Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

learn more… | top users | synonyms (1)

8
votes
0answers
153 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
7
votes
0answers
155 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
votes
0answers
53 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
votes
0answers
244 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& ...
5
votes
0answers
39 views

Primitive recursion and $\Delta^0_0$

Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter). I have ...
5
votes
0answers
109 views

Calculate $\sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \ldots $

$$ \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}+ \frac{1}{2}\sqrt{\frac{1}{2}}}} \times\ldots$$ I already know ...
5
votes
0answers
134 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 ...
4
votes
0answers
66 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
votes
0answers
63 views

On $1/7$ in base $12$

Remember something from seventh grade: \begin{align} & 142857 \\ {}+ {}& 142857 \\ \\ & 285714 \\ {}+{} & 142857 \\ \\ & 428571 \\ {}+{} & 142857 \\ \\ & 571428 \\ ...
4
votes
0answers
70 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
votes
0answers
114 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
votes
0answers
212 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
votes
0answers
94 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 + ...
4
votes
0answers
103 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}= ...
3
votes
0answers
25 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
votes
0answers
77 views

What was babylonians estimation for square root 3?

We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: ...
3
votes
0answers
35 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
votes
0answers
79 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
votes
0answers
76 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
votes
0answers
157 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 $\omega^\omega$: ...
3
votes
0answers
38 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, ...
3
votes
0answers
114 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
votes
0answers
215 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
votes
0answers
279 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
votes
0answers
103 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, ...
3
votes
0answers
2k 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
votes
0answers
21 views

Determine if the members of a set can be made to equal a given number

Is there an easy way to determine if some combination of addition, subtraction, multiplication, and division will enable the numbers in a set to equal a given number? For example, if I have the ...
2
votes
0answers
35 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
36 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
votes
0answers
34 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
votes
0answers
26 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
votes
0answers
26 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 ...
2
votes
0answers
36 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
29 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 ...
2
votes
0answers
36 views

Significance of formulas similar to summation formula

We all know formula $n(n+1)/2$ for adding up the numbers from $1$ to $n$. But I would like to know if there is any significance and use of formulas of type $n(n^{p-1}+p-1)/p$, where $p$ is a prime. ...
2
votes
0answers
45 views

Given a set of nonnegative numbers, put $\pm$ between them to minimize the magnitude of the result

Let's say I have a finite set of non-negative numbers. I have to put $+$ or $-$ between the numbers, in order to minimize the absolute sum.(i.e the sum has to be closest to 0) For example: the set: ...
2
votes
0answers
165 views

Relationship between two elements of two matrices with two numbers are not elements of the two matrices

I have two matrices, $$A= \left[ \begin{matrix} 5 & 10 & 15 & \cdots \\ 17 & 28 & 39 & \cdots \\ 35 & 52 & 69 & \cdots \\ \vdots & \vdots & \vdots & ...
2
votes
0answers
278 views

The set of all natural numbers is closed under addition

I'm trying to prove the theorem described in the title, but my proof is so obvious I doubt it is sufficient. Here's my way of proving it: Definition of addition: Let a, b, and c be natural numbers. ...
2
votes
0answers
148 views

Inserting +/- into 123456789…

I'm looking at a generalization of the problem of inserting + and/or - into the blocks $123456789$ and $987654321$ to create a formula for $100$, like this: $$123 - 45 - 67 + 89 = 100$$ $$9 - 8 + 7 ...
2
votes
0answers
110 views

Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
2
votes
0answers
52 views

A problem on generalization of a solution

Consider the following question: Two cars A and B start simultaneously from two different cities P and Q respectively and move back and forth between the cities.(As soon as car A reaches city Q ...
2
votes
0answers
93 views

Necessary criterion for a field extension to be normal

I'm working on a lemma concerning some Galois theory and arithmetics. Let $p$ be an odd prime and $K/F$ be a finite Galois extension of number fields of order prime to $p$ with Galois group $H$. Let ...
2
votes
0answers
171 views

Number of solutions for $x$ of such form that $\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$

Consider $$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$ where $y$ is an integer. In relation to solutions for $x$; How could one prove that: $(1)$: There are $y$ solutions for $x$, in total. ...
2
votes
0answers
150 views

Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
2
votes
0answers
106 views

Clarification of variable values in Arithmetic Coding algorithm

I have been trying to follow this video to implement my own Arithmetic Coding algorithm in Java. I am having a bit of trouble figuring out what some of the variables in the video should be. For ...
2
votes
0answers
92 views

Need to determine the formula to work out a reduction percentage

We currently calculate a value for a business function as follows; Inputs: A list of values such as; 12 12 6 12 We then add 1 to each of these values and we have values as follows; 13 13 7 13 ...
2
votes
0answers
71 views

How to prove that $G_3>0$ in this case?

Let $\Lambda=\{a+be^{2\pi i/3}|a,b\in Z\}$, then $G_{3}(\Lambda)=\sum_{\omega\in\Lambda-\{0\}}\frac{1}{\omega^{6}}$ should be real and nonzero, but how can one prove that it's positive? Moreover, in ...
2
votes
0answers
16 views

Decomposability in a context of size constraints on intervals

Let $F$ be a finite set of pairs of positive integers. Say that a set $A \subseteq {\mathbb Z}$ is $F$-admissible iff its intersection with any integer interval of length $a$ has cardinality at most ...
2
votes
0answers
95 views

Is there a term used to describe both an equation and inequality?

Is there a term used to describe both an equation and inequality? The closest thing I can think of is "relation".
2
votes
0answers
130 views

1/3+2/3 in double precision

When I add 1/3 and 2/3 in double precision, I ended up with $1.\boxed{111\ldots1}1\times2^{-1}$, where the boxed part is the 52-bit mantissa. By the rounding to even rule, I should round it up, right? ...