Questions on the different ways to represent a number, how to convert between those different ways, and other such questions. The usual system employed by humans is the decimal (base-10) system, but other systems like binary and hexadecimal are also in frequent use.

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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 ...
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1answer
32 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 = ...
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2answers
21 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 ...
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1answer
30 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?
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1answer
16 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?
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1answer
31 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 ...
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1answer
18 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 ...
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1answer
52 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} +... ...
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0answers
11 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: ...
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1answer
47 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 ...
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0answers
17 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 ...
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503 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., ...
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1answer
44 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 ...
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2answers
33 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 ...
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2answers
30 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 ...
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1answer
53 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!
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3answers
49 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: ...
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3answers
106 views

subtraction of two repeating decimals rationals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
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4answers
100 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 ...
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2answers
56 views
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0answers
16 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). ...
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0answers
12 views

In Roman System, what does it mean that S1 cannot be before S2 if S1<=S2*10?

My textbook(Foundations of Computer Science - Cengage Learning) says "A symbol S1 cannot come before another symbol S2 if S1 <= 10 * S2. For example, I or V cannot come before C." If ...
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1answer
82 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 ...
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1answer
53 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 ...
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1answer
72 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 ...
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2answers
67 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 ...
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1answer
86 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". ...
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1answer
38 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 ...
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0answers
41 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 ...
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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 ...
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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 ...
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0answers
33 views

Is there a conventional symbol for the set of real algebraic numbers?

The real numbers are denoted ℝ, and the algebraic numbers are conventionally denoted 𝔸. Is there such convention for the real algebraic numbers ℝ∩𝔸?
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10answers
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Can a number have infinitely many digits before the decimal point?

I asked my teacher if a number can have infinitely many digits before the decimal point. He said that this isn't possible, even though there are numbers with infinitely many digits after the decimal ...
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1answer
50 views

Finding which base number given operations

$$ (35_a + 24_a) * 21_a = 1081_a $$ Which base is the above number? Any advice on how to solve questions like these? I tried making it in to a polynomial: $(3a+5 + 2a+4) * (2a+1) = 108a + 1$ ...
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0answers
29 views

How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: ...
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1answer
57 views

Fractional parts in base number systems other than base-10?

A long-standing question I've asked myself over and over again is how one might express fractional parts of a whole in base number systems other than 10. Is it truly as simple as base-10's x/y system? ...
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2answers
36 views

Represent non-integer values on the factorial base

I want to compute the representation of the following values using the factorial number system: $\pi$ $e$ $\phi$ I know how to do it for integer values, but is it feasible for non-integer values? ...
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1answer
42 views

Are there any number systems better suited to nature?

For example, number such as $\pi$ and $e$ cannot be represented as rational numbers in our number study and extend in decimal places to infinity. QUESTION: Is there a possibility that some other ...
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23 views

How to express the quantity using significant figures to imply the stated error?

Express the quantity using significant figures to imply the stated error. $$1.77 \pm 0.06$$ I tried to factor out a $6$ and got $6(.295\pm0.01)$ but then when I try to make $.295$ have an error of ...
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1answer
39 views

Using significant figures to express stated errors?

The question reads, " express each of the following quantities using significant figures to imply the stated error". a) $2.3 \pm 0.001$ b) $1.989 \pm 0.0003$ I think the first one is 2.300 because ...
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1answer
56 views

What is the numeral system which uses the number of digits as a signifier of value called?

Our standard notation of representing numbers has an implied infinite number of zero digits on the left of all numbers. 42, 042 and 00000000042 all represent the same number. I'm thinking of the ...
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1answer
64 views

Why $0.999$… isn't the largest number before 1? [duplicate]

Why doesn't it called like that? It seems fair, $1$ called $1$ while $0.999$... being the largest number before $1$, and not called $1$ while not look like it is. Let's say it isn't, how would that ...
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0answers
54 views

Roman Numerals. What are valid Subtractives? e.g. Is VL valid for 45?

"Standard" syntax for Roman Numerals seem to always show 45 = XLV. I wrote a little program to convert roman numerals to integers, so I started wondering what is valid in subtractive notation and ...
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0answers
69 views

Ternary balance with unknown weight

Main references: Ternary (Wolfram MathWorld) Balanced ternary (Wikipedia) Weighing scale: Balance (Wikipedia) <quote> Balanced ternary has other applications besides computing. For example, a ...
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1answer
90 views

Given the normalised floating point number system, calculate smallest possible value of y - x

First I present the problem and then my workings and thoughts: Given the normalized floating point number system $(\beta, t, L, U) = (10, 7, -6, 4)$ where $\beta$ is the base and $t$ is the ...
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2answers
34 views

For any base-$n$ number system, what is the average length of a number $\leq100$?

By this I mean the amount of symbols (bits / digits / etc. ) for any number from $0$ to $100$. I don't know whether this can be answered, I'm just asking out of interest - not homework or anything.
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4answers
100 views

How to compute 1/7 in base 8?

This is probably a very basic question but nonetheless I choked when I got in a math for programmers class. I was taught how to convert from base to base but I have no idea how to convert fractions to ...
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1answer
30 views

Conversion in Number system

Here I provide the image. My question is why is it necessary to have 2 different results of the same problem. Am I doing something wrong. Please Help! Thanks in advance.
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1answer
41 views

Two conversions to base three yield different results

How are there two different conversion results for the same bases? Am I doing something wrong?
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38 views

number of trailing zeroes of factorial raise to power by another factorial

Finding trailing zeroes in any factorial is easy. Every time you pass a multiple of 10 (or something 5 mod 10) you will accumulate another 0 For example 10! has two trailing zeros, one from ...