geq 0, y \geq 0$$ where x and y are continuous variables whereas V is a binary variable. M ...
0
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1answer
74 views

How many 10-bit binary sequences begin and end with the same bit?

Can you please teach me how to solve this? There are other similar questions that I have, but I don't understand how to solve them. Another question is: How many 10-bit binary sequences contain an odd ...
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2answers
34 views

About 2-adic representation of integers

How would I express -3 in 2-adic representation? Is it just revercimal calculation of binary expression of -3? like: -3 = -11 in binary, so using revercimal, -11. in binary = 01. ?
2
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2answers
52 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: ...
0
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1answer
62 views

Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
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2answers
197 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 values 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, the numbers $1,2,4,8$ ...
5
votes
7answers
640 views

How to find the unique sums in the values 1,2,4,8,16, 32

I apologize but I'm not sure what you would even call this problem. I have some data that provide a numeric code for race as follows: hispanic(1) + american_indian_or_alaska_native(2) + asian(4) + ...
0
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1answer
33 views

What actually determines a 'borrow' in binary subtraction?

Given Byte1 and Byte2 in binary. How to determine whether Byte1 - Byte2 results in a borrow ...
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0answers
17 views

Is it possible to break a system into two BSCs to find the total capacity of a system?

I have been trying to solve this problem in manor of ways but I cannot seem to find a satisfactory solution. I have tried the basic way of calculating capacity through self information but I was ...
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1answer
127 views

How many binary numbers of length n do not contain the substring 000?

How many binary numbers of length $n$ do not contain the substring $000$? Denote this number by $Z_n$; find a relationship between $Z_n$, $Z_{n-1}$, and (something else not given) to form an ...
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0answers
15 views

How to perform binary transformations?

One of the steps in Binary Index Tree algorithm is to find a node's parent which is done by un-setting the rightmost SET bit. For example: if a node has index 1010 than it's parent is 1000. To apply ...
2
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0answers
117 views

Bounding the global intersection of a family of sets

Suppose that we have a decision tree of height $r + 1$ that describes how to increment an $n$-bit integer in the range $[0, 2^n -1]$. That is, the internal nodes are labelled with a bit position that ...
2
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2answers
755 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 ...