Neither, Infinity % 2 = NaN
Is infinity an odd or even number? (2011)
61–70 of 398 posts
Re: Is infinity an odd or even number? (2011)
#62The problem with transfinite is that you lose commutatively. Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..." ω+1 = ω+1 , i.e. "the next thing after infinity" 1+ω = ω , i.e. "the same infinity as before" 2ω = ω , i.e. "the same infinity as before", so it's even 1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap ω2 =…
> The problem with transfinite is that you lose commutatively.
Depends on which transfinite algebra you're working with. If you restrict "number" to mean "element of an ordered field" (thus excluding things like the "complex numbers" but matching the usual intuition of how numbers should behave) then you can't include Cantor's ordinals but you can include the Surreal Numbers. Those include infinite ordinals and (due to being a field) have commutative addition and multiplication operations.
Re: Is infinity an odd or even number? (2011)
#63I 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.
Turn it into a proof?
I need to revisit Cantor's proof, the argument I was taught left out key aspects of numbers, particularly how "number" and "string of digits you've printed out so far" aren't the same thing. It's really about creating a space-filling curve.
Re: Is infinity an odd or even number? (2011)
#64In 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).
Child: “Is 100 million the biggest number?”
Teacher: “Well, there’s 100 million and one”
Child: “I was pretty close then!”
Re: Is infinity an odd or even number? (2011)
#65Earlier quoted context omitted.
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.
The correct mathematical term is “real number”.
Re: Is infinity an odd or even number? (2011)
#66Infinity is equal to the product of all the primes. It is even because it has 2 as a factor.
I find that primes are an odd thing to bring into the discussion, not even mildly relevant.
Re: Is infinity an odd or even number? (2011)
#67Earlier quoted context omitted.
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 ...
I had it explained at a very early age as "three lots of three", and to imagine it like three boxes of three ice-creams. Treating the multiplication symbol as one would to indicate quantity in a list, thus calculating how many ice-creams there are.
Re: Is infinity an odd or even number? (2011)
#68Re: Is infinity an odd or even number? (2011)
#69Earlier quoted context omitted.
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.
The correct mathematical term is “real number”.
Yes, but you'll see "decimal" more in the wild, and that's what people mean by it. "You write it with a decimal point", and they do usually mean to include the irrationals. So, yes, real numbers, but the reasoning behind their usage is "you write it with a decimal point". I'd bet more people understand "decimal number" used in that sense, than understand "real number".
Re: Is infinity an odd or even number? (2011)
#70In 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?
No, it comes from the fact that floating point is binary and has limited precision. Think of it in terms of scientific notation. Here's an example in decimal. If we limit ourselves to four significant digits, then a number like:
3.101 * 10^3
is odd -- it's equivalent to 3101 (three thousand one hundred one). It's followed by 3.102*10^3 (3102), which is even, and 3.103*10^3 (3103), which is odd. But a number like:
3.101 * 10^5
which is equivalent to 310100, is even. It's followed by 3.102*10^5 (310200), which is also even, and 3.103*10^5 (310300), which is again even. If you have four significant digits and an exponent larger than 3, then you the value in the ones place will always be zero. Thus, the number is always a multiple of 10, and therefore even.
Floating point is the same, except it's binary. In a 32-bit float, you have 23 bits of mantissa after the decimal point. If the exponent is larger than 2^23, the ones place is always zero, so the number is guaranteed to be a multiple of 2, and therefore even.