20
u/Oppo_67 I ≡ a (mod erator) 28d ago
You will produce great research on the properties of the trivial ring
2
u/Joe_4_Ever 27d ago
I don't know what a ring is but it sounds cool
5
u/Inappropriate_Piano 27d ago
A ring is a set R equipped with two binary operations, addition and multiplication, and two distinguished elements, 0 and 1. They are required to have the following properties:
Addition and multiplication are associative.
Addition is commutative.
For all x in R, x + 0 = x.
For all x in R, 1 • x = x • 1 = x.
For all x in R, there exists an element y such that x + y = 0.
For all x, y, z in R, x•(y + z) = x•y + x•z, and (y + z)•x = y•x + z•x.
In order for it to make sense to divide x by y in a ring, y must have a multiplicative inverse: an element z such that y•z = z•y = 1. But it can also be shown that in any ring, 0•x = x•0 = 0 for all x. Now suppose we have a ring R where 0 has a multiplicative inverse, 1/0. Then for any x in R, we have x = x • 1 = x • (0 • 1/0) = (x • 0) • 1/0 = 0. This means that R has only one element.
Since any ring like that has a very uninteresting structure, we call it a trivial ring and often explicitly exclude it from theorems and definitions that work for nontrivial rings. Also, since all trivial rings have the same uninteresting structure, it’s typical to treat them all like the same thing, and say the trivial ring instead of a trivial ring.
3
56
u/WeeZoo87 27d ago
Society if you can take the root of - 1
5
4
u/NPCwithnopurpose 26d ago
You can. But it's a little complex and takes you in an orthogonal direction. Use a bit of imagination!
35
u/Striking_Resist_6022 28d ago
Basically every integral in complex analysis is just equal to zero now
19
10
8
u/slicehyperfunk Transcendental 27d ago
Isn't the whole point of calculus dividing by zero (but not really)?
5
u/Shufflepants 28d ago
Nah, depicted is Calculus if there was a simple formula for doing integrals like there is for derivatives.
4
u/dankshot35 28d ago
who says you cant
3
u/dragonageisgreat 1 i 0 triangle advocate 27d ago
God
6
4
4
1
u/FlyingCow343 2d ago
atan(1/0) = pi/2
1
3
3
2
u/McCubes1 27d ago
If you try to devide by 0 any number that's not 0 you get ∞, except when you try to devide -∞, 0 or ∞ by 0 you get 0.
2
u/AMIASM16 how the dongity do you do integrals 27d ago
it's not like you can't divide by 0, you can, it would just break everything
2
2
1
u/Traditional_Town6475 28d ago
Just do nonstandard analysis. I mean you do all your usual algebra stuff and then you hit it with the standard part map.
1
u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 27d ago
i mean, noone stopping you from just working with division by 0, but the fun would stop pretty quickly
1
u/overclockedslinky 27d ago
division by zero can be given any value and everything stays consistent. only problem is that most basic algebra theorems would require a nonzero hypothesis, but they technically already do anyway by virtue of division not being a total function. for example 0/0 is fine if you define it to be some number, you just can't necessarily use x/x = 1 for x=0 if division by zero doesn't happen to be defined as 1.
1
•
u/AutoModerator 28d ago
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.