Okay, scientists. I am tired of your bullshit. Every time I talk

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define 'standard form' before you ask us to prove its well defined brainlet

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There is nothing wrong with a conventional form
being "standard", nor with a standard form
being "conventional". Get over it.

>>8745111
If you are a scientist then you know what standard form is, faggot.

If you are not a scientist then gtfo my thread.

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also theorem 1 is obviously false brainlet

take n=-1

then since a>0 for 1<=a <10 and 10^b >0 for b in Z there exists no such a and b such that n=a10^b

>>8745114
Hmm. I have examined your post but I do not think that is a suitable proof.

So you are retarded. You only use standard form because your professors forced you to. Good to know, kys.

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>>8745115
i am 99% sure i have taken more math classes than you and have more math degrees than you brainlet

>>8745116
Absolute value of a then.

Sharp observation my friend.

The theorem still stands: Prove that every number has a standard form.

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>>8745121
define standard form brainlet, and then we'll talk

>>8745122
gtfo my thread if you don't even know what standard form is, faggot.

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>>8745124
>prove this thing but i won't tell you what it is
nice try brainlet

>>8745131
Here you lil faggot.

http://www.mathsisfun.com/algebra/standard-form.html

I give you a kids website because they teach standard form in high school so you must be waaay underage.

>>8745132
Hahahahahahahahahahahah, oh wow. 10/10, I haven't had this hearty of a laugh in a long while.

>>8745105
http://www.mathsisfun.com/algebra/standard-form.html
Or "scientific notation" for the rest of the world.

>Show that no two real numbers have the same standard form.
That one is easy.

Let [math]a[/math] be a real number.

We have [math]n[/math] an integer so that :

[eqn]\frac{a}{10^n} < 1[/eqn]

We can define the "standard from" as :

[eqn]s_f(a) = 10^n * \lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a}{10^k} [/eqn].

Where [math]dk_a[/math] is the [math]k[/math]-th decimal of a.

We have [math]s_f(a) = s_f(b)[/math] if and only if :

[eqn]10^n_a * \lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a}{10^k} = 10^n_b * \lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_b}{10^k}[/eqn]

In other words, both :

[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.

>>8745148
Whoops, [math]10^na[/math] is wrong, I meant : [math]10^{n_a}[/math]

>>8745148
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.

>>8745105
clever homework thread retard, but do it yourself

politely saged

>>8745167
mathematics has nothing to do with science

>>8745167
>Also, you are CLEARLY not a fucking scientist. Like any scientist could come up with that.
You're right, I'm an engineer.

>>8745174
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.

>>8745176
No fucking way. Fuck you. You are trolling.

>>8745177
>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

>>8745183
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.

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>>8745188
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?

>>8745177
>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.

>>8745195
>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

>>8745105
>for all [math] n \in \mathbb{R}[/math]

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>>8745204
is your first real analysis homework due tomorrow retard? you're starting to sound desperate

>>8745210
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?

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>>8745211
>Real analysis?
was n not supposed to be a real number?

what would you prefer to call it retard?

>>8745204
>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 :

[eqn]\lim_{m\to\infty} \sum_{k=0}^{m} \frac{dk_a}{10^k} < \lim_{m\to\infty} \sum_{k=0}^{m} {\frac{9}{10}}^k = 10[/eqn]

>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.

>>8745212
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.

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>>8745217
>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?

>>8745214
>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?

>>8745221
>which construction of the reals would you prefer, dedekind cuts?

Dedekind cuts are 10/10.

>>8745105
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

>>8745224
>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.

>>8745233
Oh right, you did a proof of that before.

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>>8745231
lel

OP btfo for the sixth(?) time this thread?

>>8745236
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.

>>8745237
>>8745231

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.

>>8745255
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

>>8745255
its still not true for 1<= |a| < 10

because then there's no standard form for 0

try again

>>8745276
>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

>>8745105
>standardize way to write any number using the base of its number system as a factor
>not the ultimate short-hand

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!

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>>8745255
>sophomore in mathematics
get oUT OF MY BOARd