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Is infinity an odd or even number? (2011)

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Re: Is infinity an odd or even number? (2011)

#11

Post by layer8 »

pxx wrote: post #8

In IEC 60559* floating-point arithmetic, pow(-1, ∞) is 1. This is because all large binary and decimal floating-point numbers are even, and thus so is infinity. *this is the successor standard to ieee-754 and shares text in recent revisions, though I don't have direct access on this phone. You can find the specific pow specification in Annex F of the C99 standard.

Is the “thus” for ease of implementation? I.e., so that all floating-point numbers comparing greater than some threshold can be considered even without having to check for infinity?

Re: Is infinity an odd or even number? (2011)

#12

Post by alexb_ »

warent wrote: post #10

In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite: 1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯ Presumably this person has no experience with 6 year olds? This explanation is horrendous haha

No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).

Re: Is infinity an odd or even number? (2011)

#13

Post by warent »

Eisenstein wrote: post #9

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

My understanding is that this is true because there are infinite decimals between every decimal, infinitely.

For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity.

In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something)

https://en.m.wikipedia.org/wiki/Aleph_number

Re: Is infinity an odd or even number? (2011)

#14

Post by irrational »

>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair.

The answer this quote came from is amazingly obtuse, but it does make me think that infinity must be even since infinity can be divided into 2 pairs, each of which is of equal size since both are infinity.

Re: Is infinity an odd or even number? (2011)

#15

Post by contravariant »

Eisenstein wrote: post #9

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

Depends what you mean by decimal. Decimal is a system of notation, does it count as a decimal number if it cannot be written in decimal notation (in finite time)?

if not then they are equal, if yes then there are more decimal numbers between 0 and 1 than integers.

Re: Is infinity an odd or even number? (2011)

#17

Post by daef »

warent wrote: post #13
Eisenstein wrote: post #9

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_numb…

You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.

Re: Is infinity an odd or even number? (2011)

#18

Post by pxx »

layer8 wrote: post #11
pxx wrote: post #8

In IEC 60559* floating-point arithmetic, pow(-1, ∞) is 1. This is because all large binary and decimal floating-point numbers are even, and thus so is infinity. *this is the successor standard to ieee-754 and shares text in recent revisions, though I don't have direct access on this phone. You can find the specific pow specification in Annex F of the C99 standard.

Is the “thus” for ease of implementation? I.e., so that all floating-point numbers comparing greater than some threshold can be considered even without having to check for infinity?

It's not that they are considered even, they just are. There's no way to encode a large odd even-radix floating point number. You have some (small, compared to the range that the exponent can encode) bits of significand and once you exhaust those all numbers are even (or divisible by ten in the rare decimal case).

Re: Is infinity an odd or even number? (2011)

#19

Post by warent »

daef wrote: post #17
warent wrote: post #13

Earlier quoted context omitted.

My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_numb…

You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.

Oh right this is only true for irrational and transcendental numbers

Re: Is infinity an odd or even number? (2011)

#20

Post by rcme »

warent wrote: post #13
Eisenstein wrote: post #9

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_numb…

It’s not because there are infinite decimals between every two decimal numbers. That applies to the rational numbers too, e.g. there are infinite rational numbers between 1/2 and 3/4. Rather, the real numbers are more dense in a way that makes them fundamentally larger than the integers / rational numbers. “Larger” means not being able to pair up the two sets one by one so that each element of both sets is the member of a pair. No matter how you pair up the integers to the reals, you can prove that some real numbers will be unpaired.
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