Okay so this is bugging me

So you believe that If a number has nothing between it and another number then it's the same number? Seems like special pleading.

That's an infinitesimal bruv

Believe it or not, there's an infinite amount of numbers between 1 and 2.

Yeah.....and?

You came to the conclusion that 3/3 is less than 1 because you believe .999 (repeating) is less than 1, right? Then how did you assert that 1/3 = 0.333 (repeating) in the first line of your math? By your assumptions, 1/3 should be bigger than 0.333 (repeating), giving you 1 = 3/3 > 0.999 (repeating), which confirms your bias.

If a and b are two numbers with distinct values, then there must exist an [math] n [/math] such that

[math]a

because its rounded off

[eqn]\sum_{n=1}^{\infty}\frac{9}{10^n}[/eqn]
converges to 1.

0.9999. . . . = 1

because 0.0000. . . = 0

There is no one at the end of those zeros, since it goes on forever.

>3/3rds is less than one according to our math?

No. What happened is that they (essentially) had to extend the = operator so that it could handle stuff like 1/3 = 0.3333...

(Technically, it's called the "limit" operator, but it basically ended up extending the way the = operator is used.)

The reason the original = operator didn't work is because you had this unsolvable problem:

1/3 = 3/10 + 3/100 + 3/1000 + 3/10000 + ...

The right side is a sum that doesn't end. But if it doesn't end, then it makes no sense to ask questions like "what does the right side equal in the end" or "what is the final sum" or anything like that, because there's no end. All those kinds of questions are self-contradictory and hence have to be thrown out -- you simply can't justify asking something that's basically "what is the END RESULT of and ENDLESS process?". It's such a nonsensical question that mathematicians had to abandon even trying.

So to fix the problem, they essentially had to extend the = operator (via the limit) so that it yields true if it can proven that no numbers exist that are strictly between the left and the right side of the equation. (The actual extension is a little more complicated than that, but you get the basic idea.)

Mathematicians have conceded that the above equation is never true under the original (simple) meaning of the = operator. (That happened in the mid 1700s, after a few decades of heated arguments among the top guys in math.) So if one side of the equation is an infinite sum (like 0.333...) then you need to activate the new extension of the = operator that looks at whether any numbers exist between the two sides.

Here's an example of how the extension works:

1: Start with this statement: "If x approaches 3, then x^2 approaches 9."

2: Rewrite it in a standard way like this:

lim (x->3) x^2 = 9

Notice that the 2nd appearance of the word "approaches" got replaced by the equal sign in the "lim" notation. That's the extension.

That doesn't seem like a modification of the equals symbol at all.
[math]\lim_{x \to \infty}\frac{1}{x}[/math] isn't some kind of special value that "kinda equals" zero, it is just zero.

>Technically, it's called the "limit" operator, but it basically ended up extending the way the = operator is used.
What? No.

>That doesn't seem like a modification of the equals symbol at all.

Of course it is.

This gets really noticeable when you have a limit that's equal to infinity. Since infinity is not a number, the original (simple) meaning of the = operator could never have claimed that anything is ever equal to infinity. It requires a special extension to the equal operator to allow it to claim "equality" with infinity.

The extension to the equal operator is really easy to see, by looking at what the limit means:

"if x approaches 0. then 1/x^2 approaches infinity"

When this gets converted to limit notation, notice what happens to that second word "approaches" in that statement -- it gets changed to an equal sign -- despite the fact that 1/x^2 is never equal to infinity at any step of the process. Changing the 2nd word "approaches" to the equal operator is the precise point at which the equal operator gets extended.

This extension to the equal operator is included entirely within in the definition of the limit. If you don't use the limit (i.e. if you refuse to set anything equal to infinity or 0.333... or the like) then you don't trigger the extension to the = operator.

i always wondered why you're not allowed to say that 1/0 = infinity
only by using the limit do you get the right to say that anything can be equal to infinity
kinda makes sense now

>This gets really noticeable when you have a limit that's equal to infinity.
Limits can't be equal to infinity. A limit which diverges to infinity doesn't equal anything, because limits only have a value when they converge. You're being incredibly sloppy, and then claiming the notation is responsible for your misunderstandings.

>When this gets converted to limit notation, notice what happens to that second word "approaches" in that statement -- it gets changed to an equal sign -- despite the fact that 1/x^2 is never equal to infinity at any step of the process.
Stop.
Please stop.

Because depending on which direction you approach 0 you reach two different limits.

>Limits can't be equal to infinity.

Of course they can. Pick up any calculus textbook, and you'll see limits that are literally set equal to infinity.

To make limits equal to infinity, they extend the set of values that are allowed to be compared using the equal operator. That extension is called the "affinely extended real number system":

en.wikipedia.org/wiki/Extended_real_number_line

In that article, see the section titled "Limits" to see how this works. This explains why calculus textbooks will show limits that are literally set equal to infinity.

> Limits can't be equal to infinity

dude, seriously

What people don't get is that .999 is effectively that series.

The notation s for repeating digits..... makes people think it is some number other than it's value. ...1

Converges to 0

>effectively

Not just effectively, it's actually the same.

You are 99.999...% correct

Technically correct. The best KIND of correct!

Fair enough. I was under the impression that was just notational shorthand, not a rigorous extension of R.

don't mix up sequences and series