Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag.

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314 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& \boldsymbol{+}&...
7
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78 views

Generators for a matrix group

Lets denote $\Gamma_0(4)$ the subgroup of $SL_2(\mathbb Z)$ : $$\Gamma_0(4):=\left\{\begin{pmatrix} a &b\\ 4c&d \end{pmatrix}\in SL_2(\mathbb Z)\right\}.$$ We also define $A$ and $B$ in $\...
7
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60 views

To what extent can the fondamental theorem of arithmetic be used to give a canonical form to non-integer numbers?

The fundamental theorem of arithmetic gives us a unique way of writing any non-zero integer. For any $n \in \mathbb{Z}^*$, we have a unique decomposition : $$n = (-1)^\epsilon \prod\limits_{i \in \...
7
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178 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
6
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151 views

Does this operation have a name?

For a field $F$, define the binary operation $\parallel :(F\mathbb{P}^1 \times F\mathbb{P}^1 \setminus\{(0,0)\}) \to F\mathbb{P}^1$ by $$a \parallel b = \frac{1}{\frac{1}{a} + \frac{1}{b}}.$$ This ...
6
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69 views

Is there a numeral system that makes both addition and multiplication easy?

Decimal positional notation, the system for writing numbers we all use every single day, makes addition very easy by transforming it from a computation to a repeated operation on individual digits (...
6
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267 views

Quotition versus partition

This new Wikipedia article may look different by the time the reader of this question sees it. For now, it says $6\div 2$ can be construed in either of two ways: "How many parts of size $2$ must be ...
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97 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ &...
4
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66 views

Sorting prime numbers on two sets of equals weights

Lets denote $(p_n)$ the sequence of all prime numbers $(p_1=2, p_2=3,\ldots)$. The conjecture is the following. For infinitely many $n\in \mathbb N_{\geq 1}$ $$\exists I \subset \{1,\ldots n\...
4
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146 views

Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
4
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70 views

imo question to be explained in a manner so as to a layman

Suppose that you mark a finite collection of points on an infinite plane in such a way that you cannot draw a straight line through any three marked points. We define a windmill to be the following ...
4
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77 views

$d$ and $d+1$ both dividing certain integers

\begin{align} & 1\cdot 72 \\ & 2\cdot 36 \\ & 3\cdot 24 \\ & 4\cdot 18 \\ & 6\cdot 12 \\ & 8\cdot 9 \end{align} When the divisors of a number are listed in this way, let us ...
4
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67 views

On $1/7$ in base $12$

Remember something from seventh grade: \begin{align} & 142857 \\ {}+ {}& 142857 \\ \\ & 285714 \\ {}+{} & 142857 \\ \\ & 428571 \\ {}+{} & 142857 \\ \\ & 571428 \\ {}+{...
4
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143 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
4
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220 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
4
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383 views

Can this be only solved by trial and error?

The following question was asked in a competitive exam Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the ...
4
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104 views

Regarding identities with sums of consecutive squares

This comic http://abstrusegoose.com/63 points out an interesting identity with sums of consecutive squares. Let us take positive integers $k, p, q$, with $p < q$, and ask if $k^2 + \ldots + (k + p)^...
4
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112 views

Square and reverse reading of an integer

For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$, we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= \sum_{i=0}...
3
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0answers
62 views

Algebraic number that exponentiated with algebraic number give $\pi$

I'm not sure if an algebraic number elevated with an algebraic exponent can give rise to a transcendental number. If that's the case does anybody know a closed form for an algebraic number that ...
3
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0answers
26 views

Sum of integers and zêta functions

I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance: $$\sum_{m \leqslant X} m ...
3
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0answers
30 views

Does anyone know of a Mental Math Game for blind students?

I'm looking for any computer game made for blind students where the math exercises are asked through a computer's speaker and answered through a microphone by the student. Unlike normal math games, ...
3
votes
0answers
73 views

Application of GRR in number theory

In Neukirch Book Algebraic Number Theory page 254, states the Grothendieck-Riemann Roch-Theorem, but missing of applications. Do you know references for applications for this theorem, or may be ...
3
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0answers
56 views

Shepherdson's model for Open Induction

In the paper "A Non-Standard Model for a Free Variable Fragment of Number Theory", Shepherdson constructs a recursive model for a fragment of arithmetic known as "Open Induction". I would like to ...
3
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0answers
38 views

Is there a name for this property of multiplication (and other functions)?

Suppose $x,y \in \mathbb{R_+}, x<y$, and $ 0 < \varepsilon \leq (y-x)/2$. It seems to me that $xy < (x+\varepsilon)(y-\varepsilon)$ and equivalently that $(x+\varepsilon)(y-\varepsilon)$ is ...
3
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0answers
34 views

Finding lower bounds for solution of linear Diophantine equation in two varuables

I am currently encountering the following arithmetical problem : given four nonzero integers $A,B,C,D$, let $\Omega=\bigg\lbrace (x,y)\in{\mathbb N_{\geq 0}}^2 \ \bigg| \ \frac{Ax+By}{Cx+Dy} \in {\...
3
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0answers
37 views

question in product

can any expert just check my solution You bought a car for $\$2500$ down and made payments of $\$299.50$ each month for $36$ months. (a) Find the amount of the payments over the $36$ months. (b) Find ...
3
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0answers
82 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { \...
3
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0answers
105 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has $...
3
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0answers
42 views

How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$?

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0,a_i\in\mathbb Z,n\in\mathbb N.$ How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$? (For example, $r=10^{-...
3
votes
0answers
253 views

How does my $10\times10$ abacus work?

How does this $10\times10$ abacus work? More specific: Counting: is it common to count from 1 to 100, or from 1 to 9.999.999? How does addition work? How does multiplication work? Are there any ...
3
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0answers
300 views

Arithmetic mean sum

Let $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ng(k)=A $$ Then for what functions $f(x)$ does $$\lim_{n\to\infty}\frac{\sum_{k=1}^n f(k)g(k)}{\sum_{k=1}^nf(k)}=A$$
3
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0answers
108 views

Least characters in a numerical representation of integers

I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, $10=2\...
3
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0answers
3k views

Is there an equivalent to the distributive law for division over subtraction and/or addition?

I understand that the the distributive law cannot be applied to division over addition/subtraction, but is there an equivalent law to expand it out. For example, I know: $$100 \times (5 + 3) = (100 \...
2
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0answers
25 views

Can you check if my final answer correct? (Compound interest problem)

Problem: Blooie has the following obligations: 100,000 due after 4 years without interest and 200,000 dues after 3 years with accumulated interest at ($j_1$ = 0.12, $m_1$ = 2). Blooie wants to ...
2
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0answers
28 views

What is the name of the typography symbol for the line under which totals/solutions are written?

I've tried googling for this, but no keywords I can come up with provide meaningful results. Specifically, the line used in basic arithmetic for addition, subtraction, and multiplication. Does this ...
2
votes
0answers
62 views

What is difference between the “usual multiplication” and multiplication?

I have been reading books in the algebra, and I noticed that some books use terms "usual multiplication" and "usual addition". Do they carry different meaning that multiplication and addition? If "...
2
votes
0answers
22 views

Bending a horizontal from 0 to infinity real number line, ninety degress counter-clickwise at 1.

Can the real number line from 0 to infinity, which of course is often represented as a horizontal straight line, also be represented as being bent ninety degrees counter-clockwise at 1? I.e., if such ...
2
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0answers
50 views

A question on ordinal arithmetic.

I have to order these two ordinals and I was just wondering if I have done it correctly. $\omega^\omega + \omega^3$ and $\omega + \omega^3 +\omega^\omega$ I have worked out that $\omega + \omega^3 +...
2
votes
0answers
23 views

Partition of an integer of a particular type

I'm working on a project, but i'm stuck because i would need to count the different partitions of an integer which verify a certain property. I've never seen anyone looking at such a kind of ...
2
votes
0answers
43 views

How many number of multiplication/addition operations are there in a multiplication of two numbers of equal length?

BACKGROUND: Note: The following question arose in my mind when watching this lecture (watch at 5:30 minutes if you will). Assumption: Just for the sake of this question, let's assume that the term "...
2
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0answers
33 views

Mathematics of two's complement

I am trying to understand the underlying mathematics of two's complement. Googling the topic gives me a lot of articles on how to invert the digits and add one, and why computers use this system ...
2
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0answers
68 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ f_2(...
2
votes
0answers
242 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. $381600^{809197},...
2
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0answers
40 views

How to reduce exponentiation expressions?

It is a simple question but I am afraid of its simplicity. Is that correct : $2^{30}+2^{30}+2^{30}+2^{30} = 2^{30}(1 + 1 + 1 + 1) = (2^{30})\cdot 4 = 2^{30}\cdot2^2 = 2^{32}$? I am doing complex ...
2
votes
0answers
39 views

The smallest prime factor with a set of digits

I was wondering if there was a way to logically/mathematically derive what the smallest possible largest prime factor to a number was, using each of the digits 1-9 only once. An example could be ...
2
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0answers
38 views

Can I have a trailing dot at the end of a number?

Is 12. a valid way to say 12.0 I was trying it with python. If I say a = 12., python will ...
2
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0answers
46 views

Arbitrarily long arithmetic progressions?

I found a theorem that states that if $A\subset \mathbb{Z}$ such that the upper Banach density is non-zero, then $A$ contains arbitrarily long arithmetic progressions, this is called Szemer├ędi's ...
2
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0answers
30 views

Doubt about semigroups in this article (anyone can help).

I need help in this article. My doubt is very arithmetical and I think follows directly from the definitions. So I think anyone could help me. The author defines what is a semigroup, gaps and non-...
2
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0answers
54 views

Why have multiplicative operators precedence over additive operators?

Considering that addition is (in my understanding) a more basic operation than multiplication, would it not make sense to give it higher priority? That is to say, we would expect to encounter more ...
2
votes
0answers
31 views

find other sums similarly under sum

In the under sum there exists all number 1,...,9. Similarly write at least 10 sums other. $$659+214=873.$$ For example we can write $259+614=873$ or $619+254=873$ or $596+142=738$. Do there exists a ...