Okay, scientists. I am tired of your bullshit. Every time I talk with one of you, you insist that I use "standard form".
You insist on it so much that you know what? Okay. I'll do it. But I will do it only if you prove that the standard form is well defined.
In other words, prove the following theorems:
Theorem 1: For all [math] n \in \mathbb{R} [/math] there exists [math] a \in [1,10) [/math] and [math] b \in \mathbb{Z} [/math] such that [math] n = a 10^b [/math]
Theorem 2: Show that no real number has two standard forms. In other words, prove that the standard form of a number is unique
Theorem 3: Show that no two real numbers have the same standard form.
Scientists, if you can prove those 3 theorems to me then I will accept your "standard form" as a well defined notion that deserves my respect. If you can't prove it then that shows that the notion of standard form is just a half assed convention you used because your professors told you to and you have no capability of thought.
[eqn]10^n_a = 10^n_b, ie n_a = n_b[/eqn] and : [eqn]\lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a - dk_b}{10^k} = 0[/eqn] In other words, I need to have [math]dk_a = dk_b[/math] because if [math](dk_a - dk_b)/10^k \neq 0[/math] for a certain k then it's relatively easy to see than [math](dk_a - dk_b)/10^k > |\sum_{i=k+1}^{\infty} \frac{dk_a - dk_b}{10^k}|[/math] because [math]dk_a - dk_b < 10[/math] for any k.
So we have too numbers so that all their decimals are equals and the numbers of integers before the point are equals. Those numbers are equals.
Dylan Ross
Whoops, [math]10^na[/math] is wrong, I meant : [math]10^{n_a}[/math]
Easton Wilson
First, you only proved that if a standard form exists, it "belongs" to a unique number.
You haven't proven that the standard form exists at all.
Also, you are CLEARLY not a fucking scientist. Like any scientist could come up with that.
If you are mathfag then GET THE FUCK OUT. This is not for you. I know you can do this. Heck, I can do this easily but this is meant to show scientists how retarded they are.
Adam Miller
clever homework thread retard, but do it yourself
politely saged
David Allen
mathematics has nothing to do with science
Nathaniel Carter
>Also, you are CLEARLY not a fucking scientist. Like any scientist could come up with that. You're right, I'm an engineer.
Jace Richardson
I know but I just want to show scientists that even their most trivial concepts are not even well defined, because no scientist has ever proven the standard form to be well defined.
No fucking way. Fuck you. You are trolling.
Ethan Evans
>I know but I just want to show scientists that even their most trivial concepts are not even well defined, because no scientist has ever proven the standard form to be well defined. But that's the mathematician's job, not the scientist's job. Why do you not see the distinction between the two? Scientists use the scientific method, most mathematicians use the axiomatic method
Liam Phillips
Okay then. Show me empirical evidence that the standard form is well defined.
I don't know... smash some particles. Whatever you brainlets do to "prove" things.
Jason Perry
You're still confused.
Mathematicians have proved the standard form exists, and so scientists use it.
They do not need empirical evidence to use the standard form because it's a mathematical construction.
Do you understand?
Aiden Adams
>Fuck you. You are trolling. But I am a filthy engineer. I even fit the stereotype to be gay.
>if a standard form exists, it "belongs" to a unique number. Standard form associates a series and an integer to a number. I've more or less shown this association to be injective. That's all that's needed.
Justin Howard
>But I am a filthy engineer. I even fit the stereotype to be gay.
If you are gay then I believe you are an engineer.
Still, you didn't prove this correctly. If you wanted to publish a proper math paper you would have to do it in the way I outlined it.
Theorem 1: Show that it exists Theorem 2: Show it is unique Theorem 3 : Show the inverse is unique
Grayson Smith
>for all [math] n \in \mathbb{R}[/math]
Oliver White
is your first real analysis homework due tomorrow retard? you're starting to sound desperate
Cooper Bell
Real analysis?
THIS IS REAL ANALYSIS FOR YOU?
Fuck man, how is life at Podunk State University? What did you do for calculus 1? Count rocks?
Gavin Allen
>Real analysis? was n not supposed to be a real number?
what would you prefer to call it retard?
Charles Myers
>Theorem 1: Show that it exists With the definition [eqn]s_f(a) = 10^n * \lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a}{10^k}[/eqn] It exists because n is an integer and so [math]10^n[/math] exists as well. [math]\lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a}{10^k}[/math] exists as well. Short proof :
>Theorem 2: Show it is unique With my "proof" (I don't think I can call it that way though) I've shown that's injective. Which means that if [math]s_f(a) = s_f(b)[/math], then [math]a = b[/math] by necessity. We are not in danger of confusing too numbers because they have the same scientific notation.
>Theorem 3 : Show the inverse is unique We have no need of the scientific notation as a bijective form.
Dominic Bailey
This is at best elementary number theory.
And when I say elementary I mean 5 year old tier elementary.
I mean, fuck. If you are not at least working with sequences don't even dare to call your shit real anal.
Brandon Foster
>If you are not at least working with sequences don't even dare to call your shit real anal. which construction of the reals would you prefer, dedekind cuts?
Zachary Diaz
>Short proof :
You can apply the monotone convergence theorem, okay, okay. I accept.
>We are not in danger of confusing too numbers because they have the same scientific notation.
This is not a proof. Wtf is this. No one is in danger if we assume the riemann hypothesis to be false either, but we still need to proof before we say anything. top kek.
>We have no need of the scientific notation as a bijective form.
But why not go the extra mile and prove it is bijective?
Brody Diaz
>which construction of the reals would you prefer, dedekind cuts?
Dedekind cuts are 10/10.
Benjamin Miller
Let n be 0 n = a10^b = 0
A product is zero if at least one of the factors is zero a is greater or equal to 1 and less than 10, therefore not zero 10 to the power of an integer can never be zero either
Therefore Theorem 1 does not hold
Blake Turner
>This is not a proof. But it is. The fact that [math]s_f[/math] is injective is more than enough. Never two different numbers will have the same standard form, neither will we make a mistake when doing the reverse.
Jackson Fisher
Oh right, you did a proof of that before.
Charles Parker
lel
OP btfo for the sixth(?) time this thread?
Tyler Lewis
If you wanted to nitpick, my proof isn't entirely valid though. It lacks proper definition for real numbers < 1.
Still kinda proud of it.
Joshua King
Okay, I can't lie.
The reason I was thinking about standard form today is because I took my first Physics I class (I am a sophomore in pure mathematics but we have science requirements and physics is the least gay science I know about) and the professor defined standard form for us quickly and it was him who sayd that a had to be bigger than 1 and smaller than 10.
Now I notice that the should have said the absolute value of a so if you want to call him a brainlet please do and I will tell him next Friday, when I meet him again, for you.
Nolan Gonzalez
He probably defined it only for strictly positive reals
No idea why he would define it in the first place in a physics class though
Luke Sanders
its still not true for 1
John Gray
>No idea why he would define it in the first place in a physics class though
Well, today was introduction day pretty much. First class is always like this. This is the entire class in a nutshell
>Says his name and asks for our names >Sits down and starts asking about how much Calculus we know. Gets very specific but the answer to all is yes >Says good, then stands up and starts narrating the story of different physicists to inspire us to care about his subject >Tell us about how Einstein conjectured relativity in 1916, it then was experimentally confirmed in 1919 and then he became a superstar >Also tells us about Newton, Higgs, Fermat, Bernoulli, etc. >Sits down again >Stands up and gives us some basic knowledge of physics. First standard form, then magnitude, then about measurement error, then statistical error >Class ends
Eli Robinson
>standardize way to write any number using the base of its number system as a factor >not the ultimate short-hand
Daniel Ortiz
Also bonus challenge for the people actually qqrecking this proof, show that it's true with base 2 and either extend the proof to either all base n in R or show the constraints for n to be true and prove where n would be false. Rigour to the winner!
