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

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6
votes
4answers
412 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 ...
1
vote
1answer
30 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
vote
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 $1,2,4,8$ ...
0
votes
0answers
30 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 ...
5
votes
5answers
409 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
1answer
50 views

Probability of 2 n-bit binary strings lining up, given that the first k bits do so

There are two rows of 10,000 bits, one on top of the other, currently all set to 0. 00000000000000000... 00000000000000000... Now imagine that in each row, 40 of the bits are randomly flipped to a ...
0
votes
1answer
29 views

A property of permutation codes

For $k\ge2$ and $M\ge k+2$ two integers, a permutation code matrix $C$ is a $\binom Mk\times M$ matrix which columns contain all distinct permutations of $M-k$ zeroes and $k$ ones. Page 44 of his ...
3
votes
2answers
83 views

How to populate a $0-$line with $1$'s?

I have a line of $n$ $0$'s like this: Zeroth index -->$000...000$ I want to populate the line with $m$ $1$'s with the following rules: (1) They have to occur after the index ...
1
vote
0answers
45 views

Learning Booth's Algorithm, I Can't Find the Issue on Final Result

I am practicing using Booth's Algorithm to multiply a positive number and a negative number (specifically the problem is $-12 \times 4$). I have included my attempt, but I can't find the issue. If ...
0
votes
3answers
287 views

Efficient algorithm for taking powers of binary numbers?

Is there any efficient algorithm for taking powers of binary numbers? Or even just squaring one? Thanks ahead of time.
4
votes
1answer
511 views

Why do we divide or multiply by 2 when converting binary?

Trying to understand the fundamentals of binary rather than just following steps, I wanted to know why do we multiply by 2 to convert a decimal (0.5, 0.25) to a binary and why do we divide by 2 when ...
1
vote
0answers
46 views

Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
0
votes
1answer
66 views

Division binary using polynomials

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
0
votes
1answer
94 views

Computing the Value of a minimax tree

I am asked to compute the value of a minimax tree, which each node labeled with its initial value. I am just unsure how to do it. I know that it is a minimax tree if: the root is a min node, the ...
0
votes
1answer
29 views

Probabilities of a Binary String

You generate a random $N$-bit binary string, and compute $X = \Sigma_1^N x_i$, where the $x_i$ are the $0$ and $1$ entries of the string. What is the probability that $X$ is odd, if $N$ is odd? What ...
0
votes
3answers
153 views

the blood test riddle (number theory)

A microbiologist has been given a set of $100$ blood vials. Exact one of those $100$ vials is positive to a concrete disease X. The microbiologist desires to send only $7$ vials for analysis. He can ...
0
votes
1answer
48 views

Bitwise ops - The relationship between $a$, $b$, $a \wedge b$, $a \vee b$ and $a \oplus b$

In computer programming, the term bitwise operation is used to denote the use of boolean operators (and $\wedge$, or $\vee$, exclusive or $\oplus$) on corresponding bits of two numbers. Bits, in this ...
0
votes
1answer
51 views

I'm trying to convert a number to binary. What am I doing wrong?

So I'm trying to convert a number to binary, but I keep getting the incorrect answer. If it matters I was roughly following this guide. Here is my work: $120 = 60 + 60$ $120 = 64 + 32 + 16 + 8$ ...
0
votes
0answers
66 views

Is there an alternative encoding scheme to binary where similarity of pattern correlates with size of number?

If I compare binary for 7 111 and binary for 8 1000 there is no correlation between these two patterns that suggests that ...
2
votes
3answers
609 views

Determining if a relation is reflexive, symmetric, or transitive [closed]

Let $A = \{0,1,2,3\}$ Define a relation $T$ on $A$ as follows: $T = \{(0,1),(2,3)\}$ Is $T$ reflexive? symmetric? transitive?
0
votes
3answers
2k views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
0
votes
1answer
142 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
4
votes
0answers
98 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 ...
1
vote
0answers
77 views

How to determine size and height balance of binary search tree?

I've been reading/ learning binary search trees and I've been stuck on the following question for a while now. I have the following tree, how do I determine the height and size balance of it? How do ...
1
vote
2answers
138 views

Recursively defining sets of strings discrete math

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set K? Can someone ...
-1
votes
1answer
125 views

Binary strings and recurrence relations [closed]

So the problem is: How many binary strings of length n contains 111? Give a recurrence relation Tn, where Tn is the number of binary strings of length n that contains 111. How could we possibly ...
1
vote
1answer
99 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
0
votes
1answer
44 views

Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
0
votes
3answers
48 views

Binary remainder not equal to the decimal remainder

I am having a weird result. I am dividing the binary number $10101010100000$ by $10011$. In binary division. I get $R= 0100$ which is 4. However, If I consider the decimal representation of the ...
0
votes
1answer
51 views

Binary Representation of the Collatz Conjecture

What is the benefit of looking at the binary representation of the collatz conjecture. I know that it makes the computation easier because there is really one operation involved which is multiplying ...
0
votes
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 ...
2
votes
1answer
61 views

What is the minimum longest repeated substring of a binary string of size n?

The longest repeated substring of 0111011 is 011 for example. My question is given the size of a binary string, what is the shortest this longest repeated substring can be. I have computed values for ...
0
votes
1answer
273 views

Evaluating the decimal equivalent of binary numbers in; sign and magnitude, one's complement and two's complement

For example, i have this binary number : 1011 1101 Now i wish to evaluate the decimal equilant using sign and magnitude, one's complement and two's complement. Now for sign and magnitude, i know the ...
1
vote
1answer
36 views

Why Are The First and Last Entries of a Binary Reflected Gray Code Sequence Adjacent?

Some Information on Gray Codes If you're rusty or unfamiliar with Binary Reflected Gray Code, but want to try and help, here is a youtube tutorial explaining what they are: ...
3
votes
1answer
107 views

Proofs involving some general formulae for trees and binary trees.

So here I have 3 tree-related questions. 1) Let $n\geq2$ and let $d_1 ≤d_2 ≤···≤d_n$ be a sequence of integers. Show that there is a tree with degree sequence $d_1,d_2,...,d_n \Leftrightarrow \sum ...
3
votes
1answer
116 views

Disjoint subsets and Number of 1's in the binary representation

For a subset $S$ of $[n]$, let $\chi(S)$ denote the $n$ bit 'characterisitc vector' of $S$, i.e., $\chi(S)=(a_1, a_2, \ldots, a_n)$ where $a_i=1$ if $i \in S$ and $a_i=0 $ if $i \notin S$. Think of ...
1
vote
2answers
50 views

Base-n arithmetic and multi-dimensional matrices

Interesting thing about binary numbers is to find their decimal value you can represent them as a multidimensional array, where each cell is indexed, starting from 0. For simplicity, let's start with ...
0
votes
0answers
30 views

Find minimum algorithm complexity

So, I have this task: Let us have square matrix $A \ size\ n\ \times\ n$ for which is true: $$A(k,l)<=A(m,p)\ if \ k <=m, l<=p$$ I need to find algorithm, which finds value X in such matrix, ...
1
vote
1answer
54 views

How Leibniz invented the Binary System?

Do you know which reasoning and observations made Leibniz invent the Binary system ? Some say that he was inspired by Chinese mathematicians do we have any record of how he came with this idea ?
1
vote
3answers
516 views

Convert from binary to quinary

How to convert a number from binary to quinary system without using decimal system ? It is possible ? I want to write a program who does it.
10
votes
2answers
370 views

Sum of series with binary parity in the numerator

I'm now stuck with this question, and I don't even know where to start: Find sum of series$$\sum_1^\infty \frac{f(n)}{n(n+1)}$$, where f(n) - number of ones in binary representation of n. I wish I ...
0
votes
0answers
31 views

Determining whether two trees are isomorphic

Is there a (probably recursive) algorithm that can be used to determine whether two not necessarily binary ordered (sub)trees are isomorphic or not?
-2
votes
2answers
2k views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...
1
vote
2answers
97 views

How to check quickly $\frac{2}{3}=.101010… $ holds?

Every $x \in [0, 1]$ can be expressed in the form $\dfrac{a_1}{2}+\dfrac{a_2}{2^2}+\dots + \dfrac{a_m}{2^m}+\dots$ , where each $a_i$ equals either $0$ or $1$. For such $x$, we have the binary ...
0
votes
1answer
823 views

Probability of a binary communication system

A binary communication system is used to send one of two messages: message A is a sequence of 0s with probability 2/3, message B is a sequence of 1s with probability 2/3, The ith received bit is ...
1
vote
3answers
82 views

modulus to find 10-bit samples in 8-bit memory

For one of my electronics projects I am using an EEPROM chip that holds up to 512k bits of information, arranged in 8-bit (1 Byte) address registers. Multiply 512 x 1024 and divide by 8 and you'll get ...
0
votes
0answers
32 views

What are the possible relations?

If there are two sets A and B. Set A has two elements so does Set B. How many different possible relations can we have from A to B. For example. Set A has (David, Max) and Set B has (x,y). Would the ...
1
vote
1answer
45 views

Why is the enumeration of binary strings misaligned?

If you enumerate the set of binary strings in ascending order, you get: ...
0
votes
1answer
47 views

haw to find the values of binary numbers in octal?

what is the value of 101111.0111 of base 2 (binary) in octal(base 8) we know that in order to translate from binary to octal we need to group the bits in groups of three and the value is going to be ...
0
votes
3answers
35 views

Binary subtraction A0 - E4

I have two numbers. The negative one is translated using 2's complement. $$ a = 160_{10} = A0_{16} = 1010 0000_{2} \\ b = -28_{10} = E4_{16} = 1110 0100_{2} $$ And I have to perform the following ...