0.99999~ = 1 ?

I guess my prayers were not answered. Can you provide a definition of 'real number' so that I do not have to look for one myself?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Which it is by definition.BTW, I have obviously not attempted to disguise the fact that we are reasoning about real numbers and not about omelets or geraniums. Your admiration of how "clever" I am in doing so, though flattering, is therefore misplaced. No it can't. That was not the change I made.If you will look at my edit and compare it to the original, you will see what change I made. The purpose of it was to prevent you from reading it as though being "distinct" could be the property of one number, or of two numbers separately, rather than of a pair of numbers.Edited by Dr Adequate, : No reason given.

DP.Edited by Dr Adequate, : No reason given.

Jon writes:

quote:

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Can you provide a definition of 'real number' so that I do not have to look for one myself?

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Well, the problem is that a "Real" number isn't nearly as easily defined as other kinds of numbers. For example, there are the "Natural" numbers: 1, 2, 3, .... Notice that "0" is not among them. If you add 0 to it, you get "Whole" numbers. Add the negatives, and you have the "Integers."A "Rational" number is any number that can be expressed as a/b where a and b are Integers.Real numbers, though, start to get complicated. One of the simplest way of constructing the "Reals" is through decimal expansion. For any number, you can converge upon its decimal representation: 2, 2.7, 2.71, 2.718, 2.7182, etc. will converge upon e, a Real number.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

LOL. I'm too busy for malarkey. My debate is with lyx2no, now. If you want to join, you can start by supporting this latest claim.Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Well, try the following definition:---Let R be the set of Cauchy sequences of rational numbers. That is, sequences x₁,x₂,x₃,... of rational numbers such that for every rational ε > 0, there exists an integer N such that for all natural numbers m,n > N, |x_m-x_n|<ε. Here the vertical bars denote the absolute value.Cauchy sequences (x) and (y) can be added, multiplied and compared as follows:
* (x_n) + (y_n) = (x_n + y_n)
* (x_n) (y_n) = (x_n × y_n)
* (x_n) ≥ (y_n) if and only if for every rational ε > 0, there exists an integer N such that x_n ≥ y_n - ε for all n > N.Two Cauchy sequences are called equivalent if and only if for every rational ε > 0, there exists an integer N such that |x_n -y_n|<ε for all n > N.This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers.---Note that the sequence 0.9, 0.99, 0.999, 0.9999 ... is indeed a Cauchy sequence, since this is a sequence of rational numbers and since for any rational ε > 0 there exists some N for which ε > 10^-N.Indeed, one can also immediately prove from the definition of the equivalence relation that 0.999999... = 1.Edited by Dr Adequate, : No reason given.

Your posts belie this claim. This declaration will not, of course, prevent me from pointing out that you are wrong.

LOL. Yes; too busy for malarkey, but I can make time for bullshit. If you wish to be obtuse, care to point out where in my rewording the meaning of your original was lost?Edited by Jon, : Mr. T

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Rrhain,Thank you for actually participating with reason and evidence. My math is a little rusty; can you explain this one to me? What does it mean for numbers to converge upon e?Thanks,
Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

So I see. But what if I do not wish to be obtuse?

I did not think 'rational' and 'real' numbers to be the same. One's a subset of the other, no? Which are...?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

A sequence of numbers x₁, x₂, x₃ ... converges on e if for every ε > 0 there is some N such that for all i > N, |x_i - e| < ε.

You're right. Why do you mention it? A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, respectively), a binary relation ≤ on R, satisfying the following properties.1. (R, +, *) forms a field. In other words,* For all x, y, and z in R, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z. (associativity of addition and multiplication)
* For all x and y in R, x + y = y + x and x * y = y * x. (commutativity of addition and multiplication)
* For all x, y, and z in R, x * (y + z) = (x * y) + (x * z). (distributivity of multiplication over addition)
* For all x in R, x + 0 = x. (existence of additive identity)
* 0 is not equal to 1, and for all x in R, x * 1 = x. (existence of multiplicative identity)
* For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)
* For every x ≠ 0 in R, there exists an element x^-1 in R, such that x * x^-1 = 1. (existence of multiplicative inverses)2. (R, ≤ ) forms a totally ordered set. In other words,* For all x in R, x ≤ x. (reflexivity)
* For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)
* For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
* For all x and y in R, x ≤ y or y ≤ x. (totalness)3. The field operations + and * on R are compatible with the order ≤. In other words,* For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
* For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ x * y (preservation of order under multiplication)4. The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words,* If A is a non-empty subset of R, and if A has an upper bound, then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.Edited by Dr Adequate, : No reason given.Edited by Dr Adequate, : No reason given.

How is this different from saying:Real numbers are numbers that meet definition X
Real numbers are defined by X
All real numbers are RealWhat purpose does the Real/Non-real distinction serve? Does it actually differentiate, or is the distinction merely made up?

Clearly the only way to define the real numbers is to supply a definition of the real numbers. Clearly there are structures which are not the real numbers. Such as the complex numbers. Or quaternions. Or, for that matter, asparagus.