.9999999999...

[IamPROJECT]

I've gotten to thinking...

1 divided by 3 is .3333 repeating.

2 divided by 3 is .3333 repeating.

3 divided by 3 is ...1?

I know it's kind of a stupid question but really think about it. 3 divided by 3 should be .9999 repeating, should it not? 
We were taught in math that anything divided by itself is 1, even decimals. But how is it that 1 divided by 3 is .3333 repeating and 2 divided by 3 is repeating, but 3 divided by 3 is not .9999 repeating. 

Thoughts? Questions? Concerns (You should be concerned, what kind of a question is this lol.)

[LAwLz]

 

[IamPROJECT]

24 minutes ago, LAwLz said:

 

Lol this pretty much answers my question. Thanks, and very insightful.

[LAwLz]

6 minutes ago, IamPROJECT said:

Lol this pretty much answers my question. Thanks, and very insightful.

I remember there being another video about it that explained it better. Vi Hart has a video on it that goes more in-depth. I think Numberphiles brought it up in some video too, but I can't find it.

[ThinkWithPortals]

1 hour ago, IamPROJECT said:

I've gotten to thinking...

1 divided by 3 is .3333 repeating.

2 divided by 3 is .3333 repeating.

3 divided by 3 is ...1?

I know it's kind of a stupid question but really think about it. 3 divided by 3 should be .9999 repeating, should it not? 
We were taught in math that anything divided by itself is 1, even decimals. But how is it that 1 divided by 3 is .3333 repeating and 2 divided by 3 is repeating, but 3 divided by 3 is not .9999 repeating. 

Thoughts? Questions? Concerns (You should be concerned, what kind of a question is this lol.)

To all intents and purposes, 0.999... does equal one. So in a manner of speaking you're right. Recurring decimals are weird like that.

[You_are_a_cunt]

Uhm. No.

 

Any number divided by itself is always equal to exactly 1.

That's how math works, else it would get real messy real soon

[Space Reptile]

1 minute ago, revsilverspine said:

Any number divided by itself is always equal to exactly 1.

what about 0

[You_are_a_cunt]

7 minutes ago, Space Reptile said:

what about 0

0 isn't a number. Sort of. It has its own set of rules

[Space Reptile]

4 minutes ago, revsilverspine said:

0 isn't a number. Sort of. It has its own set of rules

0 is what black/white is to colors it seems

[You_are_a_cunt]

5 minutes ago, Space Reptile said:

0 is what black/white is to colors it seems

Pretty much. To put it in very broad therms 

[Dash Lambda]

Just to add a little more explanation to the video:

Infinity basically just means what some pattern ultimately leads to. Like with counting, you go 1, 2, 3, 4, 5, so on, and since each number is bigger than the last, 'infinity' is the number that's bigger than everything (which isn't actually a number, so infinity is just a concept).

Let's say you just start adding progressively smaller nines together, so you get .9, .99, .999, .9999, etc.. Each time you do that, you decrease the difference between that number and 1 by a factor of 10. That means that series approaches 1, and since .(9)... is what you get if you carry that out endlessly, in other words to 'infinity', it's the same as what it approaches.

 

5 hours ago, Space Reptile said:

What about 0?

 

5 hours ago, revsilverspine said:

0 isn't a number. Sort of. It has its own set of rules.

0 is a number, at least functionally speaking. It just has unique properties.

You have to evaluate anything divided by 0 as a limit, and it can change depending the function you use.

Let's say you want to evaluate the function 1/x at x=0. If you evaluate it at x=1, x=.1, x=.01, etc., You see that it increases exponentially as x approaches 0. So, you can say that the value of the function at x=0 is unbounded, which just means it's infinitely high, and that can't be represented by an actual number, so we usually just say the result is undefined.

But if you do the same thing with x/x, you see that it's equal to 1 regardless of x. That means that if you evaluate the limit at x=0, the function is equal to 1. (you could also cancel x and say 1/1=1, but that's not useful here.)

Now, if you use the function 2x/x, you see that the function at x=0 is 0/0, but if you evaluate the limit you find that the function at x=0 is equal to 2 rather than 1, because the numerator is always twice the denominator.

 

What this means is that 0/0 can actually equal anything, so the expression itself is meaningless. It only has a definite value when it's the result of a function.

[MrDynamicMan]

9 hours ago, LAwLz said:

I remember there being another video about it that explained it better. Vi Hart has a video on it that goes more in-depth. I think Numberphiles brought it up in some video too, but I can't find it.

I love vi-harts videos.

 

But yeah,   0.9^° (can't draw the dot on top, ugh) is 1. Infinite 9s means that there is no one at the end, so the answer is infinite 0s, so the 1 at the end never comes.  

[Dash Lambda]

1 hour ago, MrDynamicMan said:

But yeah, 0.9^° (can't draw the dot on top, ugh) is 1.

You can also represent a recurring decimal with ellipses and/or parentheses, which I actually usually combine as 0.([recurring string])...

When I write them by hand, though, I use a vinculum. I've never seen someone use dots...

[Mira Yurizaki]

I think the biggest problem most people have with infinity is they think it's a value. Infinity is not a value. It's a concept. It's there to represent something so our minds can comprehend what's going on better.

 

Also you can get some fun results when you're not dealing with base 10 values. For example, 0.1 in decimal is 0.000110011... in binary.

[Space Reptile]

9 hours ago, M.Yurizaki said:

I think the biggest problem most people have with infinity is they think it's a value. Infinity is not a value. It's a concept. It's there to represent something so our minds can comprehend what's going on better.

 

Also you can get some fun results when you're not dealing with base 10 values. For example, 0.1 in decimal is 0.000110011... in binary.

on this subject , this is worth watching