# Questions tagged [number-systems]

Representations of numeric values in decimal, binary, octal, hexadecimal, and other bases; one's-complement and two's-complement signed numbers; scientific notation; floating-point numbers in digital computers; history of number systems; nonstandard number systems; algorithms for arithmetic within specific number systems or for conversions between number systems.

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### Perhaps the Problem of Incomputability Is Based on a Flawed Counting System? [closed]

I've been thinking a lot about incomputable problems, which really serve as a pet peeve of mine. The fact that such simple equations as powers and roots straight up cannot be calculated is super dumb, ...
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### How could we find the HCF of two numbers a and b without using the The Factorisation Method That we normally Use!(i meant any other new methods!)

So, How could we find the HCF of two numbers a and b without using the The Factorisation Method That we normally Use!(i meant any other new methods!) In other words let's take an example on finding ...
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### Formalizing daily use of integer representation

In daily life, it is common practice to use a sequence of number elements as an integer, e.g., 999 is a decimal number. As I am reading rigorous construction of numbers from Rudin's mathematical ...
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### How to expand $\pi$ in non-decimal bases?

I was reading about $\pi$ converted into other base systems. We all know $\pi$ in base $10$: $$\pi_{10} = 3.1415926535897932384626433(...)$$ but how has this been converted into, say, base $11$, where ...
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### On the bounds for number of digits of $n$ in factorial base representation in relation to number of digits in base-$b$

For an integer $n$ written in base-$b$ let $|n|_b$ denote the number of base-$b$ digits (i.e. from the set $\{0,1,2,b-1\}$). Also, let $|n|_!$ denote the number of digits of $n$ written in factorial ...
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### how many numbers are needed in complex number bases?

I was reading about the so called "Quater-imaginary base" numbers, meaning numbers with base $2i$, and I was wondering if it's possible to have a system with a general number base of $a+bi$, ...
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### Alternative definition of pairing function such that the numbers $\pi(k_1,k_2)$ comprise a field analogous to $\mathbb C$

Consider the pairing function $\pi:\mathbb{N\times N\to N}$ as given by wikipedia/Pairing_function, and let's just write $(k_1,k_2)\in\mathbb{N}\$ for $\pi(k_1,k_2)$. Then the canonical definition, ...
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### octal representation of '-1' on spreadsheet [closed]

I'm somewhat befuddled by this, but the decimal number '-1' converted to octal is apparently 7777777777 on spreadsheet, feels incorrect and is not straight-forward to calculate in the usual way (...
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### is my proof of the fact that every real number can be represented by a decimal expansion correct?

assume that S is a set of real numbers that don't have a decimal expansion. If S is not empty, then it must have a least element r according to the well-ordering principle. r can be expressed as the ...
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### storing decimal number into computer with finite mantissa

I am learning about numerical methods and the following link caught my attention: https://www.iro.umontreal.ca/~mignotte/IFT2425/Disasters.html So from what I understand 0.1 is not exactly ...
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### Are numbers such as 3.89000.. With reoccurring 0s neither irrational numbers nor reoccurring numbers?

I think the title sums it up pretty well. I asked my friend who has a degree in mathematics "To put it another way would you agree numbers such as 3.8900000.... With reoccurring 0s is not a ...
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### Generalization of a problem involving the conversion between two or more objects

I have come across an interesting elementary school math question which I would like to generalize. The problem is roughly as follows: There are $4$ apples and $5$ bananas. $3$ apples can used to ...
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### Why does the addition of signed integers 0xA3 + 0xF9 NOT produce overflow, but the addition of signed integers 0x9F + 0xA3 does?

I presented the Hexadecimal values for these integers in the title, just to preserve space, however I am actually trying to add them in Binary. The overflow conditions (from what I could understand) ...
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### An interesting way to explain commutative property of multiplication

Five times three is equal to three times five due to the commutative property of multiplication. Is there any interesting way to explain why this is so?
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### Knowing that the sum $1014_p + 216_p = 1232_p$, for a given base p, is correct. Which base is p?

I would like a lot to know what exactly I have to study to solve this kind of question. That was from my discrete math class, but I've been missing classes, so I don't know which subject is it exactly....
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### Choosing a decimal representation in a given domain of discourse.

Say I limit myself to a domain of discourse in the the natural numbers, would it be incorrect to use a decimal representation like $2.000$ as opposed to just $2$ or $02$? Does the use of the decimal ...
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### Cubic number system

In $\Bbb{H}$, the numbers $1$, $i$, $j$, $k$, and their respective negatives can all be seen as the vertices of a cross-polytope or orthoplex. This is true for anything that uses an orthonormal basis, ...
258 views

### Trirational numbers in the complex plane (and generalizations)

(Note: I've posted my own answer, slightly redefining trirationals to be composed of reals instead of integers and addressing the problems pointed out here. Please take note of this while reading my ...
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### Number systems and Cardinality Question (to admit a bijection)

I am trying to complete the following proof: Show that the real numbers are unique, in the sense that any complete ordered field admits a bijection with R that preserves addition, multiplication, and ...
1 vote
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### What is the formula for calculating the number of bits required to encode the number $x$ in Fibonacci coding?

What is the formula for calculating the number of bits required to encode the number $x$ in Fibonacci coding? For example: if $x=1$ then because $1$ in Fibonacci coding is "11" therefore ...
1 vote
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### Determine the number base $x$ if $(59.5)_{(x)} = (89.3125)_{(10)}.$ [closed]

How can I determine the base number system $x$ of two decimal numbers using an equation?
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### In which base number system is $23^2=562$?

This is an exercise from a math chapter called number systems where I learn how to convert decimal numbers to other number bases like binary and hexadecimal. Here I need to discover which base number ...
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### Is there any sort of relationship between the terms "base" ( number systems) and "basis" ( linear algebra) as both have similar definitions.

I was going through the topic of number systems today when I went to see the proper definition of a base. It said "the number of digits or combination of digits that a system of counting uses to ...
170 views

### Can we have a number system whose base value is r=0. [closed]

Is it possible to have a number system with 0 as its base value? I haven't found any explanation as to why only numbers more than 1 is taken as a base value. For example, binary: base value is 2. ...
29 views

### Non-terminating representations of rational numbers in base factorial

What do non-terminating representations of rational numbers look like in base factorial? In particular, are there any representations $d$ of rational numbers such that $d$ is not terminating and the ...
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### Help define laws/patterns for 2D scalar number system arithmetic - tetracoords

Tetracoordinates as a 2D scalar number system Math is generally not my specialty, but as a programmer I've been entertaining the idea of a 2D scalar number system (called tetracoordinates for now). ...
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### Is (P, +, . , < , 1) a subsystem of (I, +, . , <, 0, 1)?

I'm reading The Number Systems — Foundations of Algebra and Analysis, second edition, by Solomon Feferman. I have a doubt regarding the statement of theorem 4.22, chapter 4. Theorem: --- There exists ...
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### Decimal to base 1

Explain why base 1 radial expansions are impossible. I think If b is a natural number greater than 1, to write the abbreviated base b radial expansion the number of symbols required is b. For example, ...
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### Are there any obvious flaws to representing relative factorials of each hyperoperation this way?

Note: I use the term Hyperfactorial (relative factorial) differently than used by wikipedia and wolfram. These sources use the term to describe what I am describing here as exactly (and only) the ...
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### How does shifting a binary number to the left translate to its hexadecimal representation?

So I have a binary output that is represented in Hex that I need to format it's binary representation in a certain way. Say I have a binary output of: 000110111001 or 0x01B9 in HEX. I want to add 1 ...
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### Is there a simple test for divisibility by seventeen in base-twelve? [duplicate]

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
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### Is there a simple test for divisibility by sixteen in base-twelve?

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
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### Finding all numbers such that this algorithm calculates their square

A relative of mine found an algorithm on TikTok that could supposedly calculate the square of any two digit number. The number 35 was used as an example, so I shall use it to explain how it works: ... ### The first $0$ in a base $b$ expansion
Consider the following function $Z \colon \Bbb{Z}_{\ge2} \times [0,1) \to \Bbb{N} \cup \{\infty\}$. If $b \in \Bbb{Z}_{\ge2}$ is an integer greater than $1$ and $x \in [0,1)$, let \$Z(b,x) \in \Bbb{N} \...