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/sci/ - Science & Math

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4995201 No.4995201 [Reply] [Original]

Finally a actual Math question

What is the closest number to 1?

Is there a way to say in mathematics arbitrarily close to 1

Context:
X=
is .99...99 correct mathematically?

>> No.4995214

<span class="math">2-\sqrt[\infty]e[/spoiler]

>> No.4995225

I don't know of any numeric system were real numbers are present and one of them is closer to 1 then all of the others (except 1 itself of course).

>> No.4995232

>is .99...99 correct mathematically?
No.
Because your .99... already reperesents an infinite amount of 9's, then you try to add 99 to the end?
By doing that you are terminating the chain of 9's so it's no longer an infinite chain of 9's.

>> No.4995237 [DELETED] 

>>4995225
/thread

>> No.4995249

there is no number 'arbitrarily close to 1'. this is actually the 'axiom of archimedes', it's a consequence of the defining property of the reals, which says that every set with an upper bound has a least upper bound.

you can however state 'x is arbitrarily close to 1' (NOT the same thing as there being an 'arbitrarily small number') formally. it's very common in analysis. it goes like this: given any number greater than zero, the modulus of x - 1 is less than that number.

>> No.4995255

No OP there is no such number.

Take for example the set of numbers in the interval
]0, 1[
This is an open set, and the very definition of an open set is that for every element in the set there exists a neighborhood of elements also in the set.

In other words it's impossible to find a number in the set above that is on the "edge".

>> No.4995261

>>4995249
i should elaborate a bit here... there is no number arbitrarily close to 1 which isn't 1.

equivalently, there's no 'infinitely small number' which would be the difference between 1 and the arbitrarily close number which isn't 1.

>> No.4995263

Short answer: No.

>> No.4995265

Assume there is a closest number to 1 among numbers which are not equal to 1. Let x be this number. Then let y = (1+x)/2. The distance between y and 1 is half the distance between x and 1, and the distance is nonzero since x is not equal to 1. This makes y closer to 1 than x. From this contradiction, we conclude there is no closest number to 1 not equal to 1.

>> No.4995298
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4995298

there is no actual closest number to 1.

But you may define different limits approaching 1.

For example:

Lim sin(x) / x
x-> 0

You may define an infinite amount of functions whose limits approach 1 at different rates.

>> No.4995331

OP here,
Thank you anons for informing me
and I quite well understand that there cant be a number arbitrarily close to 1

but how can I express that X is arbitrarily close to 1?
would I have to use limits?
I want to be more specific than .99<X<1

>> No.4995351 [DELETED] 

>>4995331
If the difference between X and 1 is less than epsilon, you write <span class="math">|x-1|<\epsilon[/spoiler]. If epsilon is any number greater than zero, then X is exactly 1.

>> No.4995352

>>4995331
That would depend on what you mean by "X is arbitrarily close to 1."

>> No.4995377
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4995377

OP has taken 0.999... trolling to a whole nutha level

>> No.4995388
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4995388

>>4995352
definition of a limit... actually pretty useful
thank you

>>4995352
I apologize for my ambiguity
maybe think about it this way
in R^2 space what is the largest radius of a circle which is confined in the first quadrant. its center is at (1,1).
being confined by the axis, the largest radius must be arbitrarily close to 1, but not equal to 1

>> No.4995396

>>4995377

Just trying to learn the correct way to express myself sir

>> No.4995410

>>4995388
That's simple: there is no such radius. The radius of the circle has a least upper bound (one), but no maximum.

>> No.4995412

>>4995410
He defined the circle as having a center of (1,1)

>> No.4995416

>>4995410

If I understand you correctly, your saying there can't be a largest radius, because there is no single number it can be?

>> No.4995425

>>4995416
Precisely. For any permissible radius r, you can find a strictly larger permissible radius (say (1+r)/2), so there is no largest possible radius.

>> No.4995437

>>4995425
its so clear now
Anon I fucking love you for helping me see that

>> No.4995443

>>4995437
You're welcome.

And now it's time to go to bed, goodnight anon~

>> No.4995626

I don't get it. Why isn't 0.99.... the closest number to 1?
You can't add another digit because it's all nines.

>> No.4995675 [DELETED] 

>>4995626
because it's not a meaningful number

>> No.4995681

>>4995626
that's not a number though

>> No.4995720

for any number that you say, i can come up with a number that's closer to 1. (there is no closest number.)

>> No.4995722

>>4995720
Which number is closer to 1 than 0.99...?

>> No.4995737

>>4995722
>Which number is closer to 1 than 1

>> No.4995764

.99...
IS 1 you fucknut.

This is getting so sucking old.
1/10
I gave you a troll point for trying to put a new spin on this same old tired thread but it's time for it to die.

inb4 shifty attempt to argue that they are not equal.

>> No.4995770

The number closest to 1, is 1. You didn't exclude it.
Now, the number closest to 1 that isn't 1 -- no such thing. There are uncountably many rationals between any two other rationals.

>> No.4995784

>>4995764
>0.99.../1
Fix'd

>> No.4995788

>>4995770
What about 0.99...?
This is an irrational number, right?

>> No.4995794

>>4995770
>There are uncountably many rationals [..]
cantor would like to have a word with you

>> No.4995837

>>4995794
It was pretty obvious that he meant infinitely many.

>> No.4995850

>>4995788
it was in the goddamn op, read the thread before posting

>> No.4995852

>>4995443
guys, the thread ended here, all the questions were answered

>> No.4995853

intuition says 0.999... or 1.000...1

>> No.4995856

>>4995249
>there is no number 'arbitrarily close to 1'.
in the real numbers.

There is in the hyperreals though.

>> No.4995857

>>4995788
No, 1 is a rational number.

>> No.4995859

>>4995857
I wasn't asking for 1. I wanted you to confirm that 0.99... is irrational. I mean if you tried to write it as a fraction, it would be ...999.../1....0000.... which is not a proper fraction and thus not rational.

>> No.4995862

>>4995859
0.99... = 1
Therefore it is rational.

>> No.4995863

>>4995862
But I just proved that 0.99.. is irrational. An irrational number cannot equal a rational number. Don't troll me.

>> No.4995866

>>4995863
>But I just proved that 0.99.. is irrational.
No you didn't.

0.99... = 1/1
0.99... = 2/2

Or any other fraction x/x

>> No.4995867

>>4995866
>begging the question

If you wanted to write 0.99... as a fraction, you have to have infinite 9s in the numerator which isn't allowed in fractions.

>> No.4995869

>>4995866
Also when I divide 1 by 1, I get 1 and no infinite 9s.

>> No.4995871

This could be a trick question.

1 is closest to 1.

>> No.4995872

>>4995867
>If you wanted to write 0.99... as a fraction, you have to have infinite 9s in the numerator
No you don't.
0.99... is (one of) the decimal expansion of 1, the decimal expansion doesn't really have anything to do with what's in the numerator or denominator (as in they don't have to be the same numbers)
0.33... is a rational number (1/3)

0.99.. has been proven to be equal to 1.
1 is a rational number.
Therefore 0.99... is a rational number.

>> No.4995875

http://www.wolframalpha.com/input/?i=0.999...%3D1

Wolfram Alpha knows everything.

>> No.4995876

>>4995872
When I apply the division algorithm to 1/1, I get 1 and not an infinite sequence of nines. This means 0.999.. cannot be the decimal expansion of 1.

QED

>> No.4995878

>>4995875
Like every computer program wolfram uses approximations.

>> No.4995885

>>4995876
The results of the division algorithm doesn't determine if two numbers are equal.

>> No.4995887

>>4995878
If it used approximations it would get the answer wrong actually.

>> No.4995890

>>4995885
The division algorithm is how you get a decimal expansion. If it yields a different result, the number cannot be the decimal expansion.

>>4995887
It does get the answer wrong. They are only approximately equal.

>> No.4995893

I'd say it's 1.

>> No.4995898

>>4995890
>The division algorithm is how you get a decimal expansion.
It's one way to get a decimal expansion.
>If it yields a different result, the number cannot be the decimal expansion.
That does not follow.

>It does get the answer wrong.
Nope.
>They are only approximately equal.
They are exactly equal, they are identical.

>> No.4995902

>>4995898
Did you ever go to school? The decimal expansion is defined as the result of the division algorithm.

And they are in fact not equal. One is irrational, the other is rational.

2/10 for your trolling efforts

>> No.4995944

>>4995902
>The decimal expansion is defined as the result of the division algorithm.
But the decimal expansion has nothing to do with the notion of equality.

>And they are in fact not equal.
Yes they are.
Plenty of proofs here:
http://en.wikipedia.org/wiki/0.999......

>One is irrational, the other is rational.
They are the same number, so it's impossible for one to be irrational and one rational. They are in fact both rational.

I give you 0/10 for your math knowledge.

>> No.4996000

>>4995902
If I have an algorithm that outputs a unique largest number, 5, does that mean that 6 is less than 5?
Of course not! It means that my algorithm never outputs 6.

>> No.4996056

What is the universal set?

>> No.4996073

>>4995944
It's a simple definition, my math illiterate friend. We define the biggest number that is smaller than 1. Nothing to prove here.
What you said about decimal expasion is wrong. You should brush up your knowledge of long division, because that's how you get the unique decimal representation of a number. Where I live this is elementary school math, but maybe americlaps have to wait until college to learn it.

>>4996000
The division algorithm puts out the unique decimal expension. There is only one decimal expansion and of course the algorithm function is not surjective. Irrational numbers cannot be the result of long division of fractions.

>> No.4996076

>>4996073
>the unique decimal representation of a number
>1 = 1.0000.. = 0.999...

>> No.4996079

>>4996076
Apparently they cannot be the same because they have different decimal representations. It's just that we say they approximately equal because the difference is infinitesimal.

>> No.4996082

>>4996079
even in non standard analysis this isn't true :(

>> No.4996083

>>4996076
>sigfigs

>> No.4996100

>>4996082
It is. The decimal representation of a fraction is derived by applying the long division algorithm. If a number is not in the image of the long division function, it is irrational. The image of 1 is 1 and nothing else. Try dividing 1 by 1 and see what you get.

>> No.4996145

>>4996100
>If a number is not in the image of the long division function

That's simply untrue.

>> No.4996148

>>4996145
>, it is irrational

Forgot that bit.

>> No.4996150

>>4996145
>>4996148
Why is it untrue? The decimal expansion of a rational number is defined as the result of long division. This is what children learn in elementary school. If you missed elementary school, you should talk about math here.

>> No.4996151

>>4996150
*should not talk about

>> No.4996178

1/10 = 0.(9) = 1 => 0.(9) -> Q [0.(9) is rational]

>> No.4996183

>>4996178
>1/10 = 1

U wot m8?

>> No.4996195

>>4995261
Wait dude, one can define infinitely small numbers; take for instance surreal number. Even in that case though, for every infinitely small number there is at least one that is smaller.