Questions tagged [binary]

For questions about representing numbers in base 2 or performing arithmetic on them.

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18
votes
5answers
4k views

Signed and unsigned numbers

How would the ALU in a microprocessor differentiate between a signed number, -7 that is denoted by 1111 and an unsigned number 15, also denoted by 1111?
14
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3answers
2k views

Shannon Entropy of 0.922, 3 Distinct Values

Given a string of values $AAAAAAAABC$, the Shannon Entropy in log base $2$ comes to $0.922$. From what I understand, in base $2$ the Shannon Entropy rounded up is the minimum number of bits ...
9
votes
5answers
15k views

Why does the formula 2n + 1 find the child node in a binary heap?

I learned that when you have a binary heap represented as a vector / list / array with indicies [0, 1, 2, 3, 4, 5, 6, 7, 8, ...] the indicies of the children of element at index n can be found with $...
8
votes
4answers
6k views

How can I multiply a binary representation by ten using logic gates?

So I’m currently working on something and I have converted all decimal digits 0-9 into binary. But now I want to take say 6 in binary and increase its order of magnitude by base 10 (turning 6 into 60) ...
5
votes
1answer
262 views

Could a standardized ternary system be more efficient than the binary system? [duplicate]

Could a standardized ternary (base-3) system be more efficient than the binary (base-2) system? Binary is efficient for processing using logic gates, but can be bulky when using for file storage or ...
5
votes
2answers
95 views

How do computers *really* work? (at the most basic level)

While learning about computers I will read about RAM and Storage and the CPU, and while these explain the architecture of a computer and how parts of a computer work together, I still don't understand ...
5
votes
2answers
288 views

checking '<' for two binary numbers in a cnf-formula

I want to check whether a arbitrary binary number is less or equal to another binary number in a cnf-formula. I can already construct a formula, which is not in cnf: Lets say n and m are two-digit ...
5
votes
0answers
124 views

Partitioning through block moves to the end

Suppose we have a binary string $s$. We wish to partition this string in a series of $0$s followed by $1$s (alternatively: we wish to sort), using only one operation: moving three consecutive elements ...
3
votes
4answers
691 views

Binary 2s Compliment Applied Twice Gives Original - How?

At the moment the maths behind binary 2s compliment seems like voodoo to me. If I take 0011 (+3) and "flip and add 1", I get 1101 (-3) If I apply the same process to 1101, ("flip and add 1"), I get ...
3
votes
3answers
209 views

What are some uses of the Thue-Morse sequence in computer science?

Note: I come from a mathematics background. The Thue-Morse sequence $t_n$ is a binary sequence that takes the value $0$ at the positive integer $n$ if the number of $1$s in its binary expansion is ...
3
votes
4answers
68 views

Is there a binary representation where the encoding distance grows with the arithmetic distance?

I need to binarily encode/decode small integers (around 10 bits), in such a way that the hamming distance of the encoded number grows monotonically with the (absolute value) arithmetic distance of the ...
3
votes
2answers
1k views

Why does little endian apply to numbers and not to text strings?

The Portable Executable file format is the format that Windows EXE files use. It is a binary format. Numbers are in little endian form. Thus, the following hex represents the decimal number 256, not ...
3
votes
1answer
123 views

Is is possible to determine if a given number is xor combination of some numbers?

I have been given a number Y which is ($a$ xor $b$ xor $c$ xor $d$ xor $e$ ) of some numbers ($a$,$b$,$c$,$d$,$e$) and another no X. Now i have to determine if X is a xor combination of ($a$,$b$,$c$,$...
3
votes
1answer
869 views

Computing Hamming Distance Between Binary Vectors

Consider the following problem: Input: We are given a set $S$ of $n$ binary vectors. A binary vector is of the form $(b_0, b_1, ... b_n)$, where $b_i \in \{0, 1\}$. Therefore, there are $n$ vectors ...
3
votes
1answer
96 views

Greedy algorithm for subset sum on powers of 2

I have some $n$ numbers which are powers of $2$, say $a_1,a_2,a_3,\ldots,a_n$ which are not necessarily all distinct. I have option to give them any sign. I have to find if I can make their sum after ...
3
votes
2answers
41 views

How many unique counting functions there are for strings of N bits?

Let a counting function inc for inputs of N bits be defined as: ...
2
votes
1answer
41 views

Does a megabyte represent (2^8)^1000000 bits in base2? Isn't that more than enough combinations to represent everything?

Every introductory CompSci resource I can find goes through how 8 bits are in a byte, etc. So a byte can store some 2^8 values in binary. Then when asking about four bytes, we can assume this is (2^8)^...
2
votes
1answer
43 views

Sums of $2^{-l}$ that add to 1

Consider the following problem: You are given a finite set of numbers $(l_k)_{k\in \{ 1, ..., n \}}$ such that $\sum_{k=1}^n2^{-l_k}<1$. Describe an algorithm to find a set $(l'_k)_{k\in \{ 1, .....
2
votes
3answers
120 views

Name of binary encoding scheme for integer numbers

I once found on Wikipedia a nice technique for encoding $k \in (2^{n-1}, 2^n)$ uniformly distributed integer numbers with less then $\log_2n$ average bits/symbol, thanks to a simple to compute ...
2
votes
1answer
26 views

Why does going from 2's complement (in binary) to the positive value by completing to 1 then adding 1 work?

I'm studying Computer science and this has confused me for a long time since our professor didn't give any proof. When changing from 2's complement to the positive value, we can go in reverse (by ...
2
votes
1answer
34 views

Conversion from octal numerical system to binary numerical system

I know that in order to convert a numeral in base-8 to a numeral in base-2 I can write each octal digit as a binary word of 3 bits. I know that 2^3 = 8 that is exactly the dimension of the base of ...
2
votes
1answer
126 views

Two's complement general formula

In college, we've seen some general formulas to compute the decimal value a of a binary representation $a_{N-1} \dots a_2 a_1 a_0$ in two's complement, $N$ being the number of bits. For $a \ge 0$ we'...
2
votes
2answers
177 views
2
votes
1answer
553 views

Binary representation of 129 when using 8bits for two's complement?

I have this confusion regarding binary representation of decimal value 129 (or even 128). If 8 bits are used to represent numbers when doing the two's complement, then we know that '00000000' to '...
2
votes
1answer
24 views

What are the sufficient conditions for weighted binary code to be self compliementary?

I was going through binary codes and found out that if the weighted binary code is self complimentary then it's weights addition will be equal to 9. Is the converse true? If not then what are the ...
2
votes
1answer
64 views

Prove that the next multiple of 4 is obtained using the next formula

I was reading an assembly procedure that needed to align addresses on 4 bytes boundary for performance reasons so it has used the next statement that i formulated as a theorem to be proven. Let $s$ ...
2
votes
1answer
4k views

Confusion in 2's complement of 00000000

I'm solving the end of the chapter problems of Morris Mano's Digital Design (4th Edition, if that's relevant). In one of the problems, it is asked to simply find the 1's and 2's complement of 00000000....
2
votes
1answer
272 views

Binary code and Hamming distance

I'm learning about CRC and Hamming distance and I have three questions. Lets say we have binary code described by ($+$ refers to sum modulo $2$): \begin{alignat*}{1} a_1 &+ a_2 &+ a_3 &+ ...
2
votes
0answers
29 views

How can I optimise a 8b10b encoding for maximum alignment?

Imagine you encode an 8 bit symbol as a 10 bit symbol that is sent sequentially over a wire. The goal at the receiver is to detect the byte boundary. Since there are 4 times more encoded symbols than ...
2
votes
0answers
33 views

If 2's complement is the best way to represent numbers in computers, why worry about BCD and other binary codes?

I'm studying Digital Logic and Design. And in the first chapter, there's all this information about various binary codes. I did some searching on youtube and found this awesome video - https://www....
2
votes
0answers
178 views

K-ary tree that wrapped around

I'm not sure the best way to name this problem, but basically I need to construct a complete $k$-ary tree for $k \geq 2$ which has this nice property as $k=1$ tree that we can create a ring out of it. ...
2
votes
1answer
341 views

What exactly is a “binary file”?

I suspect this may be more of a concept / theory question, but, I haven't been able to wrap my brain around what exactly a binary file is. I've read links such as the below: https://en.wikipedia.org/...
1
vote
2answers
26 views

Why are binary numbers sometimes written with one or more leading zeros that don't change the number (quantity) represented?

Binary numbers like '0110', or '00100101' are seen very often in all contexts. What are the leading (left hand side) zeroes for? Why did the writer not write '110' and '100101', respectively?
1
vote
1answer
88 views

Binary to ASCII using logical operations

I am learning Computer Architecture from Introduction to Computing Systems (2nd Edition) and am stuck on this question. What operations can be used to convert the binary representation of 3 into ...
1
vote
1answer
79 views

How can you represent a Huffman-tree in Hex?

I was trying to store a Huffman tree in a binary file recently. I thought it wouldn't be a big problem since you actually have the binary codes already in front of you. Well, the difference is that ...
1
vote
2answers
41 views

Simple question about signed integer multiplication

If you have a signed integer int x = any And then state that x < 0 Is the following statement true or false? ...
1
vote
1answer
28 views

Where is the theory about “binary toggling games”?

Let us -- using parameters $M, N$ and $L$ -- create an ordered set of size $M$ of $N$-bit long vectors $V$ and initialize them randomly: $V_k[i] = b \sim Bin(n=1, p=0.5)\ \forall i \in \{0\ ..\ N-1\}...
1
vote
1answer
38 views

Given a binary tree $t$, prove that $size(t) < 2^{h(t)}$

I was able to prove that $size(t) \leq 2^{h(t)} - 1$ for any binary tree $t$, however I wasn't able to do anything reasonable with this statement. I know it's a proof by induction and that the ...
1
vote
1answer
698 views

Binary Floating Point Range/Precision for 4-bit Mantissa and 4-bit Exponent

I'm trying to understand binary floating point and using just a 4-bit mantissa and a 4-bit exponent (both 2s compliment) to keep things simple. As far as I can tell, the largest denary number I can ...
1
vote
1answer
169 views

what does “vector” mean in computer science?

I am trying to understand how computers work under the hood, and I came across following sentence. "Computer technology is based on electronic circuits able to process vectors of 0's and 1's. For ...
1
vote
1answer
48 views

Converting a 32-bit bit pattern into floating point number

In The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. i have read something like that, Single precision or 32-bit floating point forms the floating point number,...
1
vote
3answers
27 views

Negative Numbers in 32 bit Floating Point IEEE Numbers

So I understand the logic behind converting positive decimal numbers to IEEE 32 bit floating numbers but I'm not completely sure behind the negative one's. If for example we have a decimal number say -...
1
vote
1answer
25 views

Where does the decimal to binary “Divide by 2” algorithm come from?

I don't know if this board would be an appropriate place for this question, but I have been trying to figure out how the divide-by-2 decimal to binary conversion trick works: in which you divide the ...
1
vote
1answer
91 views

Converting Decimal Numbers between 0 and 1 to Binary

I've been playing around with a program I wrote that converts decimal numbers to binary numbers and i've noticed that eventually, after applying the algorithm (multiply by 2, subtract 1 if greater ...
1
vote
2answers
56 views

Is a file a string of zeros and ones?

Is a computer file just a string of zeroes and ones? If so, can it start with zeroes? I have not been able to find an answer, surprisingly.
1
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1answer
18 views

Conversion of fractional part of a hexadecimal number to binary

The Number is given as : (C012.25)Hexadecimal I have to convert it into octal . So I converted it into Binary First and got the result as : 1100000000010010.01000000 (Since each bit in hexadecimal ...
1
vote
1answer
34 views

2d cellular automata Wolfram binary codes

This is a similar question to How 2d cellular automata rules works? However, the answer there did not provide me what I am looking for. Specifically, I want to be able to render these 2d cellular ...
1
vote
1answer
53 views

How does computer memory store the file name?

I have this doubt for long time. When i save a notepad file, it takes the memory for the information in the file. let's say, I type 'ABC' in notepad and saved the filename as stack, it shows that the ...
1
vote
1answer
24 views

binary representation of ip prefixes

How does an ip prefix 90.0.0.0/4 translate to 0101∗ in binary and 10.0.0.0/2 to ...
1
vote
2answers
305 views

Converting IPV4 address into Binary

So given an IP address it is converted into binary via calculating the binary of all different segments and then joining them. For example, 12.32.18.01 converted into a binary it would amount to this ...