http://en.wikipedia.org/wiki/0.999...

 

 

 

Kind of 

Edited November 4, 201312 yr by lookalike1

had to quote this from wiki

 

Skepticism in education[edit]

Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

• Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[34]
• Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[35]
• Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.[36]

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999...

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'".[37]

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.[38] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a supremum definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.[39] Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."[40]

As part of Ed Dubinsky's APOS theory of mathematical learning, he and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.[41]

 

Edited November 4, 201312 yr by LifezHatred

Doesn't any fraction with the same denominator and numerator equal one? 

I'm going to explain this to you using some basic fucking Algebra II.

 

f(x) = 1 / x

 

That is a rational function with a domain of 0 and a vertical asymptote at 0 and a horizontal asymptote at 0. The graph will be a hyperbola. It will move as close to 0 on the x axis but WILL NEVER EVER EVER EVER EVER EVER HIT IT. It will move as close to 0 on the y axis as possible but WILL NEVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER HIT IT.

 

It has a domain of (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s 1) does not equal 0.

The range is (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s1) does not equal 0.

 

Do you know why it will never reach 0? Because it will never reach infinity. In case you didn't notice, your domain does not include +/- infinity. 

Edited November 6, 201312 yr by ohhungry

 

I think it does because:

 

1/9  =  .111111 (repeated)

 

5/9 = .5555555 (repeated)

 

9/9 = .9999999 (repeated)

 

but 9/9 = 1 in fraction form....

 

Thoughts?

 

You are right that it does, but your proof is incorrect. 9/9 can not be .999.. if it also equals 1.

 

Let X = 0.999...

 

Then 10X = 9.999...

 

Subtract X from each side to give us:

 

9X = 9.999... - X

 

but we know that X is 0.999..., so:

 

9X = 9.999... - 0.999...

or:

9X = 9

 

Divide both sides by 9:

 

X = 1

 

X = 0.999... = 1

 

Therefore 0.999... = 1

 

This is one solution and the answer is that 0.9999 repeating will equal 1.

Otherwise u can say that 3 thirds doesn't equal 1 cus 0.3333333333333333 repeated x3 is 0.99999 repeated and not 1. Seen this thread several times already.

I'm going to explain this to you using some basic fucking Algebra II.

 

f(x) = 1 / x

 

That is a rational function with a domain of 0 and a vertical asymptote at 0 and a horizontal asymptote at 0. The graph will be a hyperbola. It will move as close to 0 on the x axis but WILL NEVER EVER EVER EVER EVER EVER HIT IT. It will move as close to 0 on the y axis as possible but WILL NEVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER HIT IT.

 

It has a domain of (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s 1) does not equal 0.

The range is (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s1) does not equal 0.

Do you know why it will never reach 0? Because it will never reach infinity. In case you didn't notice, your domain does not include +/- infinity.

Read my post again. I clearly state it goes from -infi to 0 0 to infi

I'm going to explain this to you using some basic fucking Algebra II.

 

f(x) = 1 / x

 

That is a rational function with a domain of 0 and a vertical asymptote at 0 and a horizontal asymptote at 0. The graph will be a hyperbola. It will move as close to 0 on the x axis but WILL NEVER EVER EVER EVER EVER EVER HIT IT. It will move as close to 0 on the y axis as possible but WILL NEVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER HIT IT.

 

It has a domain of (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s 1) does not equal 0.

The range is (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s1) does not equal 0.

Do you know why it will never reach 0? Because it will never reach infinity. In case you didn't notice, your domain does not include +/- infinity.

Read my post again. I clearly state it goes from -infi to 0 0 to infi

Let me explain further. You CANNOT include infinity in your domain or range, infinity is not a specific value. First of all, by convention round brackets, i.e ( ), implies that the end values are not included in your domain, while square brackets [ ], would imply that they are included, so I assumed you meant that infinity was not in the domain.

It doesn't really matter though, infinity cannot be a value in your domain, because x will never equal infinity. This is why we express infinities through limits

Here's why 0.999999999(repeat) = 1

Let P = 0.999999999(repeat)

P x 10 = 10P

10P = 9.999999999(repeat)

9P = 9.0 (9.999999999(repeat) - 0.999999999(repeat))

P = 1.0

Source: Trust me, I'm an Engineer

I'm going to explain this to you using some basic fucking Algebra II.

 

f(x) = 1 / x

 

That is a rational function with a domain of 0 and a vertical asymptote at 0 and a horizontal asymptote at 0. The graph will be a hyperbola. It will move as close to 0 on the x axis but WILL NEVER EVER EVER EVER EVER EVER HIT IT. It will move as close to 0 on the y axis as possible but WILL NEVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER HIT IT.

 

It has a domain of (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s 1) does not equal 0.

The range is (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s1) does not equal 0.

Do you know why it will never reach 0? Because it will never reach infinity. In case you didn't notice, your domain does not include +/- infinity.

Read my post again. I clearly state it goes from -infi to 0 0 to infi

Let me explain further. You CANNOT include infinity in your domain or range, infinity is not a specific value. First of all, by convention round brackets, i.e ( ), implies that the end values are not included in your domain, while square brackets [ ], would imply that they are included, so I assumed you meant that infinity was not in the domain.

It doesn't really matter though, infinity cannot be a value in your domain, because x will never equal infinity. This is why we express infinities through limits

Exactly, Infiniti cannot be included which is why you use a parenthesis and not a bracket.

 

 

 

 

I'm going to explain this to you using some basic fucking Algebra II.

 

f(x) = 1 / x

 

That is a rational function with a domain of 0 and a vertical asymptote at 0 and a horizontal asymptote at 0. The graph will be a hyperbola. It will move as close to 0 on the x axis but WILL NEVER EVER EVER EVER EVER EVER HIT IT. It will move as close to 0 on the y axis as possible but WILL NEVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER EVER HIT IT.

 

It has a domain of (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s 1) does not equal 0.

The range is (-infi, 0)U(0, infi) but it does not include 0 as DECIMAL(infinity 0s1) does not equal 0.

Do you know why it will never reach 0? Because it will never reach infinity. In case you didn't notice, your domain does not include +/- infinity.

Read my post again. I clearly state it goes from -infi to 0 0 to infi

Let me explain further. You CANNOT include infinity in your domain or range, infinity is not a specific value. First of all, by convention round brackets, i.e ( ), implies that the end values are not included in your domain, while square brackets [ ], would imply that they are included, so I assumed you meant that infinity was not in the domain.

It doesn't really matter though, infinity cannot be a value in your domain, because x will never equal infinity. This is why we express infinities through limits

Exactly, Infiniti cannot be included which is why you use a parenthesis and not a bracket.

 

Yes, as I said, infinity is not included in your domain, hence why you can never reach 0. Your post thereby doesn't prove anything. There are however, several viable proofs that 0.999... repeating does equal 1.

Well it is one of those weird sciency kinda things you can't explain for instance.

 

If you drop a tennis ball, and a bowling ball out of a plane, which one will fall faster?

 

 

 

 

 

 

Answer: They will fall at the same speed. How? But the bowling ball is bigger!, well force doesn't work that way gravity pulls at a certain speed and that is it, everything will fall at i believe 120 mph and no faster regardless of weight and matter.

 

So does 999999 = 1, well you will never really know..

Well it is one of those weird sciency kinda things you can't explain for instance.

 

If you drop a tennis ball, and a bowling ball out of a plane, which one will fall faster?

 

 

 

 

 

 

Answer: They will fall at the same speed. How? But the bowling ball is bigger!, well force doesn't work that way gravity pulls at a certain speed and that is it, everything will fall at i believe 120 mph and no faster regardless of weight and matter.

 

So does 999999 = 1, well you will never really know..

For some reason I feel obligated to correct this post (sorry in advance)...

 

Objects do not fall at the same speed, they accelerate at the same rate. More specifically, at a rate of around 9.81 meters per second. It's not a "weird sciency thing you can't explain", it's the Law of Gravitation. 

 

On another note, if you drop a bowling ball and a tennis ball out of a plane, the bowling ball would most likely hit the ground first, as it has a greater terminal velocity.

 

 

No it will not. It will just continue getting closer and closer to 1, but will never actually become 1.

 

this

 

 

Incorrect, you are forgetting that numbers are made up. According to our number system and our math system, 0.999... = 1. View my proof above.

 

 

I don't think so. If you draw a line at 1, a horizontal line, and then you draw a line right above that, of 0.999999 till however long. Will it ever become 1? it won't.

 

 

 

No it will not. It will just continue getting closer and closer to 1, but will never actually become 1.

 

this

 

 

Incorrect, you are forgetting that numbers are made up. According to our number system and our math system, 0.999... = 1. View my proof above.

 

 

I don't think so. If you draw a line at 1, a horizontal line, and then you draw a line right above that, of 0.999999 till however long. Will it ever become 1? it won't.

 

 

You are missing the point.

 

Using a geometric series:

 

The number "0.9999..." can be "expanded" as:

 

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

 

In other words, each term in this endless summation will have a "9" preceded by some number of zeroes. This may also be written as:

 

0.999... = 9/10 + (9/10)(1/10)^1 + (9/10)(1/10)^2 + (9/10)(1/10)^3 + ...

 

That is, this is an infinite geometric series with first term a =  9/10 and common ratio r =  1/10. Since the size of the common ratio r is less than 1, we can use the infinite-sum formula to find the value:

 

0.999... = (9/10)[1/(1 - 1/10)] = (9/10)(10/9) = 1

 

So the formula proves that 0.9999... = 1.

 

Using Algebra:

 

The expression 0.999... is some number; it has some value. Call this numerical value "x", so 0.999... = x. Multiply this equation by ten:

 

x = 0.999... 

10x = 9.999...

 

Subtract the former from the latter:

 

10x - 1x = 9.999... - 0.999... = 9x = 9.000...

 

Solving, we get x = 1, so 0.999... = 1.

 

"But," you ask, "when you multiply by ten, that puts a zero at the end, doesn't it?" For finite expansions, certainly; but 0.999... is infinite. There is no "end" after which to put that zero.

 

A common objection is that, while 0.999... "gets arbitrarily close" to 1, it is never actually equal to 1. But what is meant by "gets arbitrarily close"? It's not like the number is moving at all; it is what it is, and it just sits there, looking at you. It doesn't "come" or "go" or "move" or "get close" to anything.

 

On the other hand, the terms of the associated sequence, 0.9, 0.99, 0.999, 0.9999, ..., do "get arbitrarily close" to 1, in the sense that, for each term in the progression, the difference between that term and 1 gets smaller and smaller. No matter how small you want that difference to be, I can find a term where the difference is even smaller.

 

This "getting arbitrarily close" process refers to something called "limits". You'll learn about limits later, probably in calculus. And, according to limit theory, "getting arbitrarily close" means that they're equal: 0.999... does indeed equal 1.

I remember back in high school my teacher showed us a math formula that everything was correct and the answer was 1=2. Math makes no sense, just accept it.