Is infinity an integer?
Is infinity an odd or even number? (2011)
31–40 of 398 posts
Re: Is infinity an odd or even number? (2011)
#32Infinity is equal to the product of all the primes. It is even because it has 2 as a factor.
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Re: Is infinity an odd or even number? (2011)
#33I 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.
I don't usually use that term, but I take it to mean "number you (may, and, if not using e.g. fractions, must) write using a decimal point" because that seems to always be what people intend by it.
Everybody experienced writing irrational numbers using decimal notation in school, so those definitely count.
Re: Is infinity an odd or even number? (2011)
#34Earlier quoted context omitted.
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).
IEC 60559 could have defined pow(-1, ∞) as -1 if they’d wanted to. Hence my question what the “thus” is about.
Re: Is infinity an odd or even number? (2011)
#35I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.
I have an opinion that number of decimal numbers between 0 and e is equal to number of decimal numbers between e and +Infinity, because a parabola with a=e will grow in x with same speed as in y.
Re: Is infinity an odd or even number? (2011)
#36I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.
There are, and it turns out that this is a significant mathematical concept.
The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i that is an even number, it pairs with the number i/2.
To demonstrate: 0 -> 0, 2 -> 1, 4 -> 2, 6 -> 3, ...
The decimal (real) numbers between 0 and 1 are not countably infinite, and we know this from a concept called Cantor diagonalization. What Cantor did was a proof by contradiction: assume that the numbers are countably infinite, then you can arrange them in a list. However, he then builds a number by altering the first decimal place of the first number, the second decimal place of the second number, and so on. Finally, he shows that this built number is both a real number and is not on the list. Therefore, the real numbers between 0 and 1 cannot be ordered into a list, therefore they are not countably infinite, and there are more decimal numbers between 0 and 1 than integers between 0 and infinity.
Re: Is infinity an odd or even number? (2011)
#37>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…
But zero is in the middle, so it must be odd
Couldn't you trivially say zero is in "the middle" of any even split? 2 + 0 + 2 = 4?
Edit: perhaps you meant one is in the middle?
Re: Is infinity an odd or even number? (2011)
#38Infinity, is, of course, weird.
David Deutsch expanded on Hilbert's hotel in this chapter https://publicism.info/science/infinity/9.html of one of his books, one of the funnest little discussions of (mostly countable) infinity I've seen.
Re: Is infinity an odd or even number? (2011)
#39Earlier quoted context omitted.
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).
Six-year-olds know multiplication?
My 6 year old is learning multiplication at the moment.
Re: Is infinity an odd or even number? (2011)
#40Earlier quoted context omitted.
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).
Six-year-olds know multiplication?
Some do, yes. If they have an aptitude for basic sums then pointing out that 3 x 3 is the same as 3 + 3 + 3 sets them down the right path ...