Define what a real number is.
The limit of a convergent sequence of rational numbers.
>>8030689
Define a limit without using real numbers.
>mfw normies do math with reals
>>8030684
>Define what a real number is.
Seven.
All those other numbers are fuckin' bullshit.
>>8030684
The Supremum of a bounded set
>>8030726
>mfw I read it as superman
>>8030692
That's actually pretty trivial in a metric space (X,d) where the distance between any two points in the space is a rational number.
f(t) -> L in X as t -> x in X
Iff
For any rational number p>0, there exists a rational number q>0 such that
d(f(t),L)<p whenever 0<d(t,x)<q and t in X and x in X.
This definition might involve the axiom of choice however.
Idk and idc, complex numbers are infinitely more comfy than your shitty real numbers and cartesian plane.
>>8030692
>he never took topology
>>8030684
What is the probability that you will choose any given real number at random? 1/|R| where |R| is the cardinality of the reals? What the hell is that?
>>8030802
it's 0
>>8030807
Ha that's a good point! I'm thinking about this problem but in terms of reals instead of naturals: https://www.careercup.com/question?id=12426697
If you take the function to return a float on a computer that's still a finite set of possible values so I think it's just like the case of the naturals
>>8030813
yep
>>8030793
The only topology I've taken is this:
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
That way I don't have to deal with real number voodoo.
>>8030692
Limits don't use real numbers in general, they use open sets. Check out topology. Once you have this, it's easy to define open sets on rational numbers, whence you can receive limits of rational numbers which are not rational (i.e. the space of rational numbers is not closed under limits). Why would you care about the limits of rational numbers? Consider Trying to find the circumference of a circle. It will usually require some sort of series, which will be a sequence of rational numbers, which converges to a multiple of some number which is not rational, pi. I hope this helps.
>>8030692
>What is a cauchy sequence of rationals
oh boy
>>8030689
>>8030726
>>8030774
>>8030793
>>8030802
>>8030807
>>8030813
>>8030826
>>8030892
>>8030900
None of these definitions are satisfactory. How do you intuitively define a real number? Any line I draw will always have a smallest possible distance between any two specs of ink. Thus it is ultimately discrete.
Real numbers are a facade.
>>8030953
>intuitively
>he doesn't understand it so it's not intuitive
any line you draw will be unreachable to us like anything else in the real world
>>8030961
But what even is a line? A line requires a sense of continuity. Something impossible without creating arbitrary definitions like the real numbers,
>>8030953
Okay Wildberger, let's up the rigor in mathematics by defining natural numbers as strokes on a board and real numbers as specs of ink.
>>8030976
It works.
>>8030689
Incorrect. A real number is the [math]{\it equivalence~class}[/math] of the Cauchy sequences on [math]\mathbb{Q}[/math]. Convergence is just an artifact of the fact that [math]\mathbb{R}[/math] is complete.
>>8030692
>being this retarded
[math]x[/math] is a limit point in the topological space [math]X[/math] if every neighborhood [math]x \in U[/math], [math]U \in \tau(X)[/math] intersects [math]X[/math]. You don't even need a metric for this.
>>8030987
>equivalenceclass
equivalence classes are a lie
>topological space
So instead of defining it more intuitively, you make the definition even more abstract and pointless? Next thing you know you're gonna be defining things in terms of Schemes.
>>8030953
>Any line I draw
so things in mathematics which don't apply to your specific PHYSICAL scenario are facades?
if we're going to call everything that is slightly abstract a facade then we're going to have to eliminate a lot of solid mathematics, amigo
also, you ask how do you intuitively define a real number
surely the power of mathematics comes from logical formalism - the entire point is that theorems are NOT created via intuition but instead deduced.
If your mathematical definition is reliant upon intuition, or if you demand that it fits some physical system, you're doing it wrong
>>8030991
>m-muh intuition
Fuck off retard.
>pointless
All non-empty topological spaces have points you stupid idiot.
>>8030980
So does my wiener but it's not a proof
>>8030991
spotted the pseudo intellectual
>>8030684
A member of a complete ordered field named R.
>>8031094
Lmao [math]\mathbb{F}_p[/math] is also a completely ordered field you cuckold and it's not even finite. Do you even know how to use algebra to classify [math]\mathbb{R}[/math]?
I'm curious as to why they can't be formally defined as infinite sums, sort of like an infinite decimal expansion.
Taking an axiomatic definition I think is fine to, which essentially says every decimal expansion has a limit.
Numbers are not real.
They're imaginary constructs representing an idea.
They only exist in a metaphysical space.
>>8030992
>you're doing it wrong
what you're doing is not real with real numbers.
sorry I don't have fish in this fight I just wanted to say that.
>>8030684
hol up hol up fampais
A real number is a number that can be accurately plotted on a number line. In other words, a real number is a finite one-dimensional number.
>muh dedekind cuts
>muh Cauchy sequences
>muh limits
>muh epsilon proofs
When will normies learn
>>8030684
The real problem here is that numbers in R like pi are defined as functions without order and thats just gayest thing ever.
>infinity
what you wanted to say is "an unspecified big amount wich is presumed to be endless to make our calculations less complicated"
Reals are the splitting field of the rationals for every polynomial.
>>8031506
ikr
>>8030684
Define define
Fp is in fact not an ordered field, and it is finite you fucking cuckhold
>>8031658
Good point.
>>8030996
>All non-empty topological spaces have points you stupid idiot.
kek
>>8030953
How is the topological completion of the rationals not an intuitive idea?
>>8031520
Like [math]X^2+1[/math]?
The volume of the cum i just jacked off in m^3 is a real
>>8031747
Can you express it in a simple intuitive picture? Like we can with integers or rationals? I didn't think so.
>muh topology
lol nice try loser
>>8031774
I can't put the rationals nor the integers in a picture, and neither can you.
>>8031774
Yes, you're filling in all of the holes in the rational numbers.
Im(z)=0
>>8031818
Draw picture pls
>>8031818
Define a function which takes x to x^2. This will hit the values 1 and 4, but never the value 2. There must be a hole somewhere then, because the graph of this continuous function jumps over values.
>>8031818
Alright mister "I have enough time on my hands to draw infinitely many points arbitrarily close together", please demonstrate.
>>8031826
>pfff
by god, what a beautiful argument. It answers the previous arguments with such a clarity that nothing could stand against it. You sure convinced me [spoiler]with all these hot opinions of yours[/spoiler]
>>8030684
An element of the set [math]\mathbb{R}[/math].
You dumbfuck probably meant "define [math]\mathbb{R}[/math]", which is defined as [math]\mathbb{Q}\cup\mathbb{Q^*}[/math]
>>8030684
A complex number without zero imaginary components.
>>8030987
I like the pic (except it's ugly, but I still like the idea and concept)
>>8032283
Define these "complex" numbers.
Equivalence classes of rational Cauchy sequences for the relation [math](u_n)\mathscr R (v_n) \Leftrightarrow \lim_{n \rightarrow \infty} u_n-v_n = 0[/math].
>>8032333
>n→∞
>∞
topkek
>>8031758
Lmao no, that's the complex numbers. [math]\mathbb{C} = \mathbb{R}[x]/\langle x^2+1\rangle[/math]
>>8031123
Because decimal expansions aren't unique, which gives you shit like [math]0.99\dots = 1[/math].
>inb4 I get memed on
>>8032359
>reading comprehension
He claimed R was the splitting field of *every* polynomial over the *rationals*, and I was pointing out that he's wrong. Your post has almost nothing to do with what either of us posted.
>>8032377
>0.99⋯=1
These memes man. I can't believe anyone can take real numbers seriously.
>>8030684
Any youtube math videos like wildbergers but don't involve crackpot shit?
>>8032444
Math Doctor Bob
>>8032408
0.99....=1 makes perfect sense though if you just write it as
1-(0.99.....)=0.00.......
Since the ellipses notate an infinite repeating string of decimals, we can take the above to be an infinite string of 0s after the decimal point. In other words, just zero.
I'm no mathematician (filthy physics undergrad), but I think that's sufficient explanation.
>>8032444
Are his math history videos ok, or is he gonna try to cram dumb shit down my throat?
>>8032481
>infinite
It's like you are not even trying.
>>8032487
Please forgive my fumbly (read: probably incorrect) language, but I'm pretty sure I got my point across.
>>8032305
A composite number made from pairing a real number with an imaginary real number.
>>8032496
>real number
...
>>8032499
Would it help if I said that they're paired using the ordinary addition operator?
>>8032541
>addition
stop with the memes
>>8032541
you define the real numbers as complex numbers with no imaginary part then define the complex numbers as made up of real numbers paired with an imaginary part
You really don't see any circular logic in your definition? You really think that those are meaningful definitions?
>>8030987
someone tell me what that picture is (no need to explain, i'll google). all I know is it's one of those diagrams they use in algebra. also how do i into reading these diagrams?
Everyone in this thread
>implying numbers are real
>>8032559
Literally just functions. The curved arrows generally denote an inclusion, i.e. the domain is a subset of the codomain.
Since OP is either a troll (very likely) or a monumental idiot I want to know what restrictions are set.
Basically OP:
>What are we allowed to use
>what is the goal we should approach and
>what does it actually matter, since real numbers have proved themselves fundamental to our math.
>>8032571
>What are we allowed to use
anything intuitive
> what is the goal we should approach
defining a real number
>what does it actually matter, since real numbers have proved themselves fundamental to our math
thats your opinion
>>8032559
Commutative diagrams
>>8032583
>anything intuitive
completely subjective, and there are many posts already that I would call intuitive
>>8030684
[math] \forall q [/math] and [math] p \in \mathbb{Q} [/math], [math] \exists r \in \mathbb{R} [/math] such that [math] q < r < p [/math].
>>8032559
More specifically it shows the function triangles in pic related.
The expression "on the nose" is sometimes used in mathematics in those fields. I wonder if that anon made the pic himself (I'm the guy who said it's cool but ugly/improvable)
>>8032610
double thanks
>>8032343
Name me a rational number [math] q [/math], such that
[math] q + 1 = q [/math].
https://acko.net/blog/to-infinity-and-beyond/
maybe this helps OP
>mfw all computers work with discrete values
>mfw there is absolutely no need to introduce real numbers, since we will always work with finite sets in any actual computation
>mfw the wild burger is right but for the wrong reason
>>8032635
>Name me a rational number q
>number
>q
q is a letter
>>8032667
You're right. We should also never assume that things are circles or squares.
>>8032805
kek, I remember grade-school humor
>If it doesn't exist in nature it isn't real
Getting really tired of this meme
Define what a real number is without using infinite sets
>>8033164
You first
>>8033167
OK. A real number is any string of digits with a dot/point somewhere in it where the digits to the right of the dot/point can extend infinitely. There, I used infinity, but no sets.
>>8033188
I thought a real number was just a number like 4.0 or 1,000
>>8033189
They're both strings of digits
>>8033190
So any number that isn't imaginary
>>8033192
Or complex, or any number of other possible kinds of numbers like quaternions, p-adic numbers, etc.
>>8033196
So any number without an imaginary component.
>>8033202
Nah, you can have quaternions with only j and k components, and p-adics don't necessarily have anything to do with that stuff
>>8033209
>without an imaginary component.
>quaternions with only j and k components
>>8033215
j and k are not imaginary numbers. We can call them jmaginary and kmaginary, if you want.
>>8030987
What does it mean in this context that U intersects X?
That the intersection is non-empty or something?
>>8033218
A quaternion with 0 real part is called pure imaginary. Stop making shit up based on your own interpretation.
>>8031689
That is the best description of Wildberger I've ever read.
Except I'm pretty sure I can simplify the trig I do and memorize less by using his work.
I have yet to have this happen yet though...
Wow you guys are real faggots
The Real Numbers are trivial rank 1 vector bundle. i.e. trivial line bundle
>>8031788
this
>C is algebraically closed
>starting point of algebraic geometry and the general theory of polynomials
>minimal constructions are gay
>>8033234
[math]\operatorname{Duh}[/math]
>>8033297
Is there a nice intro book that goes and constructs the complete complex numbers from the algebraic numbers?
I have a hard time understanding the construction of the polynomial rings from just Wikipedia
>>8033354
you need some knowledge of algebra
try something like "A first course in modern algebra" from rotman
>>8033333
checked
[math] \faxmachine [/math]
[math] \Faxmachine [/math]
>>8033333
witnessed
>>8033333
mane
u caught the get
>>8033333
Observed bretheren.
>>8033333
real numbers may not be real but quints sure are
