## defining a number as "real"

For the discussion of math. Duh.

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skeptical scientist
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So elements of quantum groups are real numbers because they can be physically represented in the bizarre physics of the very small?
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FiddleMath
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I think that a lot of the basic ideas in this thread were discussed, fairly precisely and at length, in this thread, to which I'll link because I'm still quite fond of my little essay, there.

Taejo
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skeptical scientist wrote:
Alky wrote:Formal definition:
http://en.wikipedia.org/wiki/Real_number

Layman's definition:
Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

So 0.999... and 1 are different as real numbers because they are different as sequences of digits?

The definition does not say that DIFFERENT digit sequences are DIFFERENT reals; only that digit sequences ARE reals. It seems to me that this definition is equivalent to the limit of sequence of rationals.
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Taejo wrote:
skeptical scientist wrote:
Alky wrote:Formal definition:
http://en.wikipedia.org/wiki/Real_number

Layman's definition:
Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

So 0.999... and 1 are different as real numbers because they are different as sequences of digits?

The definition does not say that DIFFERENT digit sequences are DIFFERENT reals; only that digit sequences ARE reals. It seems to me that this definition is equivalent to the limit of sequence of rationals.

No, this definition is a lot more like defining a real to be a cauchy sequence of rationals. You need to have precise, correct definitions, otherwise when students get confused you have no one to blame but yourself.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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You can define the reals as infinite decimal expansions if you exclude numbers that end in all 9s.

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Or, alternatively, as infinite, nonterminating expansions (that don't end in all zeros). It doesn't really matter which one you pick, just as long as you exclude one or the other.
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Yakk
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And what if I want to exclude all of the numbers that end in all 3s?

Buncha 9 and 0 haters.

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Then you will become an outcast in the mathematical community and will have to beg for bread on the street.
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Yakk wrote:And what if I want to exclude all of the numbers that end in all 3s?

Buncha 9 and 0 haters.

You working in base 4? If so, go right ahead and drop the ones that end in all 3's.
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.9 repeating and 1 belong the same equivalence partition.

QED
1010011010

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n3k5
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TheLordOfPhysics wrote:if you can physically represent it it is real. like 1 being i have one bradpeice, two bradpeices and so on and so forth because i can physiclly represent it. were as negatives are representations of debt and owning "bradpeices's" if you will.

Hmm ... there is no such thing as pi breadpieces (even if you define a breadpiece to have some exact mass). So pi is not a "real" number?

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n3k5 wrote:So pi is not a "real" number?

You have to be careful on what you think is real. Your definition of real does not fit in with mathematics definition. It's like this in all fields, scientific or not.

Pi is a real number. It's hard to define what a real number is; however, easier to define what a real number isn't.

It's like telling your friends what the perfect wife is... you know she won't be a complete... Well, you get the idea.
1010011010

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n3k5
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EradicateIV wrote:You have to be careful on what you think is real.

No I don't, I was talking about the definition of "real" in the original post. (Whatever that is *g* ... but that's the topic.)

Your definition of real does not fit in with mathematics definition.

I understand the mathematical definition, but I assumed it has already been agreen on here that that's not what we're talking about here. (N.B. I wrote ""real"", not "real".) I've said nothing about any other definition of my own, so any statements about that are meaningless ;-p

But seriously, I found the explanation of TheLordOfPhysics a bit incomplete, so I asked for a clarification. Not because I need to learn anything about real or "real" numbers, but just for the heck of it, to see what TheLordOfPhysics was getting at.

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I wasn't getting smart, just saying...

Let's not be irrational now.
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A "real" number is simply any number that's useful to think about. This probably includes all the integers and rationals, as well as all the algebraic numbers. By necessity, it would also include pi and e, and any other transcendental numbers that play a specific role in mathematics. You could probably get away with a contructionist definition such as "A "real" number is any number that can be defined by a finite statement."

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You could probably get away with a contructionist definition such as "A "real" number is any number that can be defined by a finite statement."

sqrt(-1)

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stormgren wrote:sqrt(-1)

Read back a few posts. We're talking about "real" numbers, not the set R of real numbers.

Besides, most of R can't be described finitely.
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OneTrue
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parallax wrote:A "real" number is simply any number that's useful to think about. This probably includes all the integers and rationals, as well as all the algebraic numbers. By necessity, it would also include pi and e, and any other transcendental numbers that play a specific role in mathematics. You could probably get away with a contructionist definition such as "A "real" number is any number that can be defined by a finite statement."

You might like Computable Numbers.

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OneTrue wrote:
parallax wrote:A "real" number is simply any number that's useful to think about. This probably includes all the integers and rationals, as well as all the algebraic numbers. By necessity, it would also include pi and e, and any other transcendental numbers that play a specific role in mathematics. You could probably get away with a contructionist definition such as "A "real" number is any number that can be defined by a finite statement."

You might like Computable Numbers.

Closer to his intention would probably be definable numbers.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson