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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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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Confusion about q-ary system in Zorich's Mathematical analysis

For $p \in \Bbb Z $ there is a q-ary system described which assigns to each real number x in the base q a sequence of {$\alpha_n$} such that $\sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i} \le x \lt q^{p-n} +...
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How to know without factoring whether a division will result in recurring decimals [duplicate]

Assumption: When dividing an integer n by another integer m, and representing the result as a decimal in base ...
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How can I prove the triangle inequality and the given corollaries? [duplicate]

How can I prove the following 3 claims? Or, if possible, could you provide suggestions for how I could prove them? For real numbers $x$ and $y$: Claim 1. $|x| − |y| ≤ |x − y|$ Claim 2. $|x-y| ≤ |x|+|...
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1 answer
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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: ...
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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} \...
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How can I adapt the long division algorithm to other bases?

How can I use "long division" in such a way as to get the n-ary expansion of a rational number instead of the decimal expansion? E.g. what would the long division of 866/5 look like in base ...
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What is the number base, where you would need at average add/remove the minimum amount of numbers to round the value you want?

As some example, at base 10, if you want to round to nearest number with last X digits being 0, you will need to add or remove at average a value of (10^X)/4, where X is the amount of digits you want ...
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Has anyone invented 60 symbols for base 60?

The glyphs $0,1,2,3,4,5,6,7,8,9$ represent the natural numbers from $0$ to $SSSSSSSSS0$. With them, we can write the base-10 representation of any natural number. However, has anyone invented $60$ ...
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Factorial Number System with Repetitions

Good Day I know about how Factorial Number System is useful to find permutations and also find the the index of a permutation. For example, permutation $[1, 3, 4, 0, 2]$ has index ($0$ - based) $1 \...
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Multiplicative inverse of a non-zero element "a" belongs to integers [closed]

Multiplicative inverse of a non-zero element a ∈ Z is _______. 1 -a 1/a Not defined I have search about this and found two references Reference 1 and Reference 2 and their answers are 3 and 4 ...
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How do you convert from 10s complement to SIGNED decimal?

A bit confused. I am preparing for an exam and one of the practice problems asks us to convert from 10s complement to signed decimal. Attached is the answer to the table: Table As far as I'm aware and ...
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In number systems, we use remainders for conversion and how actually this thing work out?

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Is there a number system base $a$ where $2018$ can be written as $\overline{21312}^{a}$ in it?

That's actually the whole question. Is there a number system base $a$ where $2018$ can be written as $\overline{21312}^{a}$ in it? I honestly don't know what I do more than this $2a^4 + a^3 + 3a^2 + a ...
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Failure of Hasse Principle for $x^2 - 5y^2=uv$ and $x^2-5z^2=(u+v)(u+2v)$

Can we show the failure of Hasse principle (or "local-global" principle) for these simultaneous quadratic equations? \begin{eqnarray} x^2 - 5y^2 &=& uv \tag{1}\\ x^2 - 5z^2 &=&...
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How does the $2$'s complement describe a negative number in binary?

According to my knowledge, the 2's complement is used to describe a negative number in binary representation. But I have this confusion. Example: Suppose that we are using 5 bits registers. The ...
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1 answer
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Why is $\lim_{b \rightarrow 1} concat_b(x,y) \ne x+y$ for base-$b$?

Let $x=x_N\ldots x_0$, $y=y_M\ldots y_0$ be integers with base-$b$ digits $0 \le x_i,y_j < b$. Concatenate them via $$x \otimes_b y\triangleq x_N\ldots x_0 y_M\ldots y_0=b^{len_b(y)}x+y=b^{1-\{\...
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1 vote
1 answer
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Show that "ans += resultList[i] - 2 * resultList[i-1]" in the function is the right expression to use when trying to convert ROMAN NUMBERS to INTEGERS [closed]

I was looking for creative ways in which someone could convert a valid roman number to an integer via algorithms in the web, and from all of the examples I found, there was one in particular that ...
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How to demonstrate the method used to convert decimal to any other base?

Let's say we have an arbitrary number in base $b$, $(x_3x_2x_1x_0)_b$. We can write the equivalent of this number in base $10$ as follows: $(x_3x_2x_1x_0)_b = x_3*b^3+x_2*b^2+x_1*b^1+x_0*b^0$ So, let $...
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Using Karatsuba algorithm to multiply [Edit: Answered]

I've been looking over this, and haven't seen anything wrong in my procedure. Take 2468 *3162 [Using Karatsuba multiplication]: x=2468 y=3162 x={(a)24}{(b)68} y={(c)31}{(d)62} Step 1: a*c=(24)(31)=744 ...
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1 answer
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Extension of Complex Numbers [closed]

Does there exist a number system that contains $\mathbb C$? If so, how many such systems are there? Why are they developed? Which are the larger ones?
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1 answer
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How to understand base-n expansions of numbers less than 1?

I have self-taught myself undergraduate Real Analysis using S. Abbott's Understanding Analysis. I am now moving on to learning "graduate-level" Real Analysis using Stein & Shakarchi. One ...
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Cycles in modified reverse-and-add process

Consider repeatedly adding a reverse (of digits) of a number to itself, until a palindrome is reached. Numbers that never reach a palindrome (diverging to infinity), are called Lychrel numbers. Let's ...
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Is the output of the iterative process $\sum_{n=0}^\infty 2(10^{n})$ a number?

My thought process went something like this. In computer scince we can represent 2 in binary as 10, 0010, or 0000 0010. Adding zeros to the left of a natural number dosn't chang it's value, So in ...
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How to Add the IEEE 754 single-precision floating-point numbers from hexadecimal?

Question: Add the following IEEE 754 single-precision floating-point numbers. (a) C0123456 + 81C564B7 (c) 5EF10324 + 5E039020 I know I first need to convert to binary, then add and later change into ...
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What is the algebraic structure of the natural numbers represented as words with fixed radix?

Consider the base 10 number $124273233$. This can obviously represented as $\{1,2,4,2,7,3,2,3,3\}$ where all characters are representatives of residue classes in $\mathbb Z / 10 \mathbb Z$. The ...
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What are the possible place values in balanced ternary?

Balanced ternary is an alternative number base to other bases like decimal or binary. We express an integer $n$ as $\Sigma_{i=0}^ka_iw_i$ where $a_i$ = 0, 1 or -1 (see here). Every $n$ has to be ...
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How are Ostrowski Numeration Systems built from a periodic continued fraction?

Ostrowski number systems represent integers and real numbers using the denominators of the convergents of continued fractions. One better known special case of this is Zeckendorf's Theorem and related ...
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1 answer
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Trying to understand the development of Number System [closed]

I want to understand how number system evolved from scratch. I am looking for a book which starts with natural numbers, explains their origin, explains how addition and multiplication is defined on ...
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three-dimensional number system without negative numbers

My main question is what properties, whether good or bad, am I missing from the number system that I'm about to describe. Ever since I asked about a two-dimensional number system without negative ...
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1 vote
2 answers
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Number System , Bases. Need help understanding the solution for this question [closed]

$$121x+11y+z=567$$ What is $x+y+z$, given $x,y,z$ are digits of a $3$-digit number $(xyz$). I am stuck at this step: $$ (xyz)_{11}=(567)_{10} $$ Is the above step correct or is this correct: $(xyz)_{...
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Booth algorithm for multiplying signed numbers

The algorithm of the Booth for multiplying signed numbersrs of fixed complement representation decimal point by 2 is implemented by multi-consecutive digit stain control of the multiplier, so after ...
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1 answer
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System with base $5$ - difference of importance is even

I want to show that all non zero positive integers that are multiple of $3$ and have exactly two non zero digits in their representation in system of base $5$, have even difference of importance at ...
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Triadic representation of a positive even integer

I want to show that all positive even integers have even number of digit “$1$” in their triadic representation. An even number is of the form $n = 2k$ with $k\in \mathbb{Z}$. To find the triadic ...
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3 votes
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Fractions with primes in different number systems

I noticed a pattern in fractions of 7: 1/7 = 0.142857142857142857... 2/7 = 0.285714... 3/7 = 0.428571... ... As you can see, they all share the same 6 different numbers in the same order, only ...
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2 votes
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How do we define the order relation in $\mathbb{R}$ in the construction via Cauchy sequences?

I am reading the article of Wikipedia about the contruction of $\mathbb{R}$ via Cauchy sequences, but I don't understand why the order relation that appears in the article is well-defined. It says ...
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13 votes
1 answer
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two-dimensional number system without negative numbers

Is there any existing literature for the number system that looks like this? Like the complex number system, this system exists on a plane. But instead of $i$ and $-1$, it has two numbers-- called $q$...
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Does the unary number system have fractions?

Is there a way to write a fraction in unary number system? What I mean is to write a fraction part using a "comma" notation. Writing it just as a fraction is obvious: $$ \frac {111}{1111}_{(...
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What is the total number of combinations in those numeric groups?

Extremely basic question that still confuses me but I would just like to make sure that my understanding is correct. Let's say we have 5 groups: Group1 Group2 Group3 Group4 Group5 In EACH of those ...
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4 votes
3 answers
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How to represent a decimal number into a binary expression of a different radix/base

Dear Math community, A good way to ask my question is to simply illustrate this with an example. Let's have a binary representation of the first 16 numbers {0-15} ['0000', '0001', '0010', '0011', '...
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1 answer
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Converting from Base-2 numbers to any base

Is there any general method to convert from base-2 to any base. When we convert from base-2 to base-10, we do the sum of weights ...
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3 votes
1 answer
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What is a "ratio" that allows zero on either side called?

This is a follow-up question to Ratio when one entity is 0. This question asks if ratios can have a zero, with answers that can be summarised as "It's complicated but no". That's fine, but ...
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In a floating-point system, is the unit roundoff $\epsilon_{mach}$ necessarily a machine number?

I have to answer the following questions: (a)In a floating-point system, is the unit roundoff $\epsilon_{mach}$ necessarily a machine number? (Explain your answer or give a counterexample). (b) Is ...
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difference between rational and fraction

my question is are we doing the same thing in $\frac{6}{-7}$ and $\frac{-6}{7}$? my argument is they are giving the same answer but at first we are dividing by $-7$ and second by $7$, so we are ...
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Is there a mathematical significance to the linguistic names for orders of magnitude?

This may be more a English History question as far as naming, but I'm more concerned if there are any mathematical reasons for the way we represent numbers. When written, by convention, every 3 orders ...
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All numbers from $1$ to $150$ (in decimal system) are written in base $6$ notation. How many of these will contain zero's?

Now I know that any number in decimal system which is divisible by the base $6$ will have a $0$ in the unit's place when that number in the decimal system will be converted into base $6$. This gives ...
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