Question

Something that has been on my mind lately:

We know that 0.99... ("point 9 repeating") = 1

Does this mean that a number that is infinitely close to 1 is equal to 1?

Does this mean that any number that is infinitely close to another number is that other number?

How does this relate to limits? When you take the limit of something as x approaches zero, doesn't that mean x is infinitely close to zero, and as a consequence x = 0 ? But that means we are dividing by zero!

Answer 1

0.99999...... = 1.0000000.......2 different representations for the same number.

Answer 2

"infinitely close to" means that, doesn't mean equal, the limit relies on continuity/smoothness usually

Answer 3

1) Yes.
2) Yes.
3) No. \[\text{f(x) will get as close as desired to L as x gets sufficiently close to c.}\\\ \ \ \lim_{x->c} f(x)= L\]Are you familiar with the epsilon-delta formal definition of a limit?

Answer 4

A little

Answer 5

So taking the limit something as x approaches a, doesn't mean that x is infinitely close to x?

Answer 6

to a*

Answer 7

No, it just means that you can find the value of that something get arbitrarily close to its limit for x sufficiently close to a.

Answer 8

epsilon delta is fine for proofs and formal math but useless otherwise....

Answer 9

I'd actually argue the contrary -- without a formal definition of a limit then the notion of a limit is useless.

Answer 10

Yeah formal definitions are important. But I guess that's why there's always a margin of errors with limits right? (because it's not infinitely close)

Answer 11

Right; the idea is that a limit tells you that there is a correspondence between how close f(x) is to L and x is to c such that starting with one you can derive the other.

Answer 12

feild measurements are always approximations. True math only happens with pencil and paper

Answer 13

any number can be written as nonterminating decimal

Answer 14

0.999...... is not approaching "1"; it IS "1"

if we are "constructing" it by placing 9 after each other (such that it is always in a state of termination) then it "approaches" 1

Answer 15

this might help see it in perspective

1/3 = 0.333..... ; it doesnt "approach" 1/3

1/3: 0.333...
1/3: 0.333...
1/3: 0.333...
------------
1 : 0.999.....

Answer 16

Yeah I've seen a lot of proofs for it, but that wasn't really my question that was a given

Answer 17

The more important question was, if 0.99... = 1, then can we state that it is infinitely close to 1, and if that's true, how does that relate to limits?

Answer 18

I was simply setting the stage :)

"Does this mean that a number that is infinitely close to 1 is equal to 1? ": No

Answer 19

Are you sure about that?

Answer 20

I've seen a lot of different opinions about it...

Answer 21

0.999.... is not "infinetly close to 1". it IS 1

Answer 22

The way I always think of it is like this.\[\ \ \ 1-0.9=0.1\\\ \ \ 1-0.99=0.01\\\ \ \ 1-0.999=0.001\\\ \ \ 1-0.999...=0.000...1\\\text{... however you can't place a terminating 1}\\\text{to an infinite sequence of digits.}\]

Answer 23

yeah, but same accounts for something that is infinitely close to 1. That's why I thought something that is infinitely close to 1 = 0.99... = 1

Answer 24

I agree with amistre that there is no formal "yes" to that question. So that does not mean the that a number infinitely close to one is 1 or not, it is undefined in a sense.

Answer 25

to say "infinitely close" is introducing a non-formal definition, and delta-epsilon, while good for formalizing the definition of the limit, does not resolve infinity into a workable thing.

Answer 26

for sufficiently large values of 1 :)

Answer 27

This whole infinity thing is mind boggling and a lot of people say different things, what am I to believe?

Answer 28

personally, I would believe those that know what they are talking about ;)

Answer 29

The mathematicians who are doing the math usually
some define 0^0=1
others do not

Answer 30

not all things in math are cut-and-dry like many would believe, 0^0=1 has been adopted by pure convenience many times, while I think most would regard it as undefined

Answer 31

yeah @amistre64. But different people that are good at math also have different opinions...

Answer 32

Math is a science; but there is also a philosphical aspect to it.

If we define math by what is printed, we lose what Math is. If we do not write it down and define it, we never catch a glimpse of what it can be.

Answer 33

we are taught a too young of an age that Math is absolute; and its not till you get to the higher learnings that you realize that its quite a fluid medium

Answer 34

A well-argued "constructionist" approach^

Answer 35

or rather, what we are "taught" as math (ie FOIL, "move it to the other side", muliply by a reciprical, etc) is what Math is.

Answer 36

that reads like half a thought ....

Answer 37

"When you take the limit of something as x approaches zero, doesn't that mean x is infinitely close to zero, and as a consequence x = 0 ?" No

"But that means we are dividing by zero!" No, it doesnt.

Answer 38

So, can you give me a counter-argument, why something that is infinitely close to a, is not a?

Answer 39

when my children call someone who "looks alot like me" daddy; that does not make that person their dad.

Answer 40

Infinitely close is a vague term. Are you familiar with the infinitesmal approach to calculus?

Answer 41

that's not a good argument. He doesn't look infinitely close like him :p

Answer 42

close enough to fool them for a second ... feels like an eternity to me.

Answer 43

@oldrin.bataku I am not

Answer 44

But @amistre64 there is a big difference between a finite difference and an infinite difference, as I learned from: http://www.youtube.com/watch?v=TINfzxSnnIE&feature=relmfu

Answer 45

thats a very difficult video to follow for me.

Answer 46

actually, she's agreeing with you, as I'm noticing now. Yes, she's quite hard to follow.

Answer 47

I believe it was cantor that came up with set theory.

in it, he rewrites all real numbers as nonterminating decimals

if we are used to writing a value that terminates, we can subtract 1 from the end digit and tack on the infinite 9s

Answer 48

in this manner, every real number can be written in the same manner to be viewed on equal footings

Answer 49

It just seems very weird to me that something that has infitely repeating 9 digits is not the same thing as something infinitely close to 1.

Answer 50

or to the next number

Answer 51

the term "infinitely close to" is vague. as others have pointed out. How would you define such a value?

Answer 52

what is a number that is 0.999...., that is not 1, but infinitely close to it?

Answer 53

i think alot of confusion might be in how people use the terms "infinity" and "infinite".

They construct sentences that appear valid, but have no real meaning to them. Infinite and infinity are not quantities, they are qualities. They are not values, they are magnitudes (i hope i used that right lol)

Answer 54

There should be some symbol for it to avoid confusion then

Answer 55

its a byproduct of having people teach a subject that they know nothing about, or try to dumb it down thinking that the people they are teaching will grasp it better. Then the rest of the education process is spent having to unlearn all the stupidity that they taught you to "get by".

Answer 56

:)

Answer 57

But I guess we can say that there is no real agreement on such terms? They are just defined as mathematicians see fit in order to do something else?

Answer 58

lol, we would then have to define what constitutes a mathematician ;)

Answer 59

Math is in the abstract, it is in trying to take hold of the abstract and define it in concrete terms that the Math part of it is lost.

Answer 60

So yes, it is defined as mathmatikers see fit :)

Answer 61

im currently in a physics class, and the teacher is completely butchering Math concepts. makes me cringe

Answer 62

college?

Answer 63

yeah ... the class is designed for nonMath majors.

Answer 64

What are you studying? I mean, what degree are you going for

Answer 65

He wanted me to do a problem in the board and griped at me when i dint use HIS formulas :)

Answer 66

Im going for a masters in math, just have to get thru the bachelor stuff

Answer 67

nice! But then why are you doing a non mathsy physics thing? :p

Answer 68

its just a filler course. I am not strong at applications, and thought it would be a good learning experience.

Answer 69

Aha. I'm going for a bachelor math myself, starting next year. I hope it will all work out.

Answer 70

itll be fine :)

Answer 71

http://en.wikipedia.org/wiki/Hyperreal_number
http://en.wikipedia.org/wiki/Surreal_number

Amistre is right, they just make it up as they go along...:-)

Answer 72

that youtube video is actually quite good :)