Haha we have to make a leap of faith and just accept that 0.999... = 1. It's the language of math and is defined provably so. Yet a simple counter is that 0.999... is smaller than 1. The onus is on those who state the positive.
from: http://en.wikipedia.org/wiki/User:ConMan...ot_equal_1
As you have functions that gradually curve toward a value (say 1), this function will eventually hit every possible value along .9999 repeating. As we know the curve will NEVER hit 1, then .9999 repeating is by necessity less than 1.
The curve is asymptotic, it reaches 1 at infinity. Since the .999 sequence is infinite, it is equal to 1.
Moreover, as you have a square on a plain, with sides of .9999 repeating units on a side. Will a square with sides of 1 fit inside?
[edit] What needs to be proved?
There is no need for anyone to prove 0.999... < 1 because by definition of the decimal radix system, it is. The onus of proof rests on those who claim these two values are equal. All their proofs are false: starting with the most common (as in the above example) and moving to the most complex (as in the Archimedean property). If x = 0.999... then 10x is not well-defined. In fact it is not defined at all because our rules of arithmetic for multiplication apply to rational numbers only. We extend these rules to non-rational numbers by approximation. That is to say, we can never calculate the area of a circle exactly and can never find the exact length of the hypotenuse in a right angled triangle where the remaining sides are both of unit length. So, if multiplication is defined for rationals, we must therefore be able to express 0.999... as a rational number. This is impossible for there are no numbers a and b such that a/b = 0.999...
Rational numbers were at first invented to represent only quantities that are less than 1 but greater or equal to 0. Numbers were written in a finite representation format. Recurring numbers were not considered rational until centuries later when the limit concept was introduced. This definition was later reworked (or extended) so that we could perform mixed arithmetic a lot easier. For example arithmetic involving an originally defined rational number and a whole number, eg. 2 + 2/3 = 6/3 + 2/3 = 8/3 This is an exact calculation because it can be represented finitely.
Much later very stupid men decided that a number would be defined as a limit of a Cauchy sequence. The definition they invented is in fact circular because they use a limit to define a number. In fact, if they were even remotely intelligent, they would have stated that any number is a rational number or the limit of the sum of infinitely many rational numbers where a rational number is a/b with b not zero and a < b (Was part of original definition but later dropped).
So as long as we stay with the same rational representation of numbers, our result will generally be true. eg. 1/3 + 1/3 + 1/3 = 3/3 = 1. However, when we try to use other representations such as radix or mixed rational/radix, we end up with anomalies. eg. 1/3 + 0.333.. + 0.3333... = 1/3 + 0.666.... = ? Oops, 0.666... is not finitely represented and thus cannot be used in our arithmetic except as an approximation. e.g. 1/3 + 66/100. We cannot say 0.666... = 2/3 because it is not. Sorry, duplicate representation is not allowed in any radix system. And no radix system is capable of representing all numbers.
Let's summarize the rules: In order for arithmetic to be performed, numbers must be finitely represented using the same representation. 2pi/3 + pi/3 = pi. 1 + sqrt(5) = 1 + 2.23 = 3.23
In light of the above it is evident that even debating the 0.999... and 1 equality is completely idiotic.
from: http://en.wikipedia.org/wiki/User:ConMan...ot_equal_1
As you have functions that gradually curve toward a value (say 1), this function will eventually hit every possible value along .9999 repeating. As we know the curve will NEVER hit 1, then .9999 repeating is by necessity less than 1.
The curve is asymptotic, it reaches 1 at infinity. Since the .999 sequence is infinite, it is equal to 1.
Moreover, as you have a square on a plain, with sides of .9999 repeating units on a side. Will a square with sides of 1 fit inside?
[edit] What needs to be proved?
There is no need for anyone to prove 0.999... < 1 because by definition of the decimal radix system, it is. The onus of proof rests on those who claim these two values are equal. All their proofs are false: starting with the most common (as in the above example) and moving to the most complex (as in the Archimedean property). If x = 0.999... then 10x is not well-defined. In fact it is not defined at all because our rules of arithmetic for multiplication apply to rational numbers only. We extend these rules to non-rational numbers by approximation. That is to say, we can never calculate the area of a circle exactly and can never find the exact length of the hypotenuse in a right angled triangle where the remaining sides are both of unit length. So, if multiplication is defined for rationals, we must therefore be able to express 0.999... as a rational number. This is impossible for there are no numbers a and b such that a/b = 0.999...
Rational numbers were at first invented to represent only quantities that are less than 1 but greater or equal to 0. Numbers were written in a finite representation format. Recurring numbers were not considered rational until centuries later when the limit concept was introduced. This definition was later reworked (or extended) so that we could perform mixed arithmetic a lot easier. For example arithmetic involving an originally defined rational number and a whole number, eg. 2 + 2/3 = 6/3 + 2/3 = 8/3 This is an exact calculation because it can be represented finitely.
Much later very stupid men decided that a number would be defined as a limit of a Cauchy sequence. The definition they invented is in fact circular because they use a limit to define a number. In fact, if they were even remotely intelligent, they would have stated that any number is a rational number or the limit of the sum of infinitely many rational numbers where a rational number is a/b with b not zero and a < b (Was part of original definition but later dropped).
So as long as we stay with the same rational representation of numbers, our result will generally be true. eg. 1/3 + 1/3 + 1/3 = 3/3 = 1. However, when we try to use other representations such as radix or mixed rational/radix, we end up with anomalies. eg. 1/3 + 0.333.. + 0.3333... = 1/3 + 0.666.... = ? Oops, 0.666... is not finitely represented and thus cannot be used in our arithmetic except as an approximation. e.g. 1/3 + 66/100. We cannot say 0.666... = 2/3 because it is not. Sorry, duplicate representation is not allowed in any radix system. And no radix system is capable of representing all numbers.
Let's summarize the rules: In order for arithmetic to be performed, numbers must be finitely represented using the same representation. 2pi/3 + pi/3 = pi. 1 + sqrt(5) = 1 + 2.23 = 3.23
In light of the above it is evident that even debating the 0.999... and 1 equality is completely idiotic.
... hmm...
