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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3
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0answers
32 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
votes
0answers
63 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
votes
5answers
768 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)} + ...
0
votes
0answers
21 views
+50

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 ...
15
votes
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
1
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0answers
22 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
votes
2answers
58 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 ...
1
vote
1answer
53 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
votes
1answer
180 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 ...
5
votes
4answers
169 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 ...
1
vote
1answer
20 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
votes
3answers
70 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 ...
-2
votes
2answers
173 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
votes
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
votes
1answer
35 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 ?
1
vote
1answer
44 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
vote
0answers
37 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
votes
2answers
141 views

Last digit of $3^{459}$.

I am supposed to find the last digit of the number $3^{459}$. Wolfram|Alpha gives me ...
6
votes
1answer
102 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 ...
9
votes
3answers
720 views

How many 0's are in the end of this expansion?

How many $0's$ are in the end of: $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$ The answer is supposed to be $1100$ but I have absolutely NO clue how to get there. Any advice?
1
vote
2answers
56 views

How can we represent $10$ in a decimal system?

This may sound like a silly question to begin with but I'm having problems finding a proper answer. The question is generally targeting numeral systems of any base, but for simplicity, I will ...
1
vote
1answer
41 views

Showing that every positive integer can be represented in this form

How can we prove that for every pair $N \in \mathbb{N}$, and natural number $\beta\in [2, \infty)$ there exists a unique set of integers $x_i \in [0, \beta -1]$, $k\in [0,\infty)$ such that: $$N = ...
0
votes
2answers
26 views

Representing number $X$ in base $r$

In general, let $X = (X_{n−1}X_{n−2}...X_0)_r$ be an n-digit number in base r. Give an algorithm or explain in English how to represent $X$ in base $r^2$. I ...
0
votes
1answer
39 views

How can I add the following 32-bit IEEE floating-point numbers?

How can I add the following two 32-bit IEEE floating-point numbers in binary? FEDCBA98(base 16) + 89ABCDEF(base 16) = a 33-bit binary number. How can this be possible?
0
votes
1answer
18 views

How would you convert the following 32-bit IEEE floating-point to decimal form?

I have got -1.101 1100 1011 1010 1001 1000 * 2^(9) How can I convert this to decimal form?
2
votes
1answer
71 views

Convert the following decimal number into 32-bit IEEE floating-point form.

I am given a negative decimal -1234.875. I understand the normal process of solving a question like this, except I am uncertain about handling the negative. What I do is find the binary form of 1234 ...
0
votes
1answer
21 views

For the following number, state the base represented as t?

$1011 \textrm{(base }t) = 4931 \textrm{(base 10)}$ I have to find $t$, which is the base of 1011. I do the following: $4931 \textrm{(base 10)} = 4 \times 10^3 + 9 \times 10^2 + 3 \times 10^1 + 1 ...
1
vote
1answer
63 views

every number $n\in \mathbb{Z} $ can be represented as sum of different powers of $2$

Using generating function prove that every number $n\in \mathbb{Z} $ can be represented as sum of different power of $2$, I mean, that for every $n\in \mathbb{Z}$ $$n=2^{k_1} +2^{k_2} +2^{k_3} +... ...
0
votes
0answers
15 views

Number Systems and Floating Point Arethmetic

I am confused about the term of finite precision number, is that the same with the floating point number? Moreover, I have a number system: ...
0
votes
1answer
56 views

Does every real number have a decimal expansion?

Can every real number be written in decimal expansion? I mean, can every real number $a$ be expressed as follows: $$\text{For }\, a \in \mathbb {R}^{+},\quad ...
1
vote
0answers
24 views

smallest poisitive integer does not belong in floating point system F

A floating-point number system F is a subset of real numbers whose elements have the form: $x=\pm(m/\beta^t)\beta^e$ base (beta), precision (t), and exponent range (emin and emax). The mantissa m ...
6
votes
1answer
549 views

What is the Scientific Notation of Zero?

This question was asked here, where the answer uses this description. The last line reads: "The special case of $0$ does not have a unique representation in scientific notation, i.e., ...
3
votes
1answer
57 views

Let x and y be two positive real numbers with x<y. Using only the axioms for real numbers, show that 0 < 1/y < 1/x

Let x and y be two positive real numbers with x < y. Using only the axioms for real numbers, show that 0 < 1/y < 1/x I apologize for how it looks, but I'm not very good with formatting. How ...
0
votes
2answers
45 views

The perfect Number system

I was thinking of the number system presently in use(the decimal system) and its shortcomings. One of them is that all numbers cannot be represented accurately, for example the value of any irrational ...
3
votes
3answers
64 views

How can I prove that the square of an even number ends in 0/4/6?

I am trying to prove that the last digit of the square of an even number is either 0, 4, or 6 but I'm completely lost and have no idea how to tackle this problem.
3
votes
2answers
31 views

Even integer in ternary representation

Suppose $(d_0,d_1...d_k)_3$ is the ternary representation of a even integer $n$. Show that there is an even number values $d_0...d_k$ that are odd, whenever $n$ is even. I have tried decomposing ...
0
votes
1answer
69 views

Does anyone knows about Ethiopian numerals?

I'm wondered if anyone could give me best resource about Ethiopian(Geez) Numerals and Could you write Geez on LaTex? Thank you very much!
1
vote
3answers
52 views

Convert 2620.123 base 10 to base 5

I understand the integer part, keep dividing by 5 and I get 40440, but for the fraction part I did a calculation, but it does not match with the answer given. I do the fraction part like this: ...
3
votes
5answers
192 views

Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
0
votes
2answers
67 views
0
votes
0answers
18 views

How do I efficiently budget for digits when converting from one base to another?

I'm trying to work out a formula that gives me the minimum number of digits required to express a number in a target radix (r2) knowing only the source radix (r1) and number of significant digits (d). ...
0
votes
1answer
193 views

Converting double precision IEEE 754 hex to base 10 with repeating decimals

The number is 0x4001 8CCC CCCC CCCC. So far I have the stored exponent as 1000000000 which equals ...
4
votes
1answer
55 views

Numerals vs. Numbers

I've noticed three different approaches in general use: Numbers are abstract and numerals are used to express numbers following a particular set of rules. The number of days in two weeks is a ...
4
votes
1answer
84 views

Systematic way to represent any irrational number

I'm wondering if there's a way to symbolically (or is there a more lose constraint?) represent ANY irrational number in a systematic way. You can represent any rational number as two integers and I ...
0
votes
2answers
83 views

Base 4 Mathematics

I have an homework question but I'm having hard time to understand the context. Here is the question: Assume that you are using 3-digit number system with base r = 4 (and n = 3). Assume also ...
1
vote
1answer
91 views

Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
0
votes
1answer
46 views

How do I detect whether a rational number has a repetend when expressed with a different radix?

My original and basic question is: How do I detect whether a rational number has a repetend when expressed with a different radix? Here is my question restated as an example: Given ...
3
votes
0answers
65 views

Is multiplication in mixed radix numeral systems complicated?

The wikipedia article on mixed radix numeral systems says Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. This sounds like "naive ...
1
vote
1answer
25 views

Determine the $n$th string among those of a given length in alphabetical order starting at a given string and using a given character set?

When building strings using a particular character set (the set can change), such as in a brute-force password cracking, how would I determine which string occurs in the $n$th position when the ...
0
votes
1answer
30 views

Is there a conventional symbol for the set of radical expressions?

There is already a question about the name of such a set: Name for numbers expressible as radicals My question is related. The rational complex numbers might be denoted ℚ(i), and the algebraic ...