# ℚ − ℤ

• Is there a name for rational numbers (fractions) that are not integers?

• Not that I know.

But I don't know a lot of number theory.

• It seems like people who don't know the definition of decimal call them "decimals". Which is wrong and I hate it.

•  Or a float.

Just go with float.

• @Ben L. said:

Is there a name for rational numbers (fractions) that are not integers?

The closest I can find is "Non-Integer Rational Numbers";  As all integers are also Rational numbers this is a tough category to identify.

• @Ben L. said:

Is there a name for rational numbers (fractions) that are not integers?

My Real and Complex Analysis book has no definition for ℚ \ ℤ * so I doubt there's a name for it.

*A \ B is the math nerd notation for all elements of a set not in another set ... IIRC subtraction is not an operation defined for sets

• Quotient

The set of quotients is defined as numbers that can be represented as p/q where p and q are integers and p mod q is nonzero, of maths degree is still knocking about up there.

Nope the set of quotients is an outdated term for the set of rational numbers and my lecturer was wrong

Quotient

The set of quotients is defined as numbers that can be represented as p/q where p and q are integers and p mod q is nonzero, of maths degree is still knocking about up there.

Why would you necro a topic that's SEVERAL hours old?

• All part of the service

My Real and Complex Analysis book has no definition for ℚ \ ℤ
Or c ∩ ℚ. Irregardless, the concept probably has so few uses that there's been no need for a name for it.

• What's your context, I mean if you're looking for a mathematical definition it would appear there isn't an accepted one, if you're looking for a way to explain it concisely without ambiguity, which is what mathematical definitions are really there for, then I'd just say a non-integer rational number, or if it's to someone who wouldn't know what rational numbers were, a non-whole mixed fraction, which is essentially the same thing but using vocabulary taught in about Year 1 over here in the UK.

• @PJH said:

My Real and Complex Analysis book has no definition for ℚ \ ℤ
Or c ∩ ℚ. Irregardless, the concept probably has so few uses that there's been no need for a name for it.

Oops, should have been ℤ \ ℚ. Maybe that's why I did so bad in that class ....

@PJH said:
My Real and Complex Analysis book has no definition for ℚ \ ℤ
Or c ∩ ℚ. Irregardless, the concept probably has so few uses that there's been no need for a name for it.

Oops, should have been ℤ \ ℚ. Maybe that's why I did so bad in that class ....

No, I think you were right the first time...

• @Ben L. said:

Is there a name for rational numbers (fractions) that are not integers?

Proper rational number.

• @pjt33 said:

@Ben L. said:
Is there a name for rational numbers (fractions) that are not integers?

Proper rational number.

Nope - a proper rational number is p ∈ ℚ | -1 < p < 1

Queue bad wikipedia reference. Apparently it's a "Mixed number".

@pjt33 said:
@Ben L. said:
Is there a name for rational numbers (fractions) that are not integers?

Proper rational number.

Nope - a proper rational number is p ∈ ℚ | -1 < p < 1

Queue bad wikipedia reference. Apparently it's not even a "Mixed number".

Fix'd - it could be a proper fraction too but a mixed number is specifically an improper fraction, but it seems fraction doesn't exclude integers - I can't beleive there's no name for it though - have you tried posting this on math.stackexchange.com probably under group theory?

• @PJH said:

@PJH said:
My Real and Complex Analysis book has no definition for ℚ \ ℤ
Or c ∩ ℚ. Irregardless, the concept probably has so few uses that there's been no need for a name for it.

Oops, should have been ℤ \ ℚ. Maybe that's why I did so bad in that class ....

No, I think you were right the first time...
I feel a case of the Mondays coming on ....

@pjt33 said:
Proper rational number.

Nope - a proper rational number is p ∈ ℚ | -1 < p < 1

No, that's a proper fraction. The term "proper rational number" occurs precisely 0 times in the Wikipedia page you link.

• Yes and a rational number is a number that can be expressed as a fraction of two integers, also known as a quotient, which means, fraction in German, which is why the symbol for the set of rational numbers looks like a Q

My Real and Complex Analysis book has no definition for ℚ \ ℤ * so I doubt there's a name for it.

Fun fact: Even though ℤ is a strict subset of ℚ, ℚ \ ℤ has the same size as ℚ (namely 0).
Another fun fact: 0.̅9 is a confusing notation for a number in ℤ.

• @Faxmachinen said:

My Real and Complex Analysis book has no definition for ℚ \ ℤ * so I doubt there's a name for it.

Fun fact: Even though ℤ is a strict subset of ℚ, ℚ \ ℤ has the same size as ℚ (namely ℵ0).
Another fun fact: 0.̅9 is a confusing notation for a number in ℤ.

One can also prove that the size of ℚ is the same as the size of ℤ. And that ℤ is a strict subset of ℚ. Therefore, x - x = x, x ≠ 0, x < x

Math is weird.

• [sigh/]

ℚ is infinite, ℤ is infinite, but they're not the same infinity, in fact you can compare two infinities. Think of the series `y = x` and `z = x + 1` - both y and z tend to infinity as x tend to infinity, however z - y is 1 for any value of x, and if you take `w = z/y` that'll tend to 1.

ℚ is infinite, ℤ is infinite, but they're not the same infinity, in fact you can compare two infinities.

Do you know something the rest of mathematics doesn't? This could be revolutionary! Old theorems thrown out!

Think of the series `y = x` and `z = x + 1` - both y and z tend to infinity as x tend to infinity, however z - y is 1 for any value of x, and if you take `w = z/y` that'll tend to 1.

Oh, nevermind. I'm not sure what you're trying to say here, but I suspect it is a demonstration of bad counting leading to incorrect conclusions.

Indeed. I guess IHBT, eh?

• @boomzilla said:

Indeed. I guess IHBT, eh?

I'm guessing from your response that you were not being sarcastic. My example above was not trickery but actually a fairly simple proof of the mathematical concept of different infinities using limit theory, but you'll have to forgive me for forgetting not every software developer has to be familiar with limit theory. Infinity is not a number, but a more abstract concept than that.

The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals. This is not a new concept, however it is only really useful to think of it in the field of pure mathematics. 0 is a bit of a funny concept too when you poke around, but by ignoring that we use the concept every day without thinking too hard about it.

The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals.

Oh, I get it now. You were writing about ℚ and ℤ, but you were thinking about ℝ.

Did you know that 0.9 equals 1?

• @boomzilla said:

The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals.

Oh, I get it now. You were writing about ℚ and ℤ, but you were thinking about ℝ.

Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

@boomzilla said:
The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals.

Oh, I get it now. You were writing about ℚ and ℤ, but you were thinking about ℝ.

Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.

@boomzilla said:
The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals.

Oh, I get it now. You were writing about ℚ and ℤ, but you were thinking about ℝ.

Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

Technically, no it isn't 1, .
Oh look - someone's wrong on the internet again.

@boomzilla said:
Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.

It's always interesting to see novel proofs. theheadofabroom seems to be into limits (even if he's a bit sloppy). But the 0.9 equals 1 proof as I originally learned it has always amused me, so I'll repeat again...

```x = 0.9
10x = 9.9
10x - x = 9.9 - x
9x =  9.9 - 0.9
9x = 9
x = 1
```

Infinities are weird, but fun. I wonder what it means to be indistinguishable but distinct.

• @boomzilla said:

@boomzilla said:
Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.

It's always interesting to see novel proofs. theheadofabroom seems to be into limits (even if he's a bit sloppy). But the 0.9 equals 1 proof as I originally learned it has always amused me, so I'll repeat again...

```x = 0.9
10x = 9.9
10x - x = 9.9 - x
9x =  9.9 - 0.9
9x = 9
x = 1
```

Infinities are weird, but fun. I wonder what it means to be indistinguishable but distinct.

Another one is that 0.9 can be thought of as 1 - 0.01, which can be thought of as the value of y as x approaches infinity in the series y = 1 - E-x. Seeing as E-infinity is 0, that makes y == 1.

I realise I was a little sloppy in some of my previous posts, but let's put the onus on those who think that 0.9 != 1. We've given several proofs that it isn't, some even without glaringly obvious holes. If you can prove that it isn't, we'll believe you.

I realise I was a little sloppy in some of my previous posts, but let's put the onus on those who think that 0.9 != 1. We've given several proofs that it isn't, some even without glaringly obvious holes.

Is it context-dependent? When working with real numbers, 0.9 = 1; but when working with surreal numbers I'm pretty sure you get 0.9 = 1 - ε.

• @pjt33 said:

I realise I was a little sloppy in some of my previous posts, but let's put the onus on those who think that 0.9 != 1. We've given several proofs that it isn't, some even without glaringly obvious holes.

Is it context-dependent? When working with real numbers, 0.9 = 1; but when working with surreal numbers I'm pretty sure you get 0.9 = 1 - ε.
No, no, no. If you're working with surreal numbers, 0.9 = giraffe.

Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.

Sorry, but you're all mistaken. 0.̅9 is a number, not a process. You don't even need to prove that 0.̅9 equals 1, because it is 1 by definition. I do agree that the notation is non-intutive.

If 1 and 0.̅9 were processes, they would obviously not be equal.

• @boomzilla said:

@boomzilla said:
Did you know that 0.9 equals 1?

Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1

Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.

It's always interesting to see novel proofs. theheadofabroom seems to be into limits (even if he's a bit sloppy). But the 0.9 equals 1 proof as I originally learned it has always amused me, so I'll repeat again...

```x = 0.9
10x = 9.9
10x - x = 9.9 - x
9x =  9.9 - 0.9
9x = 9
x = 1
```

Infinities are weird, but fun. I wonder what it means to be indistinguishable but distinct.

GAIZ I found the spot where .999.... becomes one!

```<html><body style='color:#000000; background:#ffffff; '>double nine         = 0.9999999999999999;
double oneMoreNine  = 0.99999999999999999;
double one = 1.0;
Console.WriteLine(one == nine);
Console.WriteLine(one == oneMoreNine);
```

Technically, no it isn't 1,

• 1/9 = 0.111...

8/9 = 0.888...

1/9 + 8/9 = 0.999...

9/9 = 0.999...

1 = 0.999...

1/0 = 0.999.../0

Inf = Inf

Inf/Inf = Inf/Inf

NaN = NaN

false

Proven: 1 ≠ 0.999...

• @dhromed said:

Technically, 0.9... is entirely equal to 1.

That's nice and all, but he put the defined-equal operator in the completely wrong place in "The real proof" and is essentially stating that S := lim(S).

• @Faxmachinen said:

That's nice and all, but he put the defined-equal operator in the completely wrong place in "The real proof" and is essentially stating that S := lim(S).

If you are right, then I will non-sarcastically say hat he might be interested in your correction.

• Also, if numbers and sequences were truely interchangeable, then

```0.9999... + <var>P</var> = 1
P > 0```

would hold for

```0.̅9 = (10n - 1) / 10n
P = 1 / 10n
1 = 10n / 10n
0 = 0 / 10n```

Since they aren't, P has to be 0.̅0 (which is 0 by definition), and thus not greater than itself.

• @Faxmachinen said:

@dhromed said:

Technically, 0.9... is entirely equal to 1.

That's nice and all, but he put the defined-equal operator in the completely wrong place in "The real proof" and is essentially stating that S := lim(S).

I don't see that...exactly. It looks like he just tries to show what the expansions look like, thus removing N in the expansion, but not from the limit notation. Not terribly rigorous, but probably a good strategy for communicating to a lay person.

• @Faxmachinen said:

Also, if numbers and sequences were truely interchangeable, then

That's not what the proof was using. It was using a series, which is the sum of a sequence. Your statement makes no sense for anyone to say.

@Faxmachinen said:

```0.9999... + P = 1
P > 0```

would hold for

```0.̅9 = (10n - 1) / 10n
P = 1 / 10n
1 = 10n / 10n
0 = 0 / 10n```

Since they aren't, P has to be 0.̅0 (which is 0 by definition), and thus not greater than itself.

I can't make heads nor tails of what you're trying to say, but I think I agree with you that was you said was nonsense.

• @boomzilla said:

That's not what the proof was using. It was using a series, which is the sum of a sequence.

You're right. Let's do this with series, then.

`P = 1 - 0.̅9`

P is equal to 1 - 0.̅9 at any given point:

```(1 = 1.̅0 = 1 +   0 +    0 +     0 + ...) -
(0.̅9 =     0 + 0.9 + 0.09 + 0.009 + ...) =
(P =       1 - 0.9 - 0.09 - 0.009 - ...)```

P is larger than zero at any given point:

```(P =       1 - 0.9 - 0.09 - 0.009 - ...) >
(0 = 0.̅0 = 0 +   0 +    0 +     0 + ...)
```

• @Faxmachinen said:

P is larger than zero at any given point:

You demonstrated that 0.9... is not equal to 1 if and only if you stop the series at any given point.

• @dhromed said:

@Faxmachinen said:
P is larger than zero at any given point:

You demonstrated that 0.9... is not equal to 1 if and only if you stop the series at any given point.

Fucking infinities! How do they work?

• @dhromed said:

You demonstrated that 0.9... is not equal to 1 if and only if you stop the series at any given point.

Yes. How do you propose we compare two series without stopping at points?

• @Faxmachinen said:

@dhromed said:
You demonstrated that 0.9... is not equal to 1 if and only if you stop the series at any given point.

Yes. How do you propose we compare two series without stopping at points?

• @boomzilla said:

If you have any sources that prove or define that lim(S) = lim(T) implies S = T, I'd like to know about it. The first wikipedia article mentions it, but provides no citations.

• @Faxmachinen said:

@boomzilla said:

If you have any sources that prove or define that lim(S) = lim(T) implies S = T, I'd like to know about it. The first wikipedia article mentions it, but provides no citations.

You're leaving stuff out, so I'm not sure if I'm reading your mind correctly, but it's totally possible to have two series have the same limit where the series are different. Since I didn't cite any wiki articles, I don't even know which one you're talking about.

Did you know that glass is a super cooled liquid?

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