# 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 ...
• 95
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### 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 ...
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1 vote
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### 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'...
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 ...
1 vote
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### 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 ...
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 ...
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1 vote
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 ...
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 ...
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### 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, ...
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1 vote
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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. ...
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 ...
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### 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 ...
1 vote
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 ...
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### '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 ...
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### 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 ...
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### 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, $x^3+y^5=z^{10}$ ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
### 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. $\forall x \ (0 \neq S ( x ))$ $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$ ...