# Tagged Questions

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

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### Binary arithmetic - overflow and carryout at same time?

In binary arithmetic, When you subtract 2 signed numbers you must discard the carry out. My question is, is it possible for overflow to occur and a carry out? So, on paper there would be two extra ...
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### Convert hexadecimal to binary scientific notation using IEEE 754 single-precision floating point

I am trying to convert these numbers to binary scientific notation, but I cannot figure out the process. Could someone please the process of going about solving this? For IEEE 754 single precision ...
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### What points in $[0,1)$ will have two binary expansions?

What points in $[0,1)$ will have two binary expansions? I know that $\frac{1}{2}$ has the two expansions $0.1\bar{0}$ and $0.0\bar{1}$ $0.1\bar{0}$ is found by starting with $\frac{1}{2}$ and ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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 A+...
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### 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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### 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. ...
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### 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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### 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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### 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. ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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. ?
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### 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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### 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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### 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) + ...
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### 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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### 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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### 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 ...
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### 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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### How to find the number of subsets with a given length and XOR?

I have A (0<A<500000) elements (up to 10^6) in the set. I need to find in how many ways can I remove a subset, the size of ...
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### Why not, first order logic to DNF conversion?

There seems to be huge amount of discussion about converting "first order logic to CNF". But don't see much about "first order logic to DNF" conversion. What is the reason?
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 ...