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

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How do I add multiple binary numbers without using a partial sum?

I know how to add binary numbers but what I normally do is add the first 2 binary numbers and then add the 3rd one to their sum. It is really slow. $$ 111_2 + 111_2 + 111_2 + 111_2 $$ Here is ...
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0answers
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Efficient way to compute the binomial using $(2^k+1)^{k+1}$

The following web page: "http://introcs.cs.princeton.edu/java/78crypto/" (at Exercise 28) effectively says that: "Pascal's triangle. One way to compute the $n$-th row of Pascal's triangle (for $n ...
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0answers
41 views

Is there a way to get the n-th of bits of 2^k

I have a large number N=O(2^k). For simplicity, let's say that N=n^k. However, I only need the n-th bits of N, say for example the 10-th to 16-th bit of N... without calculating the full expansion... ...
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42 views

Binary division algorithm

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1answer
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Relations between NIM-addition and addition

I will note $\oplus$ the NIM-addition. This is a commutative group law. To obtain $a \oplus b$, you decompose a and b in binary, and you sum like this : 0+0=0 ; 1+0=1 ; 1+1 =0 (it's the xor ...
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2answers
118 views

Algorithm for creating binary rational numbers

I read here an algorithm to convert a decimal rational number to binary by multiplications by 2, but although it's very simple to carry on, I still haven't managed to explain myself why it works. ...
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1answer
230 views

Binary expansion

I am trying to get my head around the left and right shift for binary expansion. The rules are: Shifting to the right ...
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1answer
32 views

Coloring a binary tree

Working through a problems practice coloring, I have found a problem that has me stumped. The problem states: For $n \in \mathbb{R}_{>o}$ the binary tree is defined recursively as follows. The ...
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1answer
22 views

Negative Binary Convertion

How do I go about converting -2 decimal to two's compliment in 7 bits? I know in 8 bit representation -2 is 11111110 but for 7 bit I am confused?
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1answer
81 views

Prove a identity with Fibonacci sequence and binary sequence.

Let us: $\{f_k\}$ Fibonacci sequence; $\{u_k\}$ binary sequence i.e. $u_k=$ 0 or 1; $\rho$ real positive number. Is there a binary sequence $v_k$ such that: $$\sum_{k=2}^{n+1}\frac{u_{k-1} ...
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1answer
24 views

property holds for all bases except 2

For any base $n$, for $n > 2$ the following holds: $1/(n-1) = 0.111...$ However in base 2 this doesn't hold. It's just 1. It's obvious why that is you have $1/1$, but I always get uneasy with ...
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3answers
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Product of “reversed” numbers

Consider any 2 binary numbers, e.g.: 10101011 ; 11111101 and their product, say P. "Reverse" (mirror image) all the digits of the 2 numbers, e.g.: ...
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1answer
18 views

squaring binary doubles the nr of bits?

If a is a 10 bit number, then is $$|a^2|=20? $$ And then $$|a^3|=30? $$Or is that at least an upper bound? Does it work this way? I want to explain how with very high probability 4096 bit number is ...
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1answer
26 views

Decimal to binary with limited fractional and exponent bits

I am trying to show the binary representation of a couple numbers using scientific notation. Using 8 bits for the fractional part, and 4 bits for the exponent. The exponent is stored using 2's ...
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1answer
41 views

Simple binary subtraction with decimals

so let's say I am trying to subtract 75.442 by 43.646. I have 43.646 = 00101011.1010, and 75.442 = 01001011.0111 from 2's ...
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1answer
78 views

Why exactly are NAND and NOR the only universal binary logic functions?

We know there are 16 possible binary logic functions: ...
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1answer
12 views

Binary variable syntax?

Is there a more compact styntax to represent binary variables, such as: $3a+2b+c=5$ where $a,b,c$ are either "0" or "1" ? I've tried setting domains, ($a\le1$ and $a\ge 0$, etc) but that only works ...
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0answers
57 views

Why is this number the smallest positive normalised binary value?

In the AQA A2 Computing textbook (Bond and Langfield, 2009), they say that this number is the smallest positive normalised value, given a 10 bit mantissa and a 6 bit exponent: ...
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2answers
49 views

DFA Construction with three strings in language

Draw DFA that recognizes the following language, with the alphabet {0, 1} {0011, 11, 0101} I'm having a lot of trouble with this, because I know DFA have to have a determined path from each state ...
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2answers
968 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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0answers
48 views

Upper limit for adding N numbers in grid of N processors in parallel

We must show how N binary numbers, each consisting of N bits each, can be added in a linear grid of N+logN processors. Suppose that the binary digits can be inserted directly in each processor of the ...
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4answers
1k views

Convert from base 10 to base 5

I am having a problem converting 727(base 10) to base 5. What is the algorithm to do it? I am getting the same number when doing so: $7*10^2 + 2*10^1+7*10^0 = 727$, nothing changes. Help me figure it ...
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3answers
164 views

Base 16 to base 10 number conversion

I know that if we want to convert from base 16 to base 10 we do as follows (for example): Given : $15C$ in base $16$ Conversion to base 10: $12 \times 16^0 + 5 \times 16^1 + 1 \times 16^2 = 348$ in ...
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2answers
24 views

Equal number of 1s and 0s in number of n digits

How many ways could one create a binary number of n digits where the number of 1s and 0s are equal? For example, if n was 8 then we could have: 10101010 or 11110000 In addition to this, I may ...
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0answers
15 views

how to write a lattice $[\alpha,\beta ]$ in the form [$a,b+c\omega _7$]

$\fbox{1}$ if we write [$2-\sqrt{7},5+3\sqrt{7}$] in the form [ $a,b+c\omega _7$],what is the value of $a,b,c$ $\omega=\sqrt{7}$,since $ 7\equiv 3\mod 4$ $N(2-\sqrt{7})=4-7=-3$ $N( ...
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2answers
80 views

Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
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1answer
39 views

Number of Positive Definite Binary Matrices

How may positive definite matrices (over finite field- $F_p$) are possible? What is the criterion in getting those?
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2answers
81 views

What is the relation between this binary number with no two 1 side by side and fibonacci sequence?

I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side. Now as an example ...
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2answers
67 views

Probability of sum over a window of binary vector

I have a vector of one's and zero's of length n with a probability p of observing a one and 1-p of observing a zero. I slide a (overlapping) window of size $k$ across this vector and take the sum ...
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1answer
41 views

My Proof for the Cardinality of a Particular Binary Distribution

my question reads as follows: I have constructed a proof and am concerned about 2 things: 1) The validity of my proof. 2) The construction of my proof. I am asking for someone to read through ...
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1answer
38 views

Distance measures for binary data

I was wondering what are some good distance measures for binary data that have the following properties. I know that there are measures like the Jaccard index and the Dice Index, but they don't ...
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1answer
82 views

How many ways are there to arrange 1's and 0's with no two 1's in a row? [duplicate]

Given n spaces, how many ways are there to fill up the spaces with 1's and 0's such that no two 1's are together. For example, let's say n = 3 (_ _ _). There are 5 ways to fill up the spaces such ...
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1answer
35 views

Cubing a Minterm

I've just been going over some stuff I should actually already know - and I've come to a question that has really stumped me, (my math skill are lacking) and I don't actually know where to begin on ...
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1answer
42 views

Decimal to Binary

I am struggling to understand Decimal to Binary using the following method, given on the second last page of http://www.ling.ohio-state.edu/~scott/teaching/2008/spring/384/handouts/decimal-binary.pdf. ...
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1answer
24 views

Show that the following holds;

Let $h(p) = -p \log p-(1-p)\log (1-p)$ denote the binary entropy of a Bernoulli distribution when the probability of observing a zero is $p$, where $\log$ denotes the logarithm to base 2. Show, using ...
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1answer
803 views

Alien Abduction Brain Teaser [duplicate]

Okay Here is the brain teaser. 10 people have been abducted by aliens because they view them as being representative of the human race. They are going to test them to see if the human race is worthy ...
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0answers
99 views

Why doesn't this base 10 number x mod 2^y work for converting base 10 to binary

Okay I tried to convert 1 million to binary by dividing by a power of 2 and taking the remainder and dividing that by a power of 2 and so on and I got this: 1111010000100100000 Google says 1 million ...
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0answers
73 views

What is the result of the calculation: $8.375 - 0.375-0.875$?

Given the base $\beta=2$ (binary) and $t=4$ digits to represent the number in computer with hidden bit representation and symmetric rounding, what is the result of the calculation in floating point ...
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2answers
55 views

Convert from binary (with floating point) to decimal?

I have to convert the number (111011,01) in binary (2) to decimal (10). I looked up many guides on YouTube but none really explained how it's done, or it's very hard to hear. I would appreciate if ...
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1answer
64 views

What makes a floating point number finite?

We started studying about the binary floating point system, and talked mainly about the limitations when trying to represent fractions on computer systems, and how mostly what we get when manipulating ...
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3answers
87 views

How to borrow from zeroes in binary subtraction (no complements)b?

I know you can use complements to subtract, but I want to subtract these two binary numbers without that. I am unsure of how to borrow numbers from zero, check it out: 1000000 -101100 How do I ...
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1answer
26 views

Partioning Mystery

Who has the wisdom to answer the following: 9 distinct marbles distrubted into 4 distinct bags with each bag receiving at least 1 marble,how many ways can this be done? Thankyou for contributing! ...
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1answer
43 views

What's the proper way to subtract a negative number with two's complement?

Title says all. Is it just ordinary addition? Because that seems like it's too simple.
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54 views

Biased form with hidden bit

A computer stores a number of 16 bits word using floating-point arrangement. Given that the first bit is reserved for the sign and followed by 6 bits for the exponent using biased form. The remaining ...
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1answer
179 views

Time efficiency of brute force algorithm as a function of number of bits?

This is homework help so advising how to solve such a problem is appreciated. The question reads as follows: What is the time efficiency of the brute-force algorithm for computing $a^n$ as a ...
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1answer
106 views

Data Representation Question

A computer stores a number of $16$ bits word using floating-point arrangement. Given that the first bit is reserved for the sign and followed by $6$ bits for the exponent using biased form. The ...
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2answers
175 views

Binary Code Decimal (BCD)

I need help with the following question, my try is at the bottom: Question : Using Binary Code Decimal 8-4-2-1 representation, calculate 6789 + 7156 - 365 My Answer : 1101010000101 + 1101111110100 = ...
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1answer
63 views

Base change system

the people in aldeeran planet use a base six system. a]show how the aldeeran people represent each of the following: i) 291.5 (base 10) ii) 1010111 (base 2) iii) 71.6 (base 8) iv) 10A.4 (base 16) ...
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1answer
211 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
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conversion of number from base 10 to base 16

I have trouble finding an example of a conversion of a number from base 10 to base 16, where there is a loss of significant digits when using the Euclidean algorithm due to division by large numbers, ...