/sci/ - Science & Math

>>9874762
y=mx+b comes from the slope formula.
If you know a point and a slope, you can get the line that passes through this point by setting one of the y's and one of the x's as a variable and solving for y(variable).

>>9874762
how would you define the set of all even natural numbers? y=2x

how would you define the set of all odd natural numbers? y=2x-1

its just a basic relationship between two numbers

>>9874762
The Ancient Greeks used one dimensional lines to represent distances and numbers. Algebra (in the sense of just manipulating symbol and solving equations) was started by the Greek Diophantus and also independently developed in the East. It was Descartes who first introduced the second coordinate y and third coordinate z to represent a plane and 3D space. He combined algebra with his vision of geometry to start "analytical geometry" and the algebra you're used to seeing.

>>9874762
Descartes figured that out. It isn't that complicated but still very non-trivial. Today you can easily discover something like that with vector algebra.

>>9874778
>using 2x-1 instead of 2x+1

>>9874787
0 isn't an element in the set of natural numbers

>>9874790
Yes it is. 0, successor function, and N are given. I construct my Peano-axioms with 0 not 1.

>>9874762
Read Decartes.

>>9874792
>0, successor function, and N are given
Fuck off von Neumann.

>>9874792
But why would you define N with 0? Think about it. It is impossible to define N without including 1 as then you can't define arithmetic with the successor function. But 0? Sure, you can add it, but it is not beautiful because it is not fundamental to the nature of N. Defining N with 0 is as retarded as it would be for someone to define N with -1. Or define N to include all the numbers bigger than -500. Absolutely pointless.

>>9874793
Descartes*.

>>9874798
>But why would you define N with 0?
It allows me to define odd numbers as 2n+1

>>9874798
>>9874797
>>9874792
>>9874790
Tao includes 0

>>9874805
Absolutely pointless. You have no concept of mathematical beauty. You are like a pig that doesn't mind eating shit.

>I define N with -1 because then I can define numbers congruent to 1 mod 4 as 4n + 5

>>9874808
Well, he is wrong.

>>9874809
Where's your additive identity with no 0?

>>9874813
it's a set, not a group.
idiot
(N_0,+) is a group

>>9874813
N doesn't have an additive identity

>>9874813
You don't need an additive identity for anything in N. That is why it is pointless.

Any proof about the nature of N that invokes 0 can be modified to sidestep that. However, you can't do that with 1. That is why 1 is the fundamental starting point of N.

>>9874815
>>9874816
>>9874818
I'm not the only one who constructs N from 0, that's how I was taught to do it and I shall continue thank you very much.

>>9874820
I know that some people use 0. This is mainly done in amateur texts. Anyone with any formal training in logic can recognize why the construction of N should begin with 1 to maximize mathematical beauty. N must be constructed with 1 because the nature of N is directly related to the nature of 1. To give an easy example, induction (the literal axiom) is a very 1-centric proposition.

Then 0 must be added in a follow-up construction known as [math] \mathbb{N}_0 [/math] which is the smallest semigroup that contains [math] \mathbb{N} [/math].

Then the negatives in a follow-up construction to construct the smallest group that contains [math] \mathbb{N} [/math], and so on. But all I've said presupposes that you care about mathematical beauty, but if you barely know math then appreciating its beauty is impossible. So you add 0 to please the brainlets.

>>9874820
you construct N from ... 1?
1, 1+1, ... , 1+n, ... , 2n, ... etc
N0 from including such an element e=0, so that for any n from N n+e=n
Z, so that a+a'=e, ie a'=-a
that's just to define a group under addition

for multiplication you need to define Q, R and C, all with Z, N0

CAN YOU AUTISTS FUCK OFF, THIS THREAD WAS FOR MATH HISTORY

>>9874833
nice dubs

>>9874833
Well, not my fault you said that 0 is a natural number. That's like posting a picture of an anal gangbang and then expecting people to just ignore that a literal gangbang was posted in a blue board.

>>9874825
tao uses 0, he is no brainlet nor is he ignorant to the beauty. You pointlessly build from 1, there's no point. I understand your point about induction but to me that has more to do with the successor function than 1.

>>9874836
0 is a natural number boi, I construct my N from 0 as do the greats

>>9874838
1 is the fucking manifestation of the successor function.

>>9874825
Well put. It's rare to see a good post on /sci/ these days.

>>9874842
If you construct N from 0 when it is not necessary, what is your reason for not constructing N from -1? Or from -50? Or fuck it , -500000?

>>9874838
Tao is a fucking meme mathematician.

>>9874843
yes, it is the successor of 0
>>9874845
why would I start from any of those?

>>9874848
>why would I start from any of those?
Why would you start from 0?

>>9874848
0 is the successor of -1.

>>9874833


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REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

>>9874852
>that face when you haven't had sex with a white guy in months so you started craving that but then you go to the beach and get reminded of why you went black in the first place.

>>9874854
>>>/pol/

>>9874815
>(N_0,+) is a group
no, its a monoid. a group requires inverses