.999999 = 1

>.999999 = 1
This meme is deeply unsettling to me

Mathematics doesn't need to make sense in a physical structure like the real world. That's why we don't consider mathematicians to be scientists.

Open a book moran.

>Mathematics doesn't need to make sense in a physical structure like the real world.

yes it does, two different things can not be equal
an infinitely re-occurring decimal can not be the same as a finite number

Would you agree that 0.9999 is "closer" to 1 than 0.9? If you do then what's your problem?

.999999 = .999999

now .999999... = 1

.999999... ~ 1

0.1111111 = ?

wrong

0.111... = 1/9
0.333... = 3/9 = 1/3
0.999... = 9/9 = 1

Wrong

If they equal they aren't two different things. They are two of the same thing.

you guys are fucking annoying

if two numbers are unequal, then there exists other numbers between them

find a number between 0.999.... and 1

there isn't one. since there are no numbers between the two of them, they are the same

0.999.... = 1

>if two numbers are unequal, then there exists other numbers between them
prove it faggot

there is a number between them tho
0.9999.... + 0.1111....

If [math] x \neq y [/math] then either [math] x < y [/math] or [math] x>y [/math]. Take the former case then: [eqn] 2x < x+y \implies x < \frac { x + y } { 2 } \\ \text { Likewise } ~ x + y < 2y \implies \frac { x + y } { 2 } < y \\ \therefore x < \frac { x + y } { 2 } < y [/eqn]

should have left it as an exercise to the brainlet

Probably. But I'm procrastinating.

An infinitely re-occuring decimal can actually be the same as a finite number.

>1.111... is between 0.999... and 1

You don't say, Sherlock!
n.000...=n

I meant
0.9999.... + 0.0000....1

Let x = 0.999...

9x = 10x-x = 9.999...-0.999... = 9

=> x = 1

>0.0000....1

kek every time

The number 0.000.....1 doesn't exist. There's no such thing as the smallest non-zero number. I can always add more zeroes.

Learn hyperreals retard nigger

Can their be a number that = 1-(infinitely small)? yes but you would have to use some kind of hyperreal number

>hyperrreals
>being cared about by anyone

There are a number of proofs of this, just google it. The problem is that our intuitive grasp of infinity is tenuous.

Here's a very simple way to think of this.

[math] 0.999... = 0.333... \cdot 3 = \frac{1}{3} \cdot 3 = \frac{3}{3} = 1[/math]

They're the same thing with different representations. You absolute brainlet.

These are just different representations. You could write 1,a 'finite number' in your terms, as 1.000..., where there is an infinite series of 0's after it. Your distinction between a 'finite number' and an infinitely recurring decimal is not grounded in how mathematics truly works.

I've made a few points on this thread, but here's another one. The real numbers are not discrete, which entails that you can always find a real number between two numbers. But there's no way to find a number between 0.999... and 1. You can't just add a 1 at the end of the the decimal extension of 0.999... since it extends to infinity. The fact that no real exists between 0.999... and 1 is another reason to concede that both are in fact the same number.

what about 1.000000000000... (repeating infinitely)

is that different from 1?

>94293flat,800x800,075,t[1].jpg (47 KB, 800x8
you tell me

0 is not a number

1.1111.... is not the same as 1

>0 is not a number
Since fucking when?

What exactly is 0.000…1 an infinitely long string of zeros with a 1 at the end?

nope

Since always

0 is a digit, not a number

The number "one" can be represented by the digit 1. Those are two different things, but from context you can usually judge which one is meant. I can write 0 and mean the number written as 0.

0 is both a number and a numerical digit. In fact, all digits are also numbers.

>0 is a digit, not a number
wow time to die

>when your math are actual shit
T.brainlet
1/3 ≈ 0.333...

the three little dots imply infinite recurrence

get out of here with your mathematical semantics or prove that 1/3 != .333...

>yes it does, two different things can not be equal

literally f(x) != g(y)

lmao, just end it you fucking brainlet

>not understanding the concept of infinity

It's okay that you have a hard time with your intro proofing class, we can help you on the way.


1. Assume .999... is different from 1
2. conclude that the difference of the two is strictly positive
3. construct a number of the form 0.000 ... 1 ... 000... that is strictly smaller than the difference
4. conclude that subtracting this number from 1 gives you a number that is smaller than .999...
5. explain why this is contradicting with the fact that we view the reals as a (complete) totally ordered field
6. conclude that .999... must logically equal 1


Good luck. If any step causes you trouble feel free to ask for more help, you could also visit your professor during office hours.

Plebs don't understand 1/3 can't be represented in base 10.
Literally primary school math level

.999...8=.999...