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

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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 ...
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1answer
377 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 ...
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1answer
27 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
38 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?
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1answer
13 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 $y$ ...
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1answer
38 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. ...
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3answers
26 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 ...
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1answer
343 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 ...
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0answers
57 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 ...
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0answers
8 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
20 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 ...
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1answer
303 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 ...
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5answers
138 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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3answers
597 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 ...
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1answer
35 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 ...
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1answer
19 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 ...
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1answer
15 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: ...
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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
180 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 ...
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3answers
126 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 ...
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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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1answer
42 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 ...
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1answer
33 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
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1answer
43 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 ...
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0answers
25 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 ...
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3answers
443 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 ...
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1answer
67 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 ...
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0answers
71 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 ...
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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: ...
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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 ...
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4answers
216 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 ...
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0answers
64 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 ...
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3answers
90 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
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2answers
321 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.
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2answers
51 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 $1,2,4,8$ ...
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5answers
102 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 ...
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0answers
19 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 ...
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1answer
39 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 ...
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0answers
11 views

Is there a standard to follow when writing a weighted binary code (ex. 6-3-1-1 or 2-4-2-1)?

For example, in 6-3-1-1 code, 7 can be represented as 1001 or 1010. Similarly in 2-4-2-1 code, 6 can be represented as 1100 or 0110. My professor said that there is a standard to follow and there can ...
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1answer
26 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 ...
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4answers
1k views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
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2answers
81 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 ...
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0answers
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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 ...
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3answers
37 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.
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1answer
72 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 ...
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1answer
23 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 ...
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0answers
32 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 ...
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2answers
248 views

Longest sequence of 1s in binary representation of a number & Average Sequence length for number with N digits

I originally posted this to rec.puzzles in October 1992. I subsequently solved the problem (at least I think I did...) and published an article about it in early 1993. The article is nowhere to be ...