# 10c - c

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One last time. Mathematicians do not define 0.999... to be equal to one. They prove that it is equal to one by a number of means, some more rigorous than others. What mathematicians define is the concept of a convergent series. With this definition, the real numbers are defined as the set of all convergent Cauchy sequences. Nowhere do mathematicians define .999... to be equal to one.

Hey Ben! Isn't it about time to lock this thread? We have a full-fledged crackpot on our hands here. This has deteriorated into yet another thread about $$0.\overline 9 \ne 1\ .$$

Defining real numbers as convergent Cauchy sequences proceeds by constructing real numbers based on rationals. It begins by assuming repeating decimals are valid expressions of rationals.

Defining real numbers as convergent Cauchy sequences proceeds by constructing real numbers based on rationals. It begins by assuming repeating decimals are valid expressions of rationals.

To complete this point - once you accept that repeating decimals are valid expressions of rational numbers, .9r = 1 is fixed as a feature. It is no longer left to be proved or disproved.

My whole point is the way that limits require you to assume that infinity is a number you actually get to.
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Okay, I'll start at the top again. My argument is that it is not possible for classical mathematics to prove .9r = 1, because it assumes .9r = 1.
I once shared an office with a very experienced high school math teacher, and we were discussing what to do about different levels of math ability, and specifically how to tell if a certain student was ready for calculus.

It was her opinion that attempting to drill calculus into a student's brain when that brain wasn't developmentally ready was useless at best, that students varied considerably in their manner and rate of development independently of their overall intelligence or intellectual potential, and so therefore some way should be found to delay calculus instruction for many students without impugning their intelligence or discouraging their intellectual (even mathematical) efforts in general.

Her personal criterion was the ability of a student to grasp the concept of a limit. Some can, some can't, some who can't at one moment can a few months later in their lives.

And it has nothing necessarily to do with intelligence in a sense - even mathematical intelligence. As we see:

It's funny that this started as an effort to understand 10c - c = 9c, and has spun into me no longer believing in rational numbers.

The calculus argument is the worst for me. Calculus is something you teach 18 year olds, and that's where I think the most confusion comes from.

Calculus allows you to use derivatives and limits to calculate the actual value of something (the area under a curve). And it blurs the distinction between "the limit, based on the derivative, approaches the real area under the curve" and what it means in other contexts for something to approach something else.

To follow - yes, if you could break an area into an infinite number of pieces, the sum of the area of those pieces would equal the actual area under a curve. That doesn't make it possible to break an area into an infinite number of pieces. That you can calculate the actual area by calculating what the sum of the areas would be if you COULD break it into an infinite number of peices doesn't mean that breaking it into an infinite number of pieces is real.

It does? This comes to me as a very great surprise. Would you care to explain your reasoning here?

Well, such a question is posted throughout many forums. Do i have this wrong?

It's one of those questions, that really holds no reason. :shrug:

And it's also one of those questions that gets boring very fucking quickly.

Oh bah! Meh!

Defining real numbers as convergent Cauchy sequences proceeds by constructing real numbers based on rationals.
Correct.

You have to start somewhere. The starting point is the Peano arithmetic. The rationals are constructed from the integers by as the ratio of a pair of integers. The integers are constructed from the non-negative integers by subtraction. Finally, the Peano arithmetic defines the non-negative integers.

Note well: There are many different ways to represent the rational 1/2: 1/2, 2/4, 3/6, 4/8, ... Every rational number can be expressed in an infinite number of ways. There are also many different ways to represent the integer -1: 0-1, 1-2, 3-2, ... There are similarly many different different ways to represent any real number. For example, $$1+\sum_{n=1}^{\infty}\frac 0 {10^n}$$ and $$\sum_{n=1}^{\infty}\frac 9 {10^n}$$ are different representations of the real number 1.0.

It begins by assuming repeating decimals are valid expressions of rationals.
Incorrect. It begins by assuming the concept of a limit, which I strongly urge you to understand.

Incorrect. It begins by assuming the concept of a limit, which I strongly urge you to understand.

The concept of "the limit represents a real value" is validated by assuming repeating decimals are valid expressions of rational numbers.

This is the mobius strip of proving .9r = 1.

We have had long discussions on this topic before. Those threads were locked, too.

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