0

What prompted this question is this graph I made in Desmos:

enter image description here

I had come across a comment online where someone said in some contexts 2+2=5, which immediately made me think of modular arithmetic and I wondered in what mods 5 is congruent to 4, hence I plotted the above function. As you can see, Desmos has computed fractional roots, which doesn't make much sense to me. I assume it must be using some generalized definition of the mod function, rather than the usual one based on remainders from integer division. But how can that definition be generalized?

I did find one question on here asking about how to do algebra when there's real-valued version of of modular arithmetic involved and someone answered saying something about the reals being an abelian group, which I assume meant there is some standard extension of modular arithmetic to the reals, but it's not clear to me what that specifically has to do with the reals being abelian.

Bill Dubuque
  • 272,048
Mikayla Eckel Cifrese
  • 289
  • 1
  • 12

1 Answers1

0

How can modular arithmetic be extended to the rational or real numbers?

Perhaps the best question to start with is "Can modular arithmetic be extended to the rational or real numbers?", which leads us to "What is modular arithmetic?", which I like to think is "The arithmetic of some ring 'passed down' to its quotients, by means of modding by some ideal". This in turn leads us to a negative answer: there's no modular arithmetic for the rings of rational or of real numbers simply because they are fields, and these (are exactly the rings that) don't have non-trivial ideals

But the story doesn't have to end there: what would "The Fundamental Theorem of Modular Arithmetic" be? Most certainly it's the Chinese Remainder Theorem, and in the case of the integers, I like to think that

$\mathbb{Z}/ \langle 2 \cdot 3 \cdot ... \cdot p_n \rangle \cong \mathbb{Z}/ \langle 2 \rangle \times \mathbb{Z}/ \langle 3 \rangle \times ... \times \mathbb{Z}/ \langle p_n \rangle$

suggests that "adding in factors" to the RHS makes the LHS "approximate $\mathbb{Z}$ ever better/more (faith)fully". As it turns out, "passing the limit to infinity" gives us the profinite integers (see also wikipedia). And also, if instead one prefers to have their fields algebraically closed from the go, and doesn't mind replacing the more familiar and straightforward product

$\prod_p \overline{ \mathbb{Z}/\langle p \rangle }^{alg}$

with its 'weighted avarage'-like quotient known as the ultraproduct

$\prod_p \overline{ \mathbb{Z}/\langle p \rangle }^{alg}/\mathcal{U}$

we get (theorem 2.4.3 or theorem 2.13.)

$\prod_p \overline{ \mathbb{Z}/\langle p \rangle }^{alg}/\mathcal{U} \cong \mathbb{C}$

so maybe we'd always had some modular arithmetic extended even beyond the rationals and reals, all along

ac15
  • 984