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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Looking to improve an algorithm for decomposing an integer

I'm interested with primorial number system. In this playful setting, out of curiosity and for relaxation, as an amateur, not knowing the current algorithms for decomposing a $2$-almost prime number $...
Stéphane Jaouen's user avatar
3 votes
0 answers
84 views

How do you multiply in Primorial number system?

Primorial number system is a number sytem that uses primorials which are defined as follows : Let $p_1=2, p_2=3,p_3=5,p_4=7,p_5=11,...$ the primes. The sequence of primorials, noted $p_n\#$ is $$(p_n\#...
Stéphane Jaouen's user avatar
2 votes
0 answers
141 views

Help to mathematically theorize the eventual value of Primorial number system

The primorial numeral system is a numeral system whose interest I am trying to establish mathematically. Unfortunately, my mathematical knowledge is limited, so I would like to briefly present some ...
Stéphane Jaouen's user avatar
1 vote
1 answer
50 views

Using Fake Numerals to Make Real Decimal Numbers [closed]

The setup to this question is very simple: Take the numbers $\frac{1}{1}, \frac{21}{12}, \frac{321}{123},...,\frac{987654321}{123456789}$ and plot them versus the natural numbers, as seen here: https:/...
Gabriel Turner's user avatar
13 votes
4 answers
2k views

Can a decimal that is infinitely repeating in one base be nonrepeating in another?

For instance, can a number like $0.1111111\cdots$ in base $3$ be represented as $0.23515613\cdots$ (non-repeating) in base $8$? I imagine the answer would be a resounding NO but it would be ...
theboombody's user avatar
2 votes
2 answers
138 views

What is this field in $\mathbb{R}^4$ that contains both the real and complex numbers called?

Note: this question is wrong – this is not a field, though it is not obvious why it wouldn't be. So, I (first year undergraduate mathematics student) was looking around the internet and found an ...
jkan5855's user avatar
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0 answers
55 views

On average, how many more digits do octal numbers have than decimal numbers? [duplicate]

My question is, how could I calculate the expected value of the number of digits of an octal representation of an $n$-digit decimal number? How can this question be answered in general for two ...
minseong's user avatar
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1 vote
2 answers
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Reduce the base $11$ fraction $\dfrac{587}{749}$ to its lowest terms.

Reduce the base $11$ fraction $\dfrac{587}{749}$ to its lowest terms. $(\dfrac{587}{749})_{11}=\dfrac{5\times 11^2 + 8\times 11 + 7}{7\times 11^2 + 4\times 11 + 9}$ But $\dfrac{...+7}{...+9}$ can't ...
ronald christenkkson's user avatar
0 votes
2 answers
66 views

Question about the formula for switching bases of numbers

To express the base $10$ number $5213$ in base $7$, all I have to do is simply divide by $7$ as follows: $5213 \div7=744$ with remainder $5$ $744\div 7=106$ with remainder $2$ $106\div7=15$ with ...
ronald christenkkson's user avatar
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0 answers
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Benefits/uses of non-base 10 number systems?

For reference, I'm studying math and anthropology at university, and I've been dying to find some overlap of math theory and ethnomathematics (math uses/tools/systems/etc in other cultures). I'm ...
Rhinestone's user avatar
-1 votes
1 answer
143 views

Expressing Numbers Without Any Decimal Presumptions

I have long been uncomfortable with how numbers in alternative bases are expressed. Alternative bases are marketed as transcending our arbitrary base-$10$ conventions, but I wonder if they really ...
user10478's user avatar
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1 vote
1 answer
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Thoughts about allowing arithmetic with infinitesimals to (mostly) solve limits, can it be done without contradictions?

I am currently working on a project for school where I would like to create an arithmetic framework that would make it easier to solve limits. Suppose we have a function $f:\mathbb{R}\smallsetminus \{...
naytte2's user avatar
  • 454
-2 votes
1 answer
63 views

4's complement of $(23)_{4}$ [closed]

I am reading up on how to find the complement of a number of any base.I picked out the number $(23)_{4}$ . This is a 2 digit number with base 4 so the complement is $4^2-(23)_4$.This is equivalent to $...
Root Groves's user avatar
1 vote
1 answer
110 views

Is there a name for this digit?

I am currently working on a problem where it is useful to consider the index of the last digit in a number which is not a $9$. For example in the number $$a=8243848392842992999,$$ the index I care ...
tekay-squared's user avatar
4 votes
1 answer
90 views

Sum of digits of a perfect square in other bases

Sum of digits of a perfect square in other bases I noticed that in base ten, the sum of digits of $n$ is equivalent to $n\pmod{9}$, and if a number in decimal is perfect square, then the sum of digits ...
Thirdy Yabata's user avatar
1 vote
4 answers
76 views

Transform $1001.12211$ from base $3$ to base $9$.

Transform $1001.12211$ from base $3$ to base $9$. Separately transform the integer and non-integer portion. The integer portion is divided by $9$: $$ \begin{array}{ r|cc } 9&1&0&0&1 \\...
ronald christenkkson's user avatar
1 vote
1 answer
59 views

Convert $22202_3$ (in base $3$) into base $11$.

Convert $22202_3$ (in base $3$) into base $11$. I know I can just convert into base $10$ and divide by $11$. $2\times 3^4+2\times3^3+2\times3^2+2=236_{10}$ Then repeated division by $11$ yields $(1A5)...
ronald christenkkson's user avatar
0 votes
1 answer
86 views

Converting a number from base3 to base2 without going through base 10.

Could anyone guide me on how to directly convert a number from base 3 to base 2 without using base 10? For example, converting "2101" (base 3) directly to base 2. Any suggestions, method or ...
a.moussa's user avatar
  • 103
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1 answer
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write the following in sigma notation $a^k+a^{2k} +a^{4k} +a^{8k} +a^{{16}k} ...+a^{{256}k}$

I see the pattern being $2k$ on the power, and I thought the formula would just be $a^{(2k)}$, but i feel like im missing something else here ? Why is it not The summation from $a=1$ to $128 = a^{2k}$...
dith659's user avatar
2 votes
1 answer
120 views

Binary with digit 2 allowed

How many ordered 11-tuples $(a_0,a_1,a_2,...,a_{10})$ of integers satisfy the equation $$a_0+2a_1+2^2a_2+...+2^{10}a_{10}=2020$$ where $0\le a_i\le 2$ for all $0\le i\le10$ AMC Mock If $f(n)$ is the ...
Starlight's user avatar
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1 vote
2 answers
51 views

What if the number ten was a single digit in the base 10 number system? How would that work? [closed]

I've recently been studying about math, and when learning about decimals. I found myself with this question. An explanation I got from Quora was: Arithmetic operations such as addition, subtraction, ...
Stim Roe's user avatar
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0 answers
27 views

Addition and multiplication of integers

Reading the book Analysis I (Fourth Edition), by Terence Tao, I found, on page 78, Lemma 4.1.3 (Addition and multiplication are well-defined.) The Lemma says: Let a, b, a', b', c, d natural numbers. ...
Paulo Argolo's user avatar
  • 4,220
0 votes
1 answer
48 views

the logic of numeral systems [closed]

i have a question about numeral systems that use the place value concept , in the decimal system the position is a power of ten and it has set of ten unique digits, the ...
maryam's user avatar
  • 21
3 votes
0 answers
178 views

Digit Sum for Base 2 Alternate Definition

I was looking at the digit sum definition, but I also saw a simpler version that can be used for binary numbers. I'm trying to figure out and understand how the generic formula for any base can be ...
denvaar's user avatar
  • 131
4 votes
1 answer
119 views

Ancient Greek Mathematical Notation

While reading the CMU FLAC overview, I stumbled upon the above image. While I have seen Greek numerals before, I had no idea that there was an entire algebraic symbology developed in Greek (though ...
William Ryman's user avatar
0 votes
1 answer
55 views

Number systems and ring Theory

Are different base number systems defined as separate rings or single ring e.g Z? asking question in other words decimal numeral system different fundamentally from unary or binary or ternary or any ...
Sage's user avatar
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1 answer
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Does $n = n.\overline 01?$ Similar to $(n-1).\overline 9 = n$ which is just coming form the other direction of $n$.

Just like what the title says. Does: $$n = n.\overline 01?$$ For example, $1.\overline 01 = 1$? Similar to $(n-1).\overline 9 = n$, for example, $0.\overline 9 = 1$. The last statement is true but ...
MathCubes's user avatar
  • 119
5 votes
2 answers
437 views

Similarities in the digits of the powers of 2 and 5

Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2: This pattern intrigued me. I started to wonder if it exists in different number bases. I soon ...
Etienne's user avatar
  • 87
0 votes
0 answers
61 views

Using modular arithmetic to explain end around carry

I understand that end-around carry occurs because the diminished-radix complement coding has two representations of zero, but I'm having trouble understanding how it follows from the rules of modular ...
user51462's user avatar
  • 673
4 votes
1 answer
222 views

Are there number systems with fractional or irrational bases?

I'm wondering if there are number systems with bases other than integers? For example, with a fractional, imaginary, irrational, transcendental basis, or with the basis "infinity"? If there ...
Глеб's user avatar
  • 151
0 votes
0 answers
53 views

Weird way of converting to a higher base

I used to think that there is only one to convert to a higher base ie: https://math.stackexchange.com/a/111158/407302 If we want to convert $(1200)_3$ to $(45)_{10}$ we do the following: $1\cdot 3^3 + ...
ng.newbie's user avatar
  • 1,017
3 votes
2 answers
59 views

Representing a 10 radix as a $10^k$ radix - How will the arithmetic and representation differ?

Hi I am trying to understand how write code that deals with large integers greater than the predefined numeric datatypes (ie much greater than 64 bit integers). I have observed that most libraries ...
ng.newbie's user avatar
  • 1,017
0 votes
0 answers
54 views

Uniqueness of negative and positive digit representation of integers

The question is regarding to the python code for following problem. We consider two type of digits, negative and positive to represent integers in given base. We are presenting theorem (self made), ...
Pruthviraj's user avatar
  • 2,707
0 votes
0 answers
31 views

Algorithm to find a rational number between i and j, minimizing for number of significant figures after n iterations

I have a program where, a user selects a number $i$ from a list of floats, and the program adds a number k to the list such that $i < k < j$ where $j$ is the upper neighbour of $i$ in the list (...
Ventsi Radev's user avatar
0 votes
0 answers
96 views

a new numeration system

Consider the sequences recursively defined by the function: f(n)=(3^n+1)/2 such that: ...
Giovanni Russo's user avatar
0 votes
3 answers
96 views

Number system that provides alternative to real numbers (mentioned by 3Blue1Brown), wha't the name?

https://youtu.be/U_lKUK2MCsg?t=171 Grant Sanderson (3Blue1Brown channel) (auto-generated subtitles): I think the way that we extend to the real numbers there's a little bit of choice in that so there'...
Martian2020's user avatar
-2 votes
1 answer
74 views

How to prove a fraction $\frac{a}{b}$ is terminating or repeating in any base without dividing. [duplicate]

So, I would like to know how to know if fractions like $\frac{51632}{2345}$ or $\frac{6}{25}$ are terminating or repeating in any integer base $N$, not just limited to decimal, such as hexadecimal or ...
ProgrammerInProgress's user avatar
4 votes
1 answer
204 views

Have generalisations of dual numbers/complex numbers/quaternions/octonions... been studied?

Can anyone point me to any generalisations of the notions in the title? For example say you have: $$ (a_1, a_2, a_3, ...,a_n) \in \mathbb{R}^n $$ and $$ \gamma_1, \gamma_2, \gamma_3, ..., \gamma_n $$ ...
Skepta's user avatar
  • 63
1 vote
1 answer
70 views

Can the staircase paradox be resolved using divisions of the square legs into real numbers instead of rational numbers?

Every time I’ve read about the staircase paradox it refers to dividing the horizontal and vertical legs of the square by repeated integer divisions. Their lengths are of the form a/b, with a usually ...
GaryW's user avatar
  • 31
1 vote
0 answers
178 views

Do we need a new number system for inverse tetration?

I know that the simplest number system is the natural numbers ($0, 1, 2, …$). And while we can easily define addition and multiplication for naturals, subtraction doesn’t work. Because what would $2-3$...
Zachary Sakowitz's user avatar
0 votes
0 answers
50 views

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 ...
Ziqi Fan's user avatar
  • 1,816
0 votes
2 answers
42 views

Base conversion doubt

(123) in base 8 can be converted to base 10 as follows => $3*8^{0}+2*8^{1}+1*8^{2} = 83$ But when I do the conversion by taking a group of 2 numbers together like $(01)*(8^{2})^{1} + (23)*(8^{2})^{...
Fin27's user avatar
  • 958
0 votes
0 answers
47 views

Determining if a positional notation can express every natural number

Recently I've researched some positional notations. However, some of them I've invented can't express every natural number. I want to determine which of them can express every natural number, and ...
Alex-Github-Programmer's user avatar
0 votes
1 answer
99 views

Equation with numbers and subscripts

I have no clue how to find the answer in the underlined spot? What does the subscript $9$, $6$ and $3$ mean here? $$888_9+555_6+222_3= \_\_\_\_\_\_\_\_ {}_{12}$$
wonderful world's user avatar
0 votes
0 answers
89 views

Number system where negative numbers are represented as letters

Forgive me for a possibly dumb question, I'm very shallow at math, but is there any number system (or I don't know, it looks more like a number mask system) in which negative numbers are represented ...
Paul Rukavishnikov's user avatar
0 votes
1 answer
72 views

In the unary numeral system, how are the numbers between 1 and -1 labelled?

In the unary numeral system, does 1 - 1 = -1 ? Because if this is true, then how would the numbers between 1 and -1 be represented? Where on a number line would .1 and -.1 be? I am not necessarily ...
William Solomon's user avatar
0 votes
0 answers
325 views

Correct way to find 8's complement?

I'm trying to figure out how to find 8's complement, but am unable to do so. Online it shows that we need to subtract each digit from 7 when the number is in it's octal bit. It gives the 7's ...
crimsonKnight's user avatar
2 votes
1 answer
97 views

Is there a classification system where the sum of the digits are a factor?

This may be confusing. An example: 12 1+2=3 3 is a factor of 12. 135 1+3+5=9 9 is a factor of 135. 555 5+5+5=15 15 is a factor of 555. What is the name of this classification system or did I create a ...
Sumaesioso's user avatar
4 votes
2 answers
220 views

Negative golden ratio $(–φ)$ as a number system base?

As negative numbers can be used as bases for numeral systems (e.g. negadecimal), and non-integers such as the golden ratio $φ$ can also be used as bases, I have tried to find information on whether ...
Anypodetos's user avatar
0 votes
1 answer
43 views

Quaternary Sequence Generation

How can a quaternary sequence be generated without converting decimal numbers to quaternary. For example, I need to generate at n = 2 , 4^n = 16 combinations, enter image description here.For large n, ...
Sparxes's user avatar

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