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Questions tagged [arithmetic]

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. Questions about number theory (sometimes called "arithmetic") should not use this tag and should instead use (number-theory) or (elementary-number-theory).

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(When) are recursive "definitions" definitions?

This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
ac2357's user avatar
  • 95
5 votes
1 answer
83 views

Decidable but incomplete arithmetical theories?

There are celebrated examples of theories that are both decidable and incomplete (the theory of algebraically closed fields, various toy theories with only finite models). But are there any examples ...
ac2357's user avatar
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1 vote
1 answer
80 views

Shouldn’t $0$ to any power be undefined?

So one of my younger cousins asked me this today. This is the summed up version of what they said. We know that for all real numbers $x^{n-1} = \dfrac{x^n}{x}$, because $x^{n+1} = x^n \cdot x$. So let'...
limaosprey's user avatar
6 votes
1 answer
277 views

High school algebra inequality question

I'm an adult who's trying to relearn high school math for my own betterment and I'm somewhat embarrassed to be taking up space and time on this forum with a very basic question such as this, but here ...
Tedd Firth's user avatar
1 vote
1 answer
93 views

A property of $\lt$ in Primitive recursive arithmetic

In Primitive recursive arithmetic (PRA), we can introduce $\lt$ by introducing its representing function $K_{\lt}$, where $K_{\lt}(x,y) =sg(x+1-y)$. Here "sg" and "-" are the ...
nilpotent's user avatar
3 votes
1 answer
264 views

Why is the divisor function not completely multiplicative?

I cannot see why that if $gcd(a,b) =1 \implies d(ab)=d(a)d(b)$ where $d$ is the multiplicative function such that if $a= a_1^{\alpha_1}a_2^{\alpha_2} \ldots a_k^{\alpha_k}$ where the $a_i$ are the ...
Victor Hugo's user avatar
1 vote
1 answer
72 views

Is there an extension of the digit sum function to real numbers?

Is there a known way to extend the base $b$ digit sum function $s_b(\sum_{i=0}^{l-1}d_i b^i)=\sum_{i=0}^{l-1}d_i$ to a smooth function on the real numbers similarly to how one can extend the factorial ...
mathemagician99's user avatar
-1 votes
1 answer
57 views

How to combine divisions and fractions?

On my Facebook-account, regularly I see questions about calculations, like $1 \div \frac{1}{2}$, which makes me doubt if this equals $0.5$ or $2$. I remember from my childhood that such notation was ...
Dominique's user avatar
  • 2,752
8 votes
2 answers
241 views

Why is it true that $\lfloor \sqrt{n}+\sqrt{n+1}\big\rfloor = \big\lfloor \sqrt{n}+\sqrt{n+2}\rfloor$

I would like to prove that $$\big\lfloor \sqrt{n}+\sqrt{n+1}\big\rfloor = \big\lfloor \sqrt{n}+\sqrt{n+2}\big\rfloor$$ for all positive integers $n$. Interestingly enough, this is not true for $\sqrt[...
MR_BD's user avatar
  • 6,100
0 votes
0 answers
27 views

Is it possible to arrange two summations in this way?

Can I always arrange those summations in this way $$\sum_x^N f(x) \sum_y^N g(x , y) = \sum_y^N \sum_x^N f(x) \cdot g(x,y) $$ for some positive integer $N< +\infty$. EDIT: $f$ and $g$ are two ...
fcoulomb's user avatar
  • 349
0 votes
1 answer
42 views

Commutative Property with More than Two Terms

Suppose I have the expression: a+b+c+d+e If I wanted to change the positions of a and e, is this one application of the commutative property of addition, where I can swap these terms directly, or ...
Jeff00's user avatar
  • 29
2 votes
1 answer
84 views

Are there more than two rational solutions to a certain system $abcd=a+b+c+d=K$ ($K$ a given constant)?

This question is a follow-on of a previous question asked some days ago which has been deleted due to its lack of precision. In fact, I found it well explained here But, implicitly, the domain of ...
Jean Marie's user avatar
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3 votes
1 answer
65 views

sum sets are equal

I have two sequence of positive integers say $a_1, a_2, a_3, a_4, a_5, a_6$ and $b_1, b_2, b_3, b_4, b_5, b_6$ (they may be repeated) such that the multisets $\{a_1, a_1+a_2, \cdots , a_1+a_2+\cdots +...
jack's user avatar
  • 362
1 vote
1 answer
96 views

Arithmetic content of proofs [closed]

Using some variant of Gödel numbering, we can turn mathematical propositions into integers (say $m$). The existence of a proof of such a statement can then be formulated in an arithmetic way, ...
Weier's user avatar
  • 785
1 vote
4 answers
101 views

Why can't we swap $5 \times 3 + 2 \times 6$ to be $5 \times 3 \times 6 + 2$

I'm teaching Order of Operations and a pupil had answered $5 \times 3 + 2 \times 6$ to be $$15 + 2 \times 6 = 15 \times 6 + 2 = 90 + 2 = 92 $$ The pupil argued that different multiplications can be ...
rainingagain's user avatar
0 votes
1 answer
63 views

Given a natutal number N, find the maximum possible product of a set of numbers whose sum is N

Given a natutal number N, find the maximum possible product of a set of positive integers whose sum is N. I read the solution and they split the problems in 3 cases by the reminder of N. If N is ...
John's user avatar
  • 11
1 vote
1 answer
64 views

Formula for Sum of Numbers with Consecutive Digits [closed]

Let's say we have a number $n$ that consists of $b$ digits, all of which are the same digit $x$. Then we have another number $m$ that has $b - 1$ digits, also all the same digit $x$. How would you ...
Sak's user avatar
  • 21
1 vote
0 answers
41 views

Is there a Relation for Exponentiation Similar to $\leq$ or $\backslash$ (divisibility)?

I'm trying to define common mathematical options on the natural numbers. I am doing this because operations like subtraction are commonly constructed as just the addition of the inverse, however this ...
Isaac Sechslingloff's user avatar
0 votes
2 answers
72 views

Trying to understand how numbers themselves (s0, ss0, sss0, etc) are represented in Gödel numbering

Problem solved: I did not actually read the table given on page 70 of nagel and newman. s does have a Godel number. It's 7. So ss0 would be broken down into 7, 7, and 6, since 0 is given the number 6. ...
Devery Sheridan's user avatar
0 votes
0 answers
39 views

how to prove subset sums are distinct

Here we only take finite sets in account. For example for one set $K=\{6,9, 11,12, 13\}$, we can prove using exhaustion by one program. If we take induction on the size $n$ of the subset, then the ...
An5Drama's user avatar
  • 416
-3 votes
1 answer
83 views

Why can't fractions be simplified separately within an expression [closed]

Here is a simple example: $\frac{-x}{2}$ + $\frac{6}{2}$ Why can't I simplify $\frac{6}{2}$ here which would give me 3 for an answer. The actual equation was y= $\frac{-x}{2}$ + $\frac{6}{2}$ I had to ...
olivetree's user avatar
1 vote
0 answers
33 views

Could we define a fifth arithmetic operation on real (or complex) numbers that is independent of addition, subtraction, multiplication, and division?

The four basic arithmetic operations with real (or complex) numbers are addition, subtraction, multiplication, and division. the first two being inverse operations and the last two being inverses of ...
Saaqib Mahmood's user avatar
2 votes
0 answers
76 views

'Galois cohomology of elliptic curves' , Poitou-Tate sequence

This is a question about exact sequence of p5 of 'Galois cohomology of elliptic curves' by Coates Sujatha which is called Pitou-Tate exact sequence. Let $F$ be a number field. Let $S$ be a finite set ...
Poitou-Tate's user avatar
  • 6,351
0 votes
0 answers
62 views

Why gradient of dot product results in cross product?

On wiki page Vector_calculus_identities It first says about the first derivative associative properties $(\mathbf {A} \cdot \nabla )\mathbf {B} =\mathbf {A} \cdot (\nabla \mathbf {B} )$ Then in the ...
Smith's user avatar
  • 1
-2 votes
1 answer
63 views

Let $x,y,z$ and $a,b,c$ be integers where $a,b,c$ are distinct integers $\ge{2}$, then show that the equation $x^a+y^b=z^c$ has no integer solution. [closed]

Progress: Let $x,y,z$ and $a,b,c$ be integers where $a,b,c$ are distinct integers $\ge{2}$, then show that the equation $x^a+y^b=z^c$ has no integer solution. Consider for example, $x^3+y^5=z^{10}$ ...
user18724's user avatar
  • 653
3 votes
2 answers
426 views

Represent addition or subtraction of two non-negative integers without using the + or - operators

This is more of a fun puzzle than a problem born out of necessity, but I would like to know if there is a way to represent the more fundamental operations of addition and subtraction without using the ...
HerrAlvé's user avatar
  • 243
1 vote
0 answers
96 views

${a_n}$: an arithmetic progression $\sum_{x_n=1}^{x_{n+1}}{\sum_{x_{n-1}=1}^{x_{n}}{\dots\sum_{x_1=1}^{x_2}{a_{x_1}}}}$?

Question) Prove that ${a_n}: \text{arithmetic progression}$, $$\sum_{x_n=1}^{x_{n+1}}{\sum_{x_{n-1}=1}^{x_{n}}{\dots\sum_{x_1=1}^{x_2}{a_{x_1}}}} = \frac{\prod_{i=0}^{n-1}(x_{n+1}+i)}{n!}\cdot\frac{...
user1317348's user avatar
2 votes
0 answers
88 views

For each $a > 1$, $\left]\frac{3^a}{2^a},\ \frac{3^a-1}{2^a-1}\right[ $ does not contain a Natural number

I am struggling with the proof of the following lemma: $\forall a > 1, \ 3^a = 2^a q + r, \ 0 < r < 2^a \Rightarrow 1 < q + r < 2^a$ This lemma allows deriving $3^a - 1 = (2^a - 1) q + (...
Franck Pepper's user avatar
2 votes
5 answers
203 views

approximate square roots of fractions with rationals

How to compute the rational approximation of square root of a fraction, i.e. I'd like to find $ \frac{a}{b} \approx \sqrt{\frac{m}{n}}$, where $a$, $b$, $m$ and $n$ are integers. Ideally, the ...
chaohuang's user avatar
  • 6,399
0 votes
0 answers
64 views

Why is 1+2+3 = (1+2)+3 [duplicate]

Not sure if this is a stupid question, apologize if it is. I am curious why we can add the first 2 numbers, then add the third one when doing addition of 3 numbers. There is a similar (IMHO) question, ...
Steve Lau's user avatar
  • 109
-1 votes
1 answer
113 views

Why am i getting wrong answer? What is wrong with my solution?

Question - After selling 10 candles a man earns a profit of the selling price of 3 pens. While selling 10 pens, a man loses selling price of 4 candles. The numerical value of profit percent and loss ...
raj rajput's user avatar
1 vote
1 answer
69 views

How to solve the roots of the following function numerically?

I have the following function $$f(n) = \frac{\sqrt{n}+ \sqrt{n+ 16 (\sqrt{n}+1)}}{8} - \frac{1}{2}\left(\sqrt[3]{n + \sqrt{n^2 - \frac{1}{27}}}+ \sqrt[3]{n - \sqrt{n^2 - \frac{1}{27}}}\right).$$ It is ...
SHASHANK RANJAN's user avatar
0 votes
0 answers
37 views

How can I express two positive integers as the real and imaginary part of an exponential sum

In the following problem, suppose you have two positive integers $A$, $B$, $A$ odd, $B$ even and let $A^2+B^2=p$ a prime. Let $g$ be a primitive root modulo $n$. For any $b$ in $\mathbb{Z}^{\times}_n=\...
3809525720's user avatar
3 votes
1 answer
25 views

Rankine-Hugoniot equation derivation intermediate step

I'm trying to go from, this equation: $\frac{\gamma}{\gamma -1} \left( \frac{P}{ \rho} - \frac{P_0}{ \rho_0} \right) - \frac{1}{2} (P-P_0) \left( \frac{1}{\rho} + \frac{1}{\rho_0} \right)=0$ to this ...
MicrosoftBruh's user avatar
0 votes
1 answer
54 views

Split number into range of numbers

I'm coming from a dev background and need to build a specific shipping calculator. But I find it difficult to figure out the correct algorithm for my case. I'm sure someone out there has had a similar ...
Faye D.'s user avatar
  • 105
0 votes
0 answers
42 views

Does this kind of fan shaped multiplication table have a name? How was it used in teaching?

I got this fan shaped multiplication table from my parents when I was a kid in Sweden in the 80:s. I don't know exactly where it came from (the author is stated as Johan Göransson) and judging from ...
PetaspeedBeaver's user avatar
0 votes
1 answer
58 views

In the addition $\begin{array}{c|c|c}+ A \ A\ A \\ \ \ B\ B\ B \\ \hline A \ A\ A\ C \end{array}$, $A,B,C$ each denotes a digit. Find them.

In the addition $\begin{array}{c|c|c}+ A \ A\ A \\ \ \ B\ B\ B \\ \hline A \ A\ A\ C \end{array}$, $A,B,C$ each denotes a digit. Find them. The solution says, "First of all $A$ denotes $1$ ...
ronald christenkkson's user avatar
1 vote
0 answers
50 views

The relationship between different definitions of height

We know that for a rational number $x=a/b \in \mathbb Q$, its height is defined by $$ \text{ht}(x)=\log\max\{|a|,|b|\}. $$ For an algebraic number $\alpha \in \overline{\mathbb{Q}}^{\times}$ with $\...
JingHao Yang's user avatar
-1 votes
1 answer
130 views

Consistency of first-order Peano arithmetic

I recently found a proof on Wikipedia that Peano arithmetic together with the axiom of its inconsistency is a consistent system if $PA$ is. It goes as follows: "Now, the consistency of $PA$ ...
Nikolai riber skånstrøm's user avatar
0 votes
0 answers
96 views

Closed form of $a_{n+2}=a_{n+1}a_n+1$

I was given this sequence and I need to find a closed form. $$a_0=1,a_1=2$$ $$\text{and } \forall n \geq0\text{ } a_{n+2}=a_{n+1}a_n+1$$ I tried defining the following generating function: $$A(q)=\...
Nizar_141's user avatar
1 vote
4 answers
124 views

Ambiguity textbook exercise involving $\sqrt{-144}$

Consider the following questions, the whole exercise is dedicated to determining the square root of negative numbers, after introducing the complex numbers. Eg. $$\sqrt{-144}$$ Solutions for the whole ...
MTGOD's user avatar
  • 19
0 votes
0 answers
43 views

Question about exponent laws [duplicate]

I can't seem to demonstrate how $\frac{\sqrt{\sqrt{3}+2}}{2}$ is equal to $\frac{\sqrt{2}+\sqrt{6}}{4}$... even though my calculator tells me that both are equal $0.965925826$. Hoping someone can show ...
sesandc3123's user avatar
2 votes
1 answer
72 views

Is there a problem if I don't use $0$ in Peano arithmetic?

Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema. $\forall x \ (0 \neq S ( x ))$ $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$ ...
MathMan's user avatar
  • 103
0 votes
0 answers
37 views

Stationary sequence of orders in $(\mathbb{Z}/p^n\mathbb{Z})^\ast$

I am struggling to prove the following statement: Let $p, q$ be coprime numbers. For all $n \in \mathbb{N}^*$, we define $t_n$ as the order of $[q]$ in $(\mathbb{Z}/{p^n}\mathbb{Z})^\ast$. Show that $(...
Arthur Filippi's user avatar
1 vote
1 answer
76 views

Sharing of profit in a partnership

I have two theories and I was pondering on what basis we are choosing one over the other. Let me take specific example and describe the two theories and any discussion in this regard would be helpful ...
Yathi's user avatar
  • 2,470
12 votes
1 answer
348 views

The proportion of primes whose digit sum is prime, out of all primes

Background: I recently got an idea. My idea was analyzing primes whose digit sum is prime. This is a property of some primes ($> 1$ digit): The sum of its digits is prime. I will call a special ...
NotMath's user avatar
  • 429
0 votes
0 answers
19 views

How to reduce division with uprounding to integer-arithmetic operations with downrounding if the denominator is not necessarily an integer?

Let 𝑖∈ℕ₀ be an unknown variable and 𝑐∈ℚ₊ be a known constant such that 𝑐≥2 and 100𝑐 ∈ ℕ₊. (So we can pre-compute anything regarding 𝑐, e.g., represent 100𝑐 as a product of powers of primes.) We ...
AlMa1r's user avatar
  • 101
12 votes
4 answers
2k views

An exact definition of multiplication

I am looking into repeated operations, and it seems really hard to precisely define multiplication. Of course, for integer $b$ and real number $a$, we use the grade school definition we all know: $$ab ...
NotMath's user avatar
  • 429
16 votes
5 answers
3k views

Can any characterized property give a solid account of why multiplication is a harder computation operation than addition?

For humans, in general, it's far easier to do an addition than a multiplication. One might argue this is just an effect of the particular way brains are wired. But from what I find, multiplication is ...
psychoslave's user avatar
1 vote
1 answer
121 views

Question about finding cube root by hand

To find the square root of $125$ by hand: $$ \begin{array}{l|l} & 1,25.00 & 11.1 \\ &1 & \\ \hline 21 & \space \space \space \space 25 & \\ &\space \space \space \...
ronald christenkkson's user avatar

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