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 ...

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1answer
41 views

In what base would $\pi$ have the fewest decimals?

What is the numeral system in which $\pi$ would have the lowest number of decimals possible?
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32 views

which is the more accurate way of representing numbers?

When we consider calculations at tiniest of scales which number system would be more accurate, when we consider the binary number system( base 2) or the number system we generally use (base 10). The ...
1
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1answer
39 views

Convert from base 10 to base 5

I understand the integer part, keep dividing by 5 and I get 112, but for the fraction part I need a help. The number is: $$ (32.\bar 5)_{10} = (112, ??)_{5} $$ $$ 0,5 * 5 = 2,5\\ 0,5 * 5 = 2,5\\ ...
0
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0answers
28 views

Is there a name for complex numbers over affinely extended reals?

Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis ...
3
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2answers
73 views

Is a decimal number system the best to grasp mathematics?

I am often amazed by how accurate a decimal numeral system is to describe the mathematics world. A lot of things feel very logical like the ...
0
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1answer
38 views

What is the significance to our number and degrees systems? [duplicate]

I saw this video recently and it suggests that there is some "magical" reason that there are 360 degrees in a circle and that it is also connected with our number system. My question is: How did we ...
1
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1answer
31 views

Representing numbers by quasilexicographic ordered strings, formula for size of conversion between different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
0
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4answers
120 views

How many $0$s does the number $30!$ have? [duplicate]

I want to find out the number of $0$s in the number $30!$, what should I do? Is there any trick that would work for a general question of this type, like number of $0$s in $50!$ ?
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2answers
101 views

Why does our number system have only 10 different symbols? [duplicate]

To be specific, I am just curious as to why does our number system have 10 different digits?Just for an example, why did it not end at 7? (it is the greatest prime number before 10, there are 7 days ...
0
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1answer
16 views

How to remove $1$ at position $x$ ( in base $B$) from a number represented in Base $10$

I was going through a solution on code chef in which we needed to remove a $1$ from a position say $x$ (in Base $B$) from a number in Base $10$ if the representation of that number in base $B$ had a ...
1
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1answer
41 views

Is the cubic formula numerically unstable?

Are there numerical rounding issues in using the cubic formula to find roots of cubic equations? Similarly with the quartic formula? I do know for the quadratic formula to solve $ax^2+bx+c = 0$ that ...
1
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0answers
54 views

Is our Arabic number system based on a geometric design counting corners? [duplicate]

The following writer asserts that our system of Arabic numerals is a geometric design where the number of corners corresponds to the number represented: My question is: Is our Arabic number system ...
0
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2answers
23 views

Denote a number as a sum of two in terms of the base

Technically, in decimal base, $$445 = 44 \cdot 10 + 5.$$ Lets say there's a number of base $B$ where I want to segregate the number into last digit and rest of the first digit as a representation ...
0
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1answer
36 views

Can't find the radix to solve this equation

I am trying to solve the equation $$ 3894937908247641871050398074967894254 = 764008325721660_x$$ Here is my attempt $$ 7x^{14} + 6x^{13} + 4x^{12} + 0x^{11} + 0x^{10} + 8x^9 + 3x^8 + 2x^7 + 5x^6 + ...
2
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3answers
64 views

Why does the division algorithm work for converting between number bases?

I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this ...
0
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3answers
130 views

Find the smallest natural number $n$

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$. Example of the operation: $123456$ becomes ...
0
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2answers
59 views

What determines what base the right side of this base coversion will be?

Referring to this example of positional notation on Wikipedia: There are several examples $$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$ But then $$465\;\;\text{(base 7)} = ...
8
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5answers
59 views

Is it allowed to define a number system where a number has more than 1 representation?

I was just curious; is it allowed for a number system to allow more than one representation for a number? For example, if I define a number system as follows: 1st digit (from right) is worth 1. 2nd ...
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1answer
57 views

Is binary more ideal than decimal? [closed]

We only chose the decimal system because we have 10 fingers. Binary is the most basic positional numbering system, so would it make sense to say that it would be the most ideal system? Is it better ...
1
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5answers
61 views

How can we count in base eleven? [closed]

We work in base 10. We know base 3 would count 1, 2, 10, 11, 12, 20. How would we count in base 11?
2
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1answer
58 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
6
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3answers
467 views

What is the next number having the same number of bit 1s? [duplicate]

You are given a number, $A$, and you have to determine a number, $B$, such that $B>A$ and the number of $1's$ in the binary representation of $A =$ number of $1's$ in the binary representation of ...
5
votes
4answers
118 views

Constructing the natural numbers without set theory.

As I understand it the idea of defining everything as sets is a relatively new idea in mathematics. Does that mean there's a non-set theoretic definition of the natural numbers? Could there be?
5
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1answer
33 views

$x-1$ in base $x$ counting systems

Please excuse the lack of expertise. I'm not a mathematician, nor have I studied it since high school. I was thinking about how all the digits of multiples of $9$ summed equal a multiple of $9$. I ...
2
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2answers
106 views

The digit at the hundred's place of $33^{33}$

I would want to know how to start with the question. And if you get hung up somewhere there's the answer it's $5$. Any help is appreciated thanks, My approach was to look at the factors to somehow ...
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1answer
20 views

How to perform mutliplication of negative octal and hexadecimal numbers?

So I am supposed to do these two multiplication problems (-6*4)8 and (-7*8)16 I did it the same way I would do with positive ones and ended up with the wrong results. I tried these on a calculator ...
0
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0answers
53 views

what are the benefits of a factorial number system?

After reading an article about factorial number system. It tells that you can present any number in a factorial system and in if you have a number $a_{n-1}...a_2a_1a_0$ in factorial number system, you ...
2
votes
1answer
66 views

Rational Irrational Numbers

I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a ...
0
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1answer
90 views

Are there any system(s) of mathematics whose relationship between variables bears difference to that found within mainstream mathematics?

I have been reading up on boolean algebra quite recently, for those not familiar, this type of mathematical system has much to do with the way logic is represented (and is primarily applied to, though ...
6
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0answers
53 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
2
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0answers
95 views

Is there any formal or scientific use for a base 7 numeral system?

This numeral system is utterly obscure, and seems to have no use at all anywhere. So, is there any formal or scientific use for a base $7$ numeral system anywhere?
12
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5answers
896 views

What would base $0$ be? How would/could it work?

If I was trying to take the number $123$ in base $10$ and try and convert it into base zero I would do something like this: $123 = 100 + 20 + 3$ $10^{\log_0(100)} + 10^{\log_0(20)} + ...
1
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1answer
67 views

Decoding the sign expansion of surreal numbers

One way to represent surreal numbers is the sign expansion. Now Wikipedia describes how to compare them, how to convert them to the standard representation of left/right sets, how to negate them, and ...
16
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3answers
2k views

How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks
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0answers
25 views

Determine base required to accurately represent number

If you have an expression like 1/3, that can't be accurately represented in base 10. It would look like 0.3333333.... However, ...
0
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2answers
67 views

Sequences and real numbers

Based on the answers so far I restate the question: on p. 63 of his volume on Analysis (http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf), Zorich says: “We now answer the question whether some ...
0
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1answer
75 views

Number System Conversion

0 down vote favorite I have a paradox: EIGHTY is a six digit number with no repeating digits and no zeros. When divided by 19, 17, 13, 11, or H, the remainders are, respectively, 17, 13, 11, 7 and G. ...
4
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2answers
303 views

Base conversion: How to convert between Decimal and a Complex base?

My motivation for this question is exploring beyond the ideas in Project Euler Problem 508. In that problem, it is helpful to know how to convert between a decimal number and a number in base ...
1
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1answer
72 views

Pseudo-Surreal numbers are analogous to?

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|} I see that pseudo-surreal numbers seem to have an interesting branch of game theory. Still having a form of {x|y}, ...
6
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3answers
201 views

Why hyperreal numbers are built so complicatedly?

I have seen approaches at building hyperreal systems by using complicated notions like ultrafilters and the like. Why not just postulate the existence of infinitesimal element $\varepsilon$ and ...
5
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0answers
65 views

When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...
1
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1answer
21 views

Most frequent first non-zero digit of $X/Y$ in base $b$ numeral system is 1?

My question concerns the distribution of the first non-zero digit in base $b$ of $X/Y$ where $X,Y$ are two i.i.d random variables. I think that the digit $1$ is the most frequent. Let $X,Y$ be two ...
0
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3answers
84 views

Formal notation for representing decimal numbers

What is the formal mathematical notation for representing a decimal number using variables as it's digits? I am honestly surprised that this has not been asked yet on MSE. Let me clarify a ...
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2answers
527 views

Convert decimal to Binary Floating Point - 8 Bit [closed]

I am trying to convert +3.5 to binary floating point, but im struggling to find the exponent. (8 Bit) Where 1st bit is the Sign, 3 bits for Exponent and 4 bits for Mantissa. Hope somebody can explain ...
11
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4answers
1k views

Are there any bases which represent all rationals in a finite number of digits?

In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
0
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1answer
49 views

Fraction in other bases

How to convert a base 10 fraction into fraction in other bases?. For example base 10 fraction 17/94, How we convert this 17/94 into base 2 fraction ?
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1answer
46 views

What's it called when you treat intervals as numbers?

Often in physics you have to do maths with a finite amount of digits, e.g. $\pi = 3.14$, but this is not exact and without knowing the next digit this is only correct in the interval $[3.135,3.145]$. ...
1
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0answers
59 views

Digit-period in the representation of powers of $9$ on base $10$

Slightly inspired by this question: I have noticed that in the representation of powers of $9$ on base $10$, each digit repeats periodically as a function of the exponent. I assume that the same ...
8
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2answers
151 views

Last digit of $3^{459}$.

I am supposed to find the last digit of the number $3^{459}$. Wolfram|Alpha gives me ...
6
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1answer
104 views

Representation of irrationals as $\sum_{n\ge 2}\frac{x_n}{n!}$

Prove that every $x\in(0,1)\setminus\mathbb{Q}$ has a unique representation as $x = \sum_{n\ge 2}\frac{x_n}{n!}$, where $x_n\in\mathbb{Z}_n = \{0,1,2,\ldots,n-1\}$. Probably this is well known, I'd ...