Skip to main content
21 votes

Is there a generalization of factoring that can be extended to the Real numbers?

This is a question that is usually answered via Ring Theory. You can think of a ring as a set where addition, substraction, and multiplication is well defined. Here, we aren't thinking of ...
Daniel5803's user avatar
6 votes

Is there a generalization of factoring that can be extended to the Real numbers?

Obviously you can factor a real number into any set of numbers whose product is that real number. As for PRIME factorizations I know of one generalization of that probably DOESN'T work well for real ...
Mr. Nichan's user avatar
3 votes

Is there a generalization of factoring that can be extended to the Real numbers?

Let's take a different perspective than the other answers have taken so far. You can think of prime factorizations as telling you exactly how the natural numbers $\mathbb{N}$ (here I am excluding zero)...
Qiaochu Yuan's user avatar
3 votes
Accepted

Comparing the dissimilarity of the order of integer sequences

There are various (Dis)Similarity measures. (A) You have given Dot Product , which is very common. $S=\Sigma a_ib_i$ (B) Commenter Malady has indicated Vector Norm , though there are variations there. ...
Prem's user avatar
  • 12.1k
2 votes
Accepted

Counting odd integers in a consecutive sequence divisible by a given prime

If I understand the problem correctly, the answer seems to be no, because $1$ is relatively far away from an odd multiple of a prime. For example: (With $p=7$, $a=6$, $n=19$): $O_7(1,1+19)=1$ ($7$ is ...
paw88789's user avatar
  • 40.9k
1 vote
Accepted

Two's complement multiplication decomposition

Let the binary form of $x \cdot y$ be represented using $\vec{a}$ whose dimension is $2w$. Therefore, by Equation 2.3 $$ \begin{aligned} x \cdot y = B2T_w(\vec{a}) &\overset{\text{def}}{=} -a_{2w-...
silversilva's user avatar
1 vote

Is there a generalization of factoring that can be extended to the Real numbers?

Any rational $x>0$ can be uniquely written as $x=\prod{p_i}^{m_i}$ with $p_i$ the $i^\text{th}$ prime and $m_i\in\mathbb Z$ its associated multiplicity. This extends the notion of factorization to ...
fgrieu's user avatar
  • 1,768
1 vote
Accepted

Demonstration of the inverse additive of an element of Z is unique

If you are working in a structure that assumes commutativity as an axiom, then your proof is fine. You just found another way to prove the same statement. Otherwise, there are structures that do not ...
Hyperbolic Cake's user avatar
1 vote

Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$

The $>$ is not part of the language used in the Peano Axioms, so if we want to prove this statement, we'll need to add a definitional axiom. For example, we can use: $\forall x \forall x (x > y \...
Bram28's user avatar
  • 101k

Only top scored, non community-wiki answers of a minimum length are eligible