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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-7
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2answers
33 views

Write $61.84 \times 10^{-3}$ in standard form [on hold]

The answer is $6.2 \times 10^{-2}$ but how do we solve it step by step to?
0
votes
3answers
105 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
votes
2answers
54 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)} = ...
5
votes
5answers
48 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 ...
-1
votes
1answer
42 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 ...
0
votes
5answers
55 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
votes
1answer
43 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
votes
3answers
443 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
97 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
votes
1answer
30 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
votes
2answers
99 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 ...
-1
votes
1answer
16 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
votes
0answers
40 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
51 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
votes
1answer
88 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
votes
0answers
48 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
69 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
816 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
vote
1answer
55 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 ...
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
vote
0answers
23 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
63 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
votes
1answer
64 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
2answers
256 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
vote
1answer
52 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}, ...
5
votes
4answers
186 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
79 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
307 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
44 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
48 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
144 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
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 ...
9
votes
3answers
725 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
63 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
49 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
29 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
46 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
24 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
125 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
26 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
66 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
16 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
63 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
25 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 ...
7
votes
1answer
597 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
68 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 ...
1
vote
2answers
65 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 ...