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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Is there an ordering on $\mathrm Z$ to make it dense?

It is well-known that the set of rational numbers $\mathrm Q$ is dense; that is, given any two rationals, say $r\ne s,$ then there exist infinitely many rationals $r_i$ such that $r<r_i<s$ for ...
2
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0answers
37 views

Composition of permutations in factoradic notation

The Lehmer code maps between permutations and natural numbers. My question: Is a "nice" algorithm known for composing permutations directly in the factoradic form? When the permutations commute, it ...
3
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2answers
50 views

Are numbers independent of the basis we use?

So I have recently read about Kaprekar's Constant (https://en.wikipedia.org/wiki/6174_(number)) and It made me wonder If this number is really "special"? It seems to me that the notion 6174 (and the ...
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0answers
60 views

Are the digits of $\log 2$ and $\log 3$, or $\sqrt{2}$ and $\sqrt{3}$, or $e$ and $\pi$, cross-correlated?

I try to find sequences of digits (in base $b$, with $b$ not necessarily an integer) that are not cross-correlated. While the digits in base $b$ of (say) $\pi$ and $e$ do not exhibit auto-correlations ...
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2answers
53 views

Is a Centillion bigger than a g Googol? [closed]

I heard that a googol is a 10 to the 100th power, in other words, 10^100, and a centillion is a 10 to the 303rd power, in other words, 10^303. Some people disagree though.
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1answer
33 views

Evaluate the number of different numbers one can write, having at one's disposal n digits of a q-ary system.

I am working on Zorich, Mathematical Analysis I and I am stuck on a problem in section 2.2. The problem asks me to evaluate the number of different numbers one can write with n digits of a q-ary ...
1
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1answer
32 views

Determine a property of $S_b(n)$, which is the sum of the digits of $n$ when $n$ is expressed in base $b$

The original question Let $S_b(n)$ be the sum of the digits of $n$ when $n$ is expressed in base $b$. was asked by Anson Chan on Jan. 24, 2019. Since it was deemed to not have enough context, it was ...
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2answers
26 views

Octal System Conversion

So I am trying to change base ten 1,398 to an octal equivalent. I had assumed that I would find it by taking 1,398 and changing it like base-four. So 1,398 is $1*8^3 + 3*8^2 +9*8^1 + 8*8^0*$ or $512+...
3
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1answer
61 views

Proof explaining opposite of what is observed

Little time ago I started to playing with binary numbers and just have a little fun . I found some pretty interesting things. I want to ask if my own findings have some official names or no .Also I ...
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2answers
35 views

Numbers between decimal number

So there is a question that says I can choose a number between $0.00001$ and $0.1$, but the problem is what exactly comes after $0.00001$? Would it be $0.0001$, continued by $0.001$, $0.01$, $0.1$? ...
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2answers
36 views

Finding closed form of sequence $0, 2, 10, 28, 60, 110, 182,…$ [closed]

Need to find closed form of this sequence. $0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770,...$ I've tried some obvious ideas but yet to have a luck.
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1answer
42 views

Number System Divisibility by 7

X is a number formed by writing 9 for 99 times. What will be the remainder of this number when divided by 7?
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3answers
48 views

If $85_b = 58_c$, what is the smallest possible value of $b$?

I have searched and watched videos online and can't find a method to solve this problem: If $85_b=58_c$ for some positive integer bases $b$ and $c$, what is the smallest possible value for $b$?
3
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5answers
47 views

Find digits $a,b$ such that $7ab + 4ba = 1a21$.

I have to find all the digits $a$ and $b$ such that $7ab + 4ba = 1a21$. Note: there is no multiplication, those are three decimal numbers. I put this equality this way: $$7\cdot100 + a \cdot 10 + b +...
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3answers
48 views

base-52 number system conversion

I'm trying to understand number systems. Consider a base-52 number system consisted of following symbols as digits: ...
5
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1answer
39 views

The digit at position $n$ of the number $x$ in base $m$

As a solution to this question, we can define a function $f_b(x, n)$ which finds the digit in the $n$th position of $x$ in base $b$. $$ f_b(x, n) = \left\lfloor \frac{x}{b^n} \right\rfloor \bmod b $$ ...
0
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1answer
95 views

In hexadecimal calculus, the square of a positive integer x is 2 identical blocks each of length k. can k be equal to 2023?

In hexadecimal calculus, the square of a positive integer x is 2 identical blocks each of length k. can k be equal to 2023? I transferred the required number to the decimal number system and got ab .....
0
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1answer
42 views

Minimum number of symbols required for a base-N positional number system

A base-10 number system requires 10 symbols. Any less, and you would not be able to represent every number. (Is this a correct assumption?) A base-16 system requires 16 symbols. How can I tell how ...
0
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1answer
17 views

How to calculate a division/quotient in octal numeric system?

Im trying to make a division in octal system $$\frac{165_{8}}{24_{8}}$$ So I have done like this using a table $$24\times 0=0_{8}$$ $$24\times 1=30_{8}$$ $$24\times 2=50_{8}$$ $$24\times 3=74_{8}$$ $$...
0
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1answer
15 views

Number system for operation

The last digit of some number 𝑋 in base 𝑘 is 2. Last number of $12_{10}*X$ in the same system is 4. How many systems that are suitable for these conditions for any X. I m really confused for ...
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0answers
79 views

Are $\mathbb{C}$ and $\mathbb{H}$ the only continuous number systems over $\mathbb{R}^{2}$ and $\mathbb{R}^{4}$ respectively?

When I say $\mathbb{C}$ is a number system over $\mathbb{R}^{2}$, I mean that the field of complex numbers is identified with the set of dilative rotations of the affine real plane ($\mathbb{R}^{2}$) ...
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0answers
58 views

Complex Base $i-1$

For all positive integers, we can represent them uniquely in the system $\langle -1+i, Z_2 \rangle$. For example, $2$ can be represented as $(1100)_{-1+i}$. Given a number $k$, I'd like to ask how to ...
6
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4answers
150 views

Factors in a different base $\ 2b^2\!+\!9b\!+\!7\,\mid\, 7b^2\!+\!9b\!+\!2$

Two numbers $297_B$ and $792_B$, belong to base $B$ number system. If the first number is a factor of the second number, then what is the value of $B$? Solution: But base cannot be negative. Could ...
0
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1answer
23 views

Repunits whose digits in base $b$ are all $b-1$

Positive integers whose base-$b$ representation contains only the digit $1$ are called repunits in that base. But what about positive integers whose base-$b$ representation contains only the digit $b-...
2
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3answers
45 views

Difficulty understanding how Babylonian reciprocals work

According to the reciprocal tables for Babylonian's base $60$ system, dividing by $2$ is like multiplying like $30$. Dividing by $3$ is like multiplying by $20$. Dividing by $4$ is like multiplying by ...
3
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1answer
25 views

How to prove that when we change numeral system same number becomes shorter

Let's start with an example: We have a binary number $1100_2$ In decimal it will be equal to $12_{10}$ In hexadecimal it will be equal to $C_{16}$ I'm trying to find a proof that for any positive ...
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2answers
22 views

Babylonian Operations Not Working Out So Well

I'm going to denote 1 as v and 10 as < to replace the Babylonian symbols my homework says to evaluate $$(vv)+(<vvvv)+(<vv+vv)+(<<v)+(vv)$$ Because Babylonians used powers of 60 for ...
0
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1answer
15 views

Confusion about normalised floating point number systems

I was assigned the following question: List all numbers that can be represented exactly in a normalised floating- point number system with base 10, two digits in the fraction, and an exponent 0 ≤ e ≤ ...
0
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1answer
50 views

Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
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2answers
34 views

Number system conversions

I am an Electrical Engineering student but my question is related to number systems, more specificaly to conversions between octal, hexadecimal and binary systems. I know the rules of conversions but ...
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0answers
49 views

Construction of number systems

I would like to get a book on set theory that presents the construction of number systems so that one really has: $\Bbb N$ contained in $\Bbb Z$, $\Bbb Z$ contained in $\Bbb Q$, $\Bbb Q$ contained in $...
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0answers
45 views

Using Babylonian Numeration, Mayan Numeration, Base Conversions

I'm struggling with understanding and using the Babylonian system to solve problems as well as the Mayan system, and I struggle most with converting from one base to the next. For my course at the ...
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1answer
87 views

Soft Question: Is the skill of hexadecimal thinking worth getting?

I am a Computer Science undergraduate and I am currently taking a hardware module that focuses a lot on hardware processors and memory addresses, all of which works in binary. However, the inputs ...
0
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1answer
33 views

Relationship between Decimal, Hexadecimal, and Binary (Beginner)

I was studying the conversion technique between Hexadecimal and Binary where For example $4C2_{16} = 010011000010_{2}$ Can be done by substituting 4 bits for each Hexadecimal digit. Why is it ...
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2answers
48 views

Bits and digits are particular cases of what?

A bit is a _____(1) that can take two different values: usually 0 or 1. A digit is a _____(1) that can take ten different values: 0, 1, ..., 10 A _____ (2) is a _____(1) that can take 16 different ...
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3answers
49 views

How to find the solution given below through this system?

Knowing that $a^2 + b^2 = c^2 + d^2 = e^2 + f^2$ the following equalities are given $$ac + bd = ec + df = ae + bf$$ A solution to this equality is given if $a=-c-e$ and $b=-d-f$. From these ...
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1answer
29 views

NUMBER system and counting

Let $S =\{1,2,3,...,20\}$ be the set of all positive from $1$ to $20$ suppose that $N$ is the smallest positive integer such that exactly eighteen numbers from $S$ are factors of $N$ and the only two ...
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2answers
72 views

Number Systems - Proof

Can someone explain the second part of this proof? This is literally the very first example of the introduction to number systems in my textbook, and I'm failing to grasp it. In the introduction ...
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1answer
66 views

Is there any way to finitely represent all the information in pi? [closed]

Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any ...
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2answers
79 views

Can this idea work as a number system? [closed]

Each decimal number is equivalent to a function $f(x)$ in this number system. $f(x)$ is defined on the whole number line and only takes single digit integer values 0-9 everywhere. The conversion from ...
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1answer
61 views

What identities collapse number systems?

I was thinking about how many proofs involving complex numbers attempt to prove that a particular number is equal to its own complex conjugate, $c=\bar{c}$, in order to show that $c\in\mathbb{R}$. I ...
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4answers
410 views

The base in which $1065 = 13 \cdot 54$ is true

How can one find the base $r$ in which $1065 = 13 \cdot 54$ is a true statement? My attempt was constructing an equation including $r$ but because the left side has 4 digits, I got a polynomial of ...
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1answer
38 views

Find the number of $x$ digit numbers in base $n$ [closed]

I tried to understand this but I can't. It is confusing and I even don't know what the question mean. Help me out, Thanks! The number of $x$-digit numbers in base $n$ is (a) $n^x$ (b) $n^x-1$ (c) ...
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3answers
54 views

How would formula change in Base 6? [closed]

How would the following formulars change if we use a base 6 arabic number system? quadratic formula (=midnight formula) Einsteins e=mc^2 pi circle A=pi*r^2 It would be counted like this: ...
2
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1answer
41 views

bijective function $f:\mathbb{N}^{n} \to \mathbb{N}$ (a map of a number system with base eq to $W \to \infty$)

I am trying to find a bijective function (a map): $$ f: \mathbb{N}^{n} \to \mathbb{N} \\ f(x_{1}, x_{2}, x_{3},..., x_{n}) = y $$ That behaves like this: $f(0, \ 0, \ 0, \ 0, \ ..., \ 0) = A$ $f(1, ...
0
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1answer
35 views

Why multiplication by numeral system base and raising to corresponding power converts any numeral system integer to decimal only

I am not very good in math so I apologize if my question is too simple and does not belong here... Why an integer in numeral system X can be converted to decimal by multiplying it's digits to X (...
2
votes
3answers
64 views

Big Integer Base Converter?

I need to convert some really big numbers from base-10 to base-8. All online converters I found cap the conversion at around 20 digits. Are there any downloadable programs with no digit limit? I'm ...
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1answer
39 views

What is the hexadecimal representation?

A signed number in $2's$ compliment notation with $16-$bit register $P=DFA0$ What is the value of $P*4$? I have tried $P*4=0111111010000000$ but sign bit changed from $1$ to $0$. What is wrong ...
0
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1answer
253 views

How to find radix? [closed]

$\sqrt{(144)_r} =(12)_r$ I have tried $\sqrt{r^2+4r+4} =r+2$ From this, I am unable to find the value of $r$. Can anyone help me out to solve this problem?
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8answers
4k views

Why do we add a zero to dividend during long division?

Suppose we want to divide 3 by 2. By Long Division we first write 1 as the first digit as the quotient, then we subtract 2 from 3, then we add a zero to the remainder 1, and then add a decimal point ...