Kosimo

May 4th, 2009, 11:58 AM

1 minus 0.99(periodic) Is????

View Full Version : 1 - 0.99 (periodic) = X ? Ώ?

Kosimo

May 4th, 2009, 11:58 AM

1 minus 0.99(periodic) Is????

Tibuda

May 4th, 2009, 12:03 PM

1 = 0.999...., so 1 - 0.999.... = 0

Kosimo

May 4th, 2009, 12:05 PM

1 = 0.999...., so 1 - 0.999.... = 0

hmmm... :-k

I'm not convinced... the result should be bigger than 0. As we're reducing less than the total 1

hmmm... :-k

I'm not convinced... the result should be bigger than 0. As we're reducing less than the total 1

GeneralZod

May 4th, 2009, 12:07 PM

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

In particular:

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

In particular:

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

Kosimo

May 4th, 2009, 12:08 PM

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

In particular:

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

Cool! Thank you for the links

I'm gonna read them right now

In particular:

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

Cool! Thank you for the links

I'm gonna read them right now

Kosimo

May 4th, 2009, 12:15 PM

Very interesting reading:

From Wikipedia

Skepticism in education

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.[33]

# 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".[34]

# 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.[35]

# Some students regard 0.999 as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.

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 professor David O. 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 havent specified how many places there are' or 'it is the nearest possible decimal below 1'".[36]

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.[37] 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.[38] 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."[39]

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky 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

From Wikipedia

Skepticism in education

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.[33]

# 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".[34]

# 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.[35]

# Some students regard 0.999 as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.

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 professor David O. 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 havent specified how many places there are' or 'it is the nearest possible decimal below 1'".[36]

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.[37] 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.[38] 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."[39]

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky 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

Tibuda

May 4th, 2009, 12:21 PM

hmmm... :-k

I'm not convinced... the result should be bigger than 0. As we're reducing less than the total 1

No, 0.99.... is not less than 1, it's exactly 1.

I'm not convinced... the result should be bigger than 0. As we're reducing less than the total 1

No, 0.99.... is not less than 1, it's exactly 1.

Giant Speck

May 4th, 2009, 01:00 PM

1 - 0.999... = 0.000...1

gn2

May 4th, 2009, 03:08 PM

1 - 0.999... = 0.000...1

Nope 0.999... is 1

Divide one by three you get a third.

A third is written in decimal notation as 0.333...

Now multiply a third by three, what do you get?

Nope 0.999... is 1

Divide one by three you get a third.

A third is written in decimal notation as 0.333...

Now multiply a third by three, what do you get?

Giant Speck

May 4th, 2009, 03:31 PM

Nope 0.999... is 1

Where did I say otherwise? 0.000...1 can also be said to equal 0.

Where did I say otherwise? 0.000...1 can also be said to equal 0.

Namtabmai

May 4th, 2009, 04:15 PM

1 - 0.999... = 0.000...1

Er no,

0.000...1

isn't a valid mathematical notation, and even if it was

1 - 0.999.... != 0.000...1

Er no,

0.000...1

isn't a valid mathematical notation, and even if it was

1 - 0.999.... != 0.000...1

SuperSonic4

May 4th, 2009, 04:25 PM

Let x = 0.999...

Multiply by 10:

10x = 9.999...

Take x from both sides:

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

9x= 9 and so x=1

Multiply by 10:

10x = 9.999...

Take x from both sides:

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

9x= 9 and so x=1

Marco A

May 4th, 2009, 04:29 PM

.

Kosimo

May 4th, 2009, 05:31 PM

Is just a crazy concept...

But looks like 0.999... is the same than 1. Maybe we should still invent the right number to define that result anyway...

But looks like 0.999... is the same than 1. Maybe we should still invent the right number to define that result anyway...

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