Define what a real number is

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?

it's 0

Ha that's a good point! I'm thinking about this problem but in terms of reals instead of naturals: 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

yep

The only topology I've taken is this:
youtube.com/playlist?list=PL6763F57A61FE6FE8

That way I don't have to deal with real number voodoo.

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.

>What is a cauchy sequence of rationals

oh boy

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.

>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

But what even is a line? A line requires a sense of continuity. Something impossible without creating arbitrary definitions like the real numbers,

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.

It works.

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.
>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.

>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.

>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

>m-muh intuition
Fuck off retard.
>pointless
All non-empty topological spaces have points you stupid idiot.

So does my wiener but it's not a proof

spotted the pseudo intellectual

A member of a complete ordered field named R.

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.

Archinedean?
Infinite sequences are more general and require fewer definitions.

Isn't there a p-adic construction that is basically using strings on digits as a construction?

Numbers are not real.
They're imaginary constructs representing an idea.
They only exist in a metaphysical space.

>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.

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

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.

ikr

Define define

Fp is in fact not an ordered field, and it is finite you fucking cuckhold

Good point.

...

>All non-empty topological spaces have points you stupid idiot.
kek

How is the topological completion of the rationals not an intuitive idea?

Like [math]X^2+1[/math]?

The volume of the cum i just jacked off in m^3 is a real

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

I can't put the rationals nor the integers in a picture, and neither can you.

Yes, you're filling in all of the holes in the rational numbers.

Im(z)=0

i can
What "holes"?

Draw picture pls

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.

Ok I did.
pfff

Alright mister "I have enough time on my hands to draw infinitely many points arbitrarily close together", please demonstrate.

>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]

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]

lol nice try
I just can't take these ridiculous arguments seriously.

A complex number without zero imaginary components.

I like the pic (except it's ugly, but I still like the idea and concept)

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].

>n→∞
>∞

topkek

Lmao no, that's the complex numbers. [math]\mathbb{C} = \mathbb{R}[x]/\langle x^2+1\rangle[/math]

Because decimal expansions aren't unique, which gives you shit like [math]0.99\dots = 1[/math].
>inb4 I get memed on

>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.

>0.99⋯=1
These memes man. I can't believe anyone can take real numbers seriously.

Any youtube math videos like wildbergers but don't involve crackpot shit?

Math Doctor Bob

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.

Are his math history videos ok, or is he gonna try to cram dumb shit down my throat?

>infinite
It's like you are not even trying.

Please forgive my fumbly (read: probably incorrect) language, but I'm pretty sure I got my point across.

A composite number made from pairing a real number with an imaginary real number.

>real number
...

Would it help if I said that they're paired using the ordinary addition operator?

>addition
stop with the memes

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?

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

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.

>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

Commutative diagrams

>anything intuitive
completely subjective, and there are many posts already that I would call intuitive

[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].

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 user made the pic himself (I'm the guy who said it's cool but ugly/improvable)

thanks senpaitachi

double thanks

Name me a rational number [math] q [/math], such that
[math] q + 1 = q [/math].

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

>Name me a rational number q
>number
>q

q is a letter

You're right. We should also never assume that things are circles or squares.

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

You first

You can.
Try to define multiplication of two infinite decimal expansions and you'll realize it's a pain in the butt, though.