[HN Gopher] Is infinity an odd or even number? (2011)
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Is infinity an odd or even number? (2011)
Author : layer8
Score : 158 points
Date : 2023-05-02 13:43 UTC (9 hours ago)
(HTM) web link (math.stackexchange.com)
(TXT) w3m dump (math.stackexchange.com)
| taco_emoji wrote:
| Simple, infinity is even, but infinity + 1 is odd.
| steveBK123 wrote:
| feels smooth not spiky, so going with even
| lo_zamoyski wrote:
| It isn't clear to me infinity is a number in the first place.
| Reification and category mistakes are as much a danger in math as
| anywhere.
| paulddraper wrote:
| There is a definition for an infinite ordinal, omega.
|
| https://en.wikipedia.org/wiki/Ordinal_number
| snotrockets wrote:
| Infinity isn't a number, but it is an ordinal (and the answer
| does mention how you can have an even/odd property on the
| ordinals)
| breck wrote:
| The difference, explained by ChatGPT:
|
| A number is a mathematical concept used to represent a
| quantity or value. Numbers can be whole numbers (integers),
| fractions, decimals, or even complex numbers, and they are
| used in various mathematical operations like addition,
| subtraction, multiplication, and division.
|
| On the other hand, an ordinal is a term used to indicate the
| position or order of an element within a set or a sequence.
| Ordinals are generally expressed using words like first,
| second, third, or numerals with suffixes like 1st, 2nd, 3rd,
| etc. Ordinals are used to describe the rank or placement of
| an item in relation to others, rather than representing a
| specific quantity or value like numbers do.
|
| In summary, numbers represent a quantity or value, while
| ordinals represent a position or rank in a sequence.
| NegativeK wrote:
| Cardinal numbers (size) and ordinal numbers (ordering) are
| both numbers. The numbers we're familiar with represent
| both concepts, sometimes simultaneously.
|
| I really don't think that block quoting ChatGPT is a good
| contribution.
| breck wrote:
| > I really don't think that block quoting ChatGPT is a
| good contribution.
|
| It's the first time I've done it. I agree with you. But
| didn't know until I tried!
| yjk wrote:
| Which is also incorrect. See
| https://en.wikipedia.org/wiki/Ordinal_numeral vs
| https://en.wikipedia.org/wiki/Ordinal_number.
| breck wrote:
| I agree. It's an interesting intellectual exercise, but I am
| not sure if we would miss out on anything if we just had a
| symbol(s) for specific really large discrete numbers.
|
| Sometimes I wonder if there's a better math language waiting to
| be invented that eschews the non-discrete.
| bitL wrote:
| I really hate these mathematical technicalities spawned from
| material implication, chosen way of making a definition and
| vacuous truth - why can't we even consider some questions to be
| marked as non-sense/non-relevant like in relevance logic?
| Cruncharoo wrote:
| Not to be obtuse but isn't all of mathematics spawned from a
| chosen way of making a definition and vacuous truth?
| bitL wrote:
| Not really. One could argue some math is innate and we are
| just rediscovering it. See the disconnect between natural
| language and math which happened early 20th century because
| of material implication bringing vacuous truth, leading to
| "impedance mismatch". Medicine is still using counterfactuals
| precisely because of weirdness introduced by Russell in order
| to make all Boolean values defined for inference.
| [deleted]
| bentcorner wrote:
| There are an infinite number of questions, answers, and topics
| that were considered nonsense, not written about, and
| impossible to post to HN.
| bitL wrote:
| Countable or uncountable? ;-)
| otabdeveloper4 wrote:
| Uncountable. Information complexity obeys no conservation
| laws.
| LudwigNagasena wrote:
| That has nothing to do with relevance logic. Someone has
| written a smart-ass answer about transfinite ordinals to a
| question about infinity; and people upvoted it for right or
| wrong reasons. To me personally the answer looks witty but
| misleading.
| NKosmatos wrote:
| I've seen the last number at the end of infinity and I could tell
| you if it's odd or even, but by doing so I would break this
| simulation :-)
| [deleted]
| zadler wrote:
| Yes|no
| [deleted]
| pxx wrote:
| 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.
| layer8 wrote:
| 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?
| AdamH12113 wrote:
| 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.
| pxx wrote:
| 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).
| layer8 wrote:
| IEC 60559 could have defined pow(-1, [?]) as -1 if they'd
| wanted to. Hence my question what the "thus" is about.
| have_faith wrote:
| In JavaScript we're spoiled by having both. There's
| Number.NEGATIVE_INFINITY and Number.POSITIVE_INFINITY.
| pelagicAustral wrote:
| In js you could just: npm install is-odd
| const isOdd = require('is-odd');
| console.log(isOdd([?]));
| warent wrote:
| Don't forget -0 (negative zero)!
| remram wrote:
| Why, is it odd?
| otabdeveloper4 wrote:
| Floats are approximations of the reals and -0 is just the
| limit towards zero from the left.
| Sharlin wrote:
| Those are simply the bog-standard IEEE/IEC.754 floating point
| infinities, so they're the same thing in essentially every
| mainstream language.
| remram wrote:
| Both what? The question is about even or odd.
| myhf wrote:
| Negative Infinity is even, and Positive Infinity is odd.
| activiation wrote:
| Is zero a number?
| compressedgas wrote:
| Infinity is equal to the product of all the primes. It is even
| because it has 2 as a factor.
| drexlspivey wrote:
| It's also equal to the product of all primes except 2 so it's
| odd.
| kibwen wrote:
| This just implies that infinity is both even and odd, which
| means that the statement "infinity is even" is still
| technically correct by this reasoning.
| tabtab wrote:
| Next you'll tell me light is both particles and a wave.
| This scientific saucery must stop! Order it to pick one or
| jailit!
| paulddraper wrote:
| Odd is defined as not even.
| [deleted]
| jstimpfle wrote:
| I find that primes are an odd thing to bring into the
| discussion, not even mildly relevant.
| ndsipa_pomu wrote:
| More worryingly, it raises the question of whether infinity
| is prime.
|
| I suppose that if it's even, then it won't be prime, but then
| again infinity/2 is still infinity so we can't assume that
| division works the same way.
| antiquark wrote:
| Infinity is a direction.
| charlie0 wrote:
| It's both until you can observe the last number. ;)
| [deleted]
| xwdv wrote:
| Tl;dr: it's even
| ForOldHack wrote:
| Ahh, the mis-uses of infinity again. Infinity, is both simply
| because inf+1 = inf, so if infinity is odd, then infinity + 1 is
| even, which equals infinity which is then odd. Think of sets. Inf
| and -inf are in both sets. You can prove this with deltas and
| epsilons, but that is beyond the scope of explaining it to 6 year
| olds.
| elif wrote:
| Nope. Just because one infinity passes one test for evenness does
| not imply they all infinities do.
| Eisenstein wrote:
| I want to know if there are more decimal numbers between 0 and 1
| than there are integers between 0 and infinity.
| pxx wrote:
| Assuming sqrt(2)/2 is one of the elements of your set between 0
| and 1, there are! See
| https://en.m.wikipedia.org/wiki/Countable_set
| jameshart wrote:
| Sqrt(2)/2 can not be written as a decimal number.
|
| A decimal number is a rational whose denominator is an
| integer power of 10.
| zorgmonkey wrote:
| No, any rational number with a denominator that has only
| the prime factors of 2 and 5 will have a finite and exact
| decimal representation in base 10.
| jameshart wrote:
| That's exactly the same thing as I said, using different
| words.
|
| And anyway, is Sqrt(2)/2 such a number?
| [deleted]
| umanwizard wrote:
| If you consider non-repeating, non-terminal decimals (like say
| pi/4) then yes. Otherwise no.
| warent wrote:
| 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
| rcme wrote:
| 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.
| daef wrote:
| 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.
| warent wrote:
| Oh right this is only true for irrational and
| transcendental numbers
| aimor wrote:
| Maybe this is misguided cheat, but couldn't you map any
| real number (between 0 and 1) to a natural number by
| mirroring the decimal digits across the decimal point. So
| 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the
| rest of the decimal representation. This creates infinitely
| large numbers, but it's still a 1-to-1 mapping.
| dangond wrote:
| Unfortunately, numbers with infinitely many digits are
| not natural numbers. You cannot count to ?601707, even
| with an infinite amount of time.
| paulddraper wrote:
| There are more reals between 0 and 1 than integers.
|
| There are not more numbers with terminating decimals between 0
| and 1 than integers.
| ftxbro wrote:
| If you thought of this question from no real math training then
| that's pretty interesting. You should have been a
| mathematician. Your question is one of the most important and
| concisely stated questions about infinity that you can ask!
| Eisenstein wrote:
| I was thinking if Pi never repeats itself and infinity of
| integers can only go up then it seems to make sense that
| there are more decimals between any two numbers than infinite
| integers. I can't describe the thought process behind it just
| seems intuitive.
| contravariant wrote:
| 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.
| yamtaddle wrote:
| 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.
| Georgelemental wrote:
| > Everybody experienced writing irrational numbers using
| decimal notation in school,
|
| To be pedantic, we experienced writing _approximations_ of
| these numbers in decimal arithmetic.
| yamtaddle wrote:
| Do you actually think I meant otherwise? Like,
| _actually_? Do you think anyone on HN was confused about
| it? Did you _really_ think that?
| GreymanTheGrey wrote:
| Yes, I thought you meant otherwise. Yes, I was confused
| about it. Yes, I _really_ thought that. Truly.
| yamtaddle wrote:
| You truly thought I didn't realize that "3.14" is an
| abbreviated representation of p, or that I somehow missed
| years and years of using the "repeating" sign above
| various decimal representations, or all those "..."s,
| such that it was plausible I meant the obviously-wrong
| thing rather than the correct thing? This stuff is
| _hammered_ in in US K-12 school.
|
| [EDIT] Look, I don't mean to be a dick, performative
| misreading and plainly-unnecessary "correction" are just
| two of my least-favorite types of HN post. I probably
| should have just downvoted the original performative
| misreading (not yours, the one up-thread) and not Assumed
| Good Faith that the original poster genuinely doesn't
| understand what every non-math-nerd means when they say
| or write "decimal number" (it's the ones you write with a
| decimal. It's... so very simple, that's why non-math-
| nerds use that and not "real number", the definition of
| which they've long since forgotten. "Well but you can't
| actually represent irrationals them entirely in decimal
| notation" great, wonderful, has _zero_ bearing on what
| people mean by it).
| umanwizard wrote:
| It's not just pedantic! It completely changes the answer
| to the question.
| jameshart wrote:
| Nobody ever _finished_ writing an irrational number in
| school.
| FabHK wrote:
| sqrt(2)
| jameshart wrote:
| ... Using decimal notation.
|
| Goodness this is like talking to GPT at times.
| xigoi wrote:
| The correct mathematical term is "real number".
| yamtaddle wrote:
| 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".
| bombolo wrote:
| i've never seen an irrational number written in digits in
| my whole life. Have you?????
|
| I've seen them expressed as letters or formule
| someweirdperson wrote:
| > i've never seen an irrational number written in digits
| in my whole life. Have you?????
|
| I'm currently reading one. Looking good so far. I'll let
| you know after I finish.
| paulddraper wrote:
| 3.14...
| bombolo wrote:
| Isn't that equal to 157/50?
| yamtaddle wrote:
| Never. Never in school? It's totally normal in the US,
| it's the reason as many people know that p starts "3.14"
| as do.
| bombolo wrote:
| Yes never, not in school, not in analysis, and certainly
| not in numerical analysis.
|
| You've proved my point. It's either p or 3.14. Except
| that the latter is a rational number :)
| yamtaddle wrote:
| > Yes never, not in school, not in analysis, and
| certainly not in numerical analysis.
|
| Weird, I just assumed that was normal in most education
| systems. I don't know how you'd get a sense of the rough
| _scale_ of various common irrationals, without having
| some idea what they look like when represented in decimal
| notation. Such representations are normal starting not
| later than when we start seriously working with circles,
| in US school, and never really stop coming after that.
| Estimation exercises lean heavily on having some idea of
| the decimal representation.
|
| > You've proved my point. It's either p or 3.14. Except
| that the latter is a rational number :)
|
| Never claimed p is 3.14, so no, I didn't at all prove
| your point. I wrote that it's very well known that it
| _starts_ that way. When a normal person says "decimal
| number" they mean to include p, because any usefully-
| precise decimal representation of it's going to involve a
| decimal point. At least in the US, they saw it
| represented "3.14..." or "3.1459..." or whatever, many,
| many times in school. It's obviously, to a non-
| mathematician, a "decimal number". They mean "the real
| numbers" (or, perhaps, depending on context, exclusively
| the parts of the reals that _aren 't_ whole integers),
| except that name is harder to remember than the incorrect
| (but more common and intuitive) "decimal numbers".
| bombolo wrote:
| Can you write here an irrational number, not in the form
| of a letter or a formula?
|
| For me a "decimal number" is a number represented in base
| 10. Which is why I was asking. I even googled and there
| is no real definition.
| daef wrote:
| How do you define 'decimal numbers'? Do you only count
| rationals, or all real numbers?
| eimrine wrote:
| 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.
| quantified wrote:
| 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.
| dangond wrote:
| The amount of real numbers between any two distinct real
| numbers (a,b) is the same as the amount of all real numbers.
| This is true for (0,e), (0,1), and any other combination.
| tobiasSoftware wrote:
| 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.
| denton-scratch wrote:
| > The decimal (real) numbers between 0 and 1
|
| The way I parse "decimal number" in this context is a number
| expressible as a (finite?) string of decimal numerals. Those
| numbers are not reals, they are rationals.
| bombolo wrote:
| does decimal numbers mean fractions or real numbers?
|
| If it means fractions only, they are countable.
| shidoshi wrote:
| This account discrete maths. Bravo!
| quantified wrote:
| Cantor's proof is a single attempt. Suppose you could
| construct a space-filling curve that did indeed map all
| numbers between 0 and 1 to all integers? Has there been a
| proof that such a curve does not exist? The fact that his
| proof leans on a specific set of decimal places at every
| juncture has always seemed a weakness of his proof, because
| you can always map any set of numbers from 0 to 1 with any
| set of decimal places to a set of integers.
| dbtc wrote:
| "Countably infinite" makes zero sense to me.
|
| Whatever method you use to generate your decimals, you can
| just slap an integer on each step of the way. You'll never
| run out of integers.
|
| I'll put Cantor and his proof in a box, tell him to give me
| his fancy decimals quick as he can, and I can match each one
| with an integer no problem.
|
| And pairing one infinite list with another infinite list
| doesn't make either one any more countable, because however
| high you count, they keep on going.
| feoren wrote:
| > Whatever method you use to generate your decimals, you
| can just slap an integer on each step of the way. You'll
| never run out of integers.
|
| Exactly correct! This holds true of everything you can
| generate stepwise, even infinite sets. Cantor proved that
| you _cannot_ "generate" (stepwise) all Reals between 0 and
| 1. Any infinite set you can generate stepwise is Countably
| Infinite.
|
| > I'll put Cantor and his proof in a box, tell him to give
| me his fancy decimals quick as he can, and I can match each
| one with an integer no problem.
|
| Exactly correct! And then infinitely later, when you're
| "done", having generated every Real between 0 and 1, he
| will then generate a new Real not on your list. Oops! You
| have not generated all Reals between 0 and 1, even with
| infinite time.
|
| > And pairing one infinite list with another infinite list
| doesn't make either one any more countable, because however
| high you count, they keep on going.
|
| Exactly correct! Any two sets you can pair together (via a
| bijection) have the exact same cardinality. Neither is more
| infinite nor countable than the other. Cantor proved you
| _cannot_ "pair" the Reals with the Natural Numbers.
|
| You and Cantor agree completely. You're very close to
| understanding why the Reals are bigger.
| dbtc wrote:
| > And then infinitely later
|
| There can be no 'and then' after infinitely later.
|
| I don't see why stepwise is important but that must be
| the key to Cantor's proof.
|
| If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he
| gives me 1.11 and I pair with 4, that seems fine as far
| as counting is concerned.
|
| The ordering could be entirely random, I don't see how it
| makes a difference. There will always be enough integers
| to match.
|
| Is it that my black box metaphor is cheating by coercing
| a truly 'parallel' generation of decimals into a linear
| operation? But even then, if I'm getting exponentially
| bigger chunks of new decimals, I can provide equally
| large chunks of integers... so it still doesn't make
| sense to me. Infinity is infinity and you cannot count
| it.
| warent wrote:
| 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,[?],o,o+1,o+2,[?],o+o=o[?]2,o[?]2+1,[?],o[?]3,[?],o2,o2+1,[?]
| ,o2+o,[?][?]
|
| Presumably this person has no experience with 6 year olds? This
| explanation is horrendous haha
| gorkish wrote:
| I explained basically this to my 4 year old nephew recently. He
| wanted to count to infinity. I asked him what is the biggest
| problem with counting to infinity? It's too slow. I said ok
| let's take bigger steps. We counted by 2's then 10's then
| hundreds and millions and then zillions and other ridiculous
| superlative numbers. It doesn't really matter because
| everything is still too slow. So then we said ok lets make up a
| number o that is half way there, One o, Two o, done. He's
| happy. Then I told him to add one more and sent him back to
| play fetch with the dog.
| pacaro wrote:
| I taught my kid that the way to think of infinity is that
| it's like hugs, there's always one more, unlike candy, which
| is limited and can be counted, infinity cannot be counted.
| apomekhanes wrote:
| Hmm, that could potentially cause confusion later. There
| are 'countable' and 'uncountable' forms of infinity /
| infinite sets.
|
| A countably infinite set could be 'counted' (i.e., you
| could sit around labeling elements using the 'natural' or
| 'counting' numbers) in the sense that we might count candy.
| The issue for a human being is that you'd run out of time
| but not elements to count, at least, proceeding in the
| sense one might count the candy - a piece at a time. Of
| course, you can, instead, simply provide a 'bijection'
| (between the natural numbers and the set you wish to prove
| is countably infinite), and in a sense, you are done.
|
| The subject of infinity and infinite sets can be kind of
| subtle, and for years the best mathematicians made many
| mistakes and had many difficulties handling these concepts
| in ways that didn't cause potentially serious problems
| (absurdities, paradoxes, etc.). I think that with the
| development of things like Zermelo-Fraenkel set theory,
| Godel's incompleteness theorems, etc., things became a lot
| clearer. It's a lot easier, with all of the groundwork laid
| by people who worked on these, to get a good sense of what
| is possible and what isn't - what gets you into trouble and
| what doesn't. But, boy, did it twist the minds of the
| people trying to work it out at the time. In part, this is
| because it was less clear, without development in these
| areas, what math even is and what its limits are ... what
| its relationship to the structure of the universe, say,
| even is (something along those lines, in my opinion /
| experience).
| logifail wrote:
| > Hmm, that could potentially cause confusion later [...]
|
| (Q: Do you have kids?)
|
| Our experience is that _pretty much everything_ parents
| tell young children could potentially cause confusion
| later.
|
| In no particular order: Father Christmas aka Santa Claus,
| The Tooth Fairy, Where Babies Come From... it's a long
| list, our eldest is 13 and we're not done yet.
| mensetmanusman wrote:
| He adds one more and the dog freezes at the event horizon of
| a black hole.
| singularity2001 wrote:
| Transfinite ordinals also known as hyperreals should really be
| taught in school as they make many parts of math easier:
| algebraic definition of derivatives (including algebraic
| derivative of step functions without dirac 'density') and yes:
| natural addition and multiplication.
|
| https://en.wikipedia.org/wiki/Hyperreal_number
| logifail wrote:
| > Transfinite ordinals also known as hyperreals should really
| be taught in school as they make many parts of math easier:
| algebraic definition of derivatives
|
| Q: What proportion of children study maths long enough to
| understand derivatives?
| singularity2001 wrote:
| I can only speak for Germany where over 90% reach 10th
| grade, where derivatives are taught.
| atemerev wrote:
| They are not explaining to a 6 years old, they explains to
| somebody who will in their turn explain it to a 6 years old,
| which is a different task and has to be optimized in a
| different way.
| freehorse wrote:
| He has children (not sure about age right now) and discusses
| mathematics often with them. His tweets have had many
| interesting examples.
|
| I do not think he means he would use symbols to explain to
| children, but that the notion of counting natural numbers that
| children have easily generalises to counting transfinite
| numbers.
| gregschlom wrote:
| In the comments of the answer the author says they have a 4 and
| a 9 year old:
|
| "Bill, despite your emphatic comments, I know for a fact that
| counting into the ordinals is something that children can
| easily learn. I have two young children (ages 4 and 9), who are
| happy to discuss a for small ordinals a---although my
| daughter's pronunciation sounds more like Olive0, Olive1---and
| my son can count up to small countably infinite ordinals. The
| pattern below oo is not difficult to grasp. Below o2, it is
| rather like counting to 100, since the numbers have the form
| o[?]n+k, essentially two digits"
| alexb_ wrote:
| 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).
| throwawaymaths wrote:
| imagine my surprise when I got to college and learned that
| infinity + 1 was actually a number! I felt so cheated from my
| childhood.
| bhk wrote:
| I thought that most of us learn at an early age, as a result
| of this kind of exchange, that "infinity" is not "the biggest
| number" or even a number at all, as far as the ordinary
| notion of "number" goes.
| JKCalhoun wrote:
| My child mind conflated infinity and God. Or maybe I was
| correct, I have no idea now.
| ASalazarMX wrote:
| That was the adults attributing infinite and
| contradictory powers to their god. Church sermons will
| frequently mention infinity.
| NegativeK wrote:
| No math instruction I had ever discussed infinity with any
| rigor until calculus -- and even then, it was only infinity
| as a limit. Infinity as a concept was brushed off in the
| same way that the square root of negative one was brushed
| off until we were actually taught about it.
| NegativeLatency wrote:
| It came up a bit in some physics classes, when you can
| mathematically make something go to positive or negative
| infinity being able to remove it from the simplified
| calculation of something is very handy.
| tesseract wrote:
| On the one hand I get why that is - the calculus notion
| of infinity is the one that tends to be useful in applied
| math - on the other hand it's a shame because the set
| theoretic notion of infinity has more to offer to someone
| trying to ponder the nature of the infinite.
|
| Or put another way, "what's [?] + 1" basically invites
| the non-answer "that's not a well-formed question"
| whereas "what's o + 1" gives you a whole intellectual
| thread to pull on.
| NegativeK wrote:
| I've always been disappointed that number theory, set
| theory, etc aren't introduced in middle school or high
| school.
|
| It makes sense, since those are a lot less useful than
| the subjects that are taught, but something like number
| theory is incredibly approachable to a middle school
| student. And it can show students that math can be a lot
| less about memorization and a lot more about creative
| thinking w.r.t. proofs.
| warent wrote:
| Six-year-olds know multiplication?
| [deleted]
| Bootvis wrote:
| The mathematically curious probably do. I did.
| abofh wrote:
| Some even post on HN!
| hammyhavoc wrote:
| That explains some stuff!
| gus_massa wrote:
| My six-year-old likes Numberblocks
| https://en.wikipedia.org/wiki/Numberblocks
| https://www.google.com/search?q=Numberblocks . She knows a
| little more about multiplication than what I expected,
| probably 2x and 3x when x is small, (but as other sibling
| comments say not a general theory or how to calculate
| 287263 * 137167).
| ineedasername wrote:
| Number blocks is a great show, my 5yo watches and, being
| entertained by it, absorbs more than I could easily get
| him to sit still for. Then he asks me questions about
| what he watched and is more engaged with my answers as a
| result.
|
| Making a subject "fun" is alright, but making it
| _entertaining_ (IME) makes for more productive
| engagement.
| gorkish wrote:
| They dont usually know formal arithmetic multiplication but
| they well understand the concepts of repeated addition and
| subtraction. Most places in the world do start teaching
| multiplication at age 6/7.
| ddispaltro wrote:
| yeah they do counting by 5's counting by 2s, etc. So how
| many 5s in 20, they say four, yay!
| apomekhanes wrote:
| Yes, I learned long division fairly well around that time.
| I was fortunate (/ disruptive) enough to be sent to a
| "Montessori school". Long division was definitely pushing
| it when I was about 5 or 6, but, honestly, given steadier
| instruction in math starting earlier, I suspect I could
| have been entirely solid on long division by that time and
| moving on to algebra. And, I think this is true for a
| reasonable proportion of children.
|
| My experience, ultimately, was much less ... 'high-
| quality', let's say. When I left the Montessori school (by
| 3rd grade), I learned practically no math from then until
| after high school. First, in normal 'elementary' school
| (US), multiplication was still being covered in 6th grade.
| Then, suddenly (from my perspective), letters were being
| brought into the picture in 7th or 8th grade. So, in my
| arc, math started to not make sense, at all.
|
| From my perspective, we had spent multiple years on
| multiplication and long division, which I already
| understood very well by the end of 2nd grade ... so, there
| was the period where I basically didn't learn anything,
| where it seemed like we'd reached the end of math or
| something. Or, perhaps, like there were some sort of
| subtleties remaining in multiplication and division. It
| just gave me a chance to be bored with all of it, boredom
| correlates heavily with mistakes with kids with attention
| issues (IMO), this fed into some sort of doubts about my
| understanding of everything etc., and then, suddenly, there
| was new material again starting in 7th grade. Material that
| was 'mechanical', and that didn't seem to have explanations
| I could understand.
|
| Ultimately, I struggled along with that garbage through
| high school, then, after, took a course where we actually
| did PROOFS. Basic number theory stuff - modular arithmetic,
| etc. Bam, suddenly, the subject started to make sense.
|
| Typing this out actually makes me slightly angry. I'm not
| sure I previously connected it all together - why I had so
| much trouble with math for some years ... how this 'arc'
| was pretty much perfectly engineered to make math a
| problem, for me. In any case, schooling through high school
| can be a really low quality experience at times - for some
| students, subjects, etc. The math curricula, methods of
| teaching, and progression I was exposed to, worked
| together, in some sense, to make the subject a problem for
| me. To do almost the opposite of what was intended - to
| pretty well impede learning. There's no one factor in that
| story I can point to and say 'here, fix this' ... no one
| involved in the story was actively attempting to do
| anything other than what they thought was best or what they
| were required to do, but, the net result was honestly worse
| - I now believe (and believed some years ago, even without
| quite this analysis) - than if I'd just been given some
| selection of math material to pick from and been allowed
| some sort of semi-self directed coursework.
|
| Even better, though, if I'd simply had that course with
| proofs / basic number theory in, say, 8th grade ... guh,
| would have avoided so much pain, I'm pretty sure...
| jlokier wrote:
| Yes. Simple multiplication and even division and fractions
| are part of the national curriculum at ages 5 to 6 in the
| UK. Which is about the age when I remember learning them
| decades ago too. I think we learned how to add and multiply
| fractions too.
|
| By age 6 to 7 they're expected to understand that addition
| and multiplication are commutative, while subtraction and
| division are not.
| hnlmorg wrote:
| My 6 year old is learning multiplication at the moment.
| jasonjayr wrote:
| 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 ...
| 13of40 wrote:
| The technique used by the Oregon public school system in
| the 80s went something like "Hand the child a 10x10 grid
| of numbers, then tell them, absent of any other context,
| that they must be memorized." I like your way better.
| retrac wrote:
| Ontario's 1990s curriculum was pretty awesome. The idea
| of dimension and sets were both introduced simultaneously
| and joined, using multiplication. Started in the 2nd
| grade and they just kept elaborating. Number lines and
| groups of items. (Tied it into geometry, too. Square
| numbers came up by at least 4th grade.) What is 3 x 3 but
| moving 3 units, 3 times in one dimension? Now, memorize
| these tables up to 12 x 12, you won't always have a
| calculator at hand.
| mulmen wrote:
| > you won't always have a calculator at hand.
|
| I do though. I still blow minds when I put my iPhone
| calculator in scientific mode. Math education is
| important for many reasons. But teaching it as a
| practical survival skill using no tools does a disservice
| to the student. Either it is useful as a problem solving
| exercise or it is a practical skill that should take
| advantage of tools. "Just memorize this stuff" isn't
| useful because it backfires into hating learning. Nothing
| about math makes it ideal for memorization and none of my
| math teachers spent any time on study skills.
| wpietri wrote:
| Nah. As somebody who ends up doing a ton of mental math,
| I think it's valuable. Yes, they should also learn how to
| use tools. But developing a feel for numbers is valuable,
| and I think that is much harder to do if one always
| relies on a calculator. (And yes, of course, this should
| be learned in a way that doesn't involve the kids hating
| it. But that's possible.)
| gowld wrote:
| Mathematics should be taught be mesmerization, not
| memorization.
| adastra22 wrote:
| I think you need to spend more time around six year olds
| ;)
| Eisenstein wrote:
| My 6 year old nephew can do primes and I taught him to
| count and add in binary.
| ineedasername wrote:
| I'm not sure that HN readers-- a community that will
| disproportionately skew towards folks educated &/or
| employed in STEM fields-- are indicative of other
| people's contact with 6-year old kids.
| hammyhavoc wrote:
| 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.
| aaronharnly wrote:
| Found the Brit! As an American I'd never heard the "lots
| of __" phrasing until I watched Numberblocks (a British
| show) with my kid...
| scotteric wrote:
| Pharmaceuticals and other manufactured goods are
| sometimes referred to in 'lots' meaning a batch.
| [deleted]
| logifail wrote:
| > Numberblocks
|
| We don't live in the UK but our kids watch Numberblocks.
|
| Our youngest started rattling off all kinds of number
| stuff which I know for sure she hasn't yet encountered in
| school.
|
| Me: Wow ... how do you know that?
|
| Her: Numberblocks!
|
| Me: Umm ... OK!
| technothrasher wrote:
| I've never watched Numberblocks, but I do like to watch a
| good game of Numberwang!
| anotherhue wrote:
| One of my favourite SMBCs: https://www.smbc-
| comics.com/comic/2014-02-16
| vmilner wrote:
| Child: "Is 100 million the biggest number?"
|
| Teacher: "Well, there's 100 million and one"
|
| Child: "I was pretty close then!"
| MalcolmDwyer wrote:
| https://m.youtube.com/watch?v=9P2ROAbQZYw
|
| Twenty-four is the highest number! That's it. Let it go.
| logifail wrote:
| > If you ask a child what comes after infinity, "Infinity +
| 1" is pretty much the default answer
|
| (Full disclosure: have three children and plenty of STEM in
| the family)
|
| I'm not sure that's the _default_ answer, of course one might
| easily get that answer if at least one parent has a STEM
| background.
|
| Schools don't teach about infinity to young children. A pity,
| really.
| FabHK wrote:
| > Any kid who knows multiplication knows "Infinity +
| Infinity" is the same as "Infinity Times Two".
|
| Or is it "Two Times Infinity"? (Hint: It isn't, because "Two
| Times Infinity" = "Infinity", while "Infinity Times Two" =
| "Infinity + Infinity". Not sure every kid knows that.)
| colecut wrote:
| This seems to disregard the commutative property of
| multiplication
| kccqzy wrote:
| Ordinal multiplication is not commutative.
| chrisoverzero wrote:
| I think you have that backwards. "Two times infinity" is
| "infinity, two times" or "infinity, twice," which maps to
| Infinity + Infinity. "Infinity times two" is 2 + 2 + 2 + 2
| + 2... forever.
| bigdict wrote:
| What? Where does that follow from?
| sritchie wrote:
| It follows from the way addition is defined on top of set
| theory. "a + b" is implemented as "increment a (the set
| that represents a) b times".
|
| A number is represented in set theory as a set that
| contains all of the numbers before it. 0, 1, 2 is {},
| {{}}, {{} {{}}}...
|
| SO! If you start with a finite "a" and increment it
| infinite times, you still have infinity; you haven't
| broken out.
|
| But if you start with Infinity, then adding anything to
| it gives you {Infinity}, {Infinity {Infinity}}, etc...
|
| Transfinite addition is not commutative!
| RHSeeger wrote:
| Is addition defined _by_ set theory, or is set theory one
| way of defining addition? If it's the later, then there
| could be other ways of defining addition that don't have
| the same results for infinity (because our math system
| doesn't really "work" for infinity, or 0, depending on
| the circumstances).
|
| I am in no way a mathematician. My question about the
| definition of addition as it relates to set theory is
| just that; a question.
| gowld wrote:
| There are many ways of definiting _everything_. Most of
| them are equivalent in the ways that matter, which is why
| math "works" so well as the language of science. Some of
| them are different in critical ways, which opens up
| vistas of new objects and concepts.
| sritchie wrote:
| It's the latter; I'm also not a mathematician, just a guy
| who worked through Halmos's "Naive Set Theory" in intense
| detail...
|
| But your question actually hints at my most profound
| takeaway from that whole book. I think what you're saying
| is right, AND that foundations-of-mathematics folks spent
| a long intense period searching for different set theory
| axioms that did NOT lead to transfinite numbers. But
| anything anyone could come up with that included "the
| axiom of infinity" led to transfinites leaking in.
|
| Which begs the question of how to think about these
| things. Are they "real"? Are they an oddball side effect
| that we shouldn't take seriously?
|
| I think you've arrowed right to the philosophical heart
| of all of this.
| mytailorisrich wrote:
| It's math so you can start your explanation with " _Assume your
| 6 year old has a PhD_ ".
| nh23423fefe wrote:
| > I would focus on the principal idea: whether finite or
| infinite, a number is even when it can be divided into pairs.
|
| why misquote someone and claim their idea is hard to
| understand?
| areyousure wrote:
| > Presumably this person has no experience with 6 year olds?
|
| In case anyone is curious, this person has experience teaching
| children mathematics. For example, on his blog, we have
|
| http://jdh.hamkins.org/math-for-six-year-olds/
| http://jdh.hamkins.org/math-for-seven-year-olds-graph-colori...
| http://jdh.hamkins.org/math-for-eight-year-olds/
| http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cu...
|
| The most recent post in his category "Math for Kids" is in fact
| teaching how to count ordinals up to omega-squared:
| http://jdh.hamkins.org/counting-to-infinity-poster/
| ilyt wrote:
| Do we have statistics on how many pupils end up hating/loving
| math after that ?
| logifail wrote:
| > this person has experience teaching children mathematics
|
| Just as a FYI, there are plenty of countries in Europe where
| many 6 year-olds are still in kindergarten not at school, as
| a result they most likely have not have properly started
| learning numbers or reading and writing.
|
| https://www.statista.com/chart/13378/when-do-children-
| start-...
| froh wrote:
| unless the kindergarten is playfully toying with numbers
| already, usually with no obligation but as an enrichment
| for those kids who love such activities.
| version_five wrote:
| Also my first thought. I assume he's writing this to other
| people who know what transfinite ordinals are (I don't
| understand the explanation) and would frame it differently with
| an actual kid. Even in context it's a hilarious quote though, I
| think it's possible this was on purpose
| hinkley wrote:
| Richard Feynman would be making disapproving noises.
|
| Explain everything like you're talking to a fifth grader. If
| you can't, you don't understand your problem fully.
|
| He spend much of his professorship agonizing about how to fit
| all of physics into a freshman lecture. When he couldn't, he
| knew we needed to think more about that area.
| mlyle wrote:
| I think the big assumption that kids can't get "complicated"
| ideas is faulty.
|
| Sure, they lack rigor, and often will just get the sketch of
| the idea.
|
| And it's a lot more work to think about how to put things in
| the terms that a kid will understand given their knowledge so
| far.
|
| But this idea? "Infinity plus one?!@" --- this is a
| conversation elementary school kids have _on their own_.
| Pulling it a little closer to a sane footing in ordinal
| analysis is not hard. Half of six year olds can handle it.
|
| On the other hand, there's not a lot of obvious utility to
| teaching a six year old this particular concept early. On the
| gripping hand, there _is_ a cost to keeping kids in a bubble
| where you don 't talk about any big ideas (of whatever sort--
| mathematical, philosophical, historical, linguistic) at all,
| or excessively dilute them to the point where they're
| meaningless.
| 0xBABAD00C wrote:
| > In my experience with children, one of the easiest-to-grasp
| concepts of infinity is provided by the transfinite ordinals
|
| Things people say on HN :)
| faceloss wrote:
| [dead]
| KingLancelot wrote:
| [dead]
| czbond wrote:
| I enjoyed watching this Netflix "show" on Infinity.
|
| Trailer: https://www.youtube.com/watch?v=CNFm_DzHDaE
| loganc2342 wrote:
| This thread almost reads like parody to me. It perfectly
| encapsulates the Stack Exchange experience in that when a
| question is clearly asked by a beginner in a subject, they are
| likely to get responses only decipherable by experts, or at least
| people who know enough to not be asking that question.
| zw123456 wrote:
| The way I would explain it to a 6 year old would be like this:
|
| Infinity isn't a number really, it's a concept, like the word
| many or the word few. If someone says they have many of
| something, you don't think is that odd or even you just know they
| have a lot of it. Infinity is kind of like that, it explains the
| idea of things going on forever, not an exact quantity of things
| like the number 10 or 11.
| Name_Chawps wrote:
| This is true until you introduce transfinite numbers.
| zw123456 wrote:
| That might be a bit too advanced for a 6 year old perhaps.
| gowld wrote:
| On the contrary, it's entirely natural. The technical
| definition is quite intuitive.
|
| "There are many infinities! The smallest one is bigger than
| all the counting numbers, so you can't count up to it, but
| it's out there! We call it _omega_. You can make bigger
| infinities too, like omega + 1! "
|
| Kids LOVE that, and it's good math too! (But gets tricky
| quickly, because addition of transfinite ordinals is not
| commutative, and standard transfinite ordinals don't allow
| subtraction)
|
| It's easy to draw as a "number tree" too:
| root / \ / \
| 1,2,3,4... omega, omega+1,...
|
| https://en.wikipedia.org/wiki/Surreal_number#/media/File:Su
| r... (includes more numbers like rationals and reals and
| negatives and backwards counting from omega, but you can
| ignore those)
| bragr wrote:
| >On the contrary, it's entirely natural. The technical
| definition is quite intuitive.
|
| https://xkcd.com/2501/
| MengerSponge wrote:
| 6 year olds have an expert understanding in "I'm not
| touching you", so you might have a shot of teaching them
| bmacho wrote:
| The way I would explain it to a 6 year old would be like this:
|
| There are natural numbers, like 0,1,2 and so on. Natural
| numbers can be odd or even. There is no such natural number as
| infinity. Therefore the question if 'infinity' is odd or even
| is meaningless. It does not even type-check.
|
| In math people like well-formed questions, and generally don't
| like ill-formed questions.
| freehorse wrote:
| > In math people like well-formed questions, and generally
| don't like ill-formed questions.
|
| This is not so simple, though. Ill formed questions can be
| interesting as a motivation to formalise them (ie make them
| well-formed) in generalising/abstracting concepts into new
| concepts. Eg how even/odd has been generalised to transfinite
| numbers.
| gowld wrote:
| The OP clearly explains why the question _is_ meaningful.
| bmacho wrote:
| The question is not meaningful as is.
|
| If you try hard enough, you can find similar questions,
| that do type-check. You can talk with 6yo children about
| them if you want. Still, I stand with my answer. I would
| say this (also I think this is the best thing to say/I am
| capable of).
| EA-3167 wrote:
| The fallback metaphor I use in these situations or similar
| ones, "What's outside of the universe" for example, is the
| old, "What's North of the North Pole?" Then you explain that
| we can create questions and statements in our languages which
| don't have logical, mathematical or physical validity.
| Although we can often describe scientific and technical
| concepts in common languages, that's just a translation, the
| real language is math.
| gopher_space wrote:
| Carlos Castaneda is at his most interesting when he
| wrestles with "what's outside of the universe" paradoxes
| since his informants seem like they're able to not only
| hold mutually exclusive concepts but exist in a
| relationship between them. They'd have an internally
| consistent idea about what's North of the North Pole and
| could explain it to you in terms you might understand.
|
| He's given me quite a bit to think about in regard to NULL
| and the assumptions I make around the concept, which is
| fascinating in itself because his books are hot garbage.
| justinator wrote:
| I was so confused at your glowing review until the
| redemption of the last sentence.
| isitmadeofglass wrote:
| If I say someone had many of something, then I know for certain
| that they must have either an even amount or an odd amount.
| Same goes for few.
| zw123456 wrote:
| It's an analogy, meant to show the similarities between two
| things in a limited way, to illustrate an idea. They do not
| have to be exactly the same in every way.
| tsoukase wrote:
| My 6 and 7 yo's call infinity the "endless number". Well, at
| least it is a NaN number :)
|
| PS: they seem to _know_ that endless*endless > endless but do
| not dare to admit it
| thriftwy wrote:
| Infinite is just a fancy word for endless, anyway.
| short_sells_poo wrote:
| Not necessarily:)
|
| A circle is endless, and yet certainly isn't infinite.
| jdkee wrote:
| A circle is made up of an uncountably infinite set of
| points.
| Dylan16807 wrote:
| Well that fear is good because multiplication won't change
| the cardinality, you have to go exponential.
| plexer wrote:
| Neither, Infinity % 2 = NaN
| EGreg wrote:
| I don't think that aleph0 + 1 is odd, because I can make a 1-1
| correspondence between two of its subsets. They're ALL even, by
| that definition!
| DonHopkins wrote:
| I don't get mad, I get odd.
| pm2222 wrote:
| Is infinity an integer?
| bell-cot wrote:
| Nope. Nor a rational number. Nor a real number. Nor...
| yarg wrote:
| Here's one that gets me: what's the sign of infinity?
|
| What do I mean? 1/d retains the sign of d across all finite and
| infinitesimal values.
|
| 1/e = +[?], 1/-e = -[?]
|
| So what about 1/0? Neutral infinity.
| zeven7 wrote:
| 1 / 0 = +-[?]
|
| Or if you must, you do that thing where you pick the positive
| option a la [?]
| yarg wrote:
| +- is useful for ambiguities, such as the the result of the
| antifunction of a symmetrical function.
|
| But there's no ambiguity here - those signs that you're
| suggesting popped out of nowhere.
| crazygringo wrote:
| So there seems like a glaring hole in the answer, but maybe I'm
| missing something. Because:
|
| > _It is easy to prove from this definition by transfinite
| recursion that the ordinals come in an alternating even /odd
| pattern, and that every limit ordinal (and hence every infinite
| cardinal) is even._
|
| Sure, if we use the natural numbers and start at 1, then we can
| group: [1, 2], [3, 4], [5, 6], ...
|
| and prove infinity is even.
|
| But we could _also_ just as easily group: 1, [2,
| 3], [4, 5], [6, 7], ...
|
| and prove infinity is odd.
|
| It's the same if we try to split into two equal subsets, because
| we can split into: [1, 3, 5, ...] [2, 4, 6,
| ...]
|
| and say it's even. Or we can divide: 1 [2,
| 4, 6, ...] [3, 5, 7, ...]
|
| and prove it's odd because we have two equal subsets plus one
| left over.
|
| So I'm missing the reason for why the second versions aren't just
| as valid.
|
| (Of course, I'm more inclined to agree with many commenters here
| that it's just a category error, and asking whether infinity is
| even/odd is as useful as asking whether democracy is blonde or
| brunette.)
| sdenton4 wrote:
| I would weaken the definition of even/odd to say that a set is
| even if /there exists/ a way to pair things off, and odd if
| /there is no way/ to pair things off (ie, not even). So the
| countable numbers would be even.
| [deleted]
| generalizations wrote:
| But pairity is just a question of sorting the countable
| numbers into sets of size two, and the more general form even
| of that is sorting into sets of size N. It's just as easy to
| say that the countable numbers are odd if there exists a way
| to sort them into sets of size three. So I'd argue the
| countable numbers are odd.
|
| And you then you could retort with sets of size four, and I
| could use five, and then we can argue about whether we'll end
| up at the limit with more odd sets or even sets, and now
| we're arguing in circles. _Reductio ad absurdum._
| rcme wrote:
| Why? You can group 30 into sets of 3 (3 x 10), but 30 is
| still even, so your definition of odd doesn't hold.
| crazygringo wrote:
| But that seems redundant with countable/uncountable sets,
| because then every countable infinite set would be even (e.g.
| rational numbers), and every uncountable infinite set would
| be odd (e.g. real numbers).
|
| It's also not clear to me what justification there would be
| for a "preference" for the "even" category that way -- it
| seems arbitrary. Why not be odd if there exists a way to pair
| things off such that one is left over, and even if there
| isn't such a way?
| yjk wrote:
| I think the reals are also even: If x is rational pair it
| as you would in the rational case (which we assume is even
| - I haven't proven this). Otherwise pair it to -x, and thus
| the reals are even.
|
| Being "even" seems like a much more interesting (and
| simpler) property of a set. I don't see what use there
| could be to know that you could pair things off, with one
| element left over. When you extend the notion you do have
| to decide what to preserve, but to me parity is much more
| about divisibility and symmetry than it is about reminader.
| I agree that it's arbitrary, though less arbitrary than the
| odd definition.
| crazygringo wrote:
| If you can pair but not count, the reals would be odd I
| think, as long as zero is unsigned. Zero would make it
| odd.
| sdenton4 wrote:
| Because evenness is a special case of k-evenness: A set is
| k-even if it can be divided into equal sets of size k.
| Which, for finite sets, is equivalent to the size of the
| set being 0 mod k, ie, is divisible by k. There are many
| ways to be not be divisible by any particular number bigger
| than two, and only one way to be divisible.
|
| Uncountability is a particularly interesting form of non-
| divisibility, so I'm just fine calling all uncountable sets
| odd and countable sets even...
|
| (And just because we're hung up on divisibility by two, let
| us remember: All prime numbers are odd, and two is the
| oddest of them all.)
| silasdavis wrote:
| Omega is the lowest countable infinity. There's no parity
| within a countable infinite as you describe.
|
| It's only even or odd with respect to other infinities which
| the cardinal numbers can count based on the presence of a
| bijection or not. It's a kind of relative parity.
| crazygringo wrote:
| > _There 's no parity within a countable infinite as you
| describe._
|
| That directly contradicts the quoted text I included from the
| original answer though, as far as I understand. It directly
| asserted that "every infinite cardinal is even".
|
| > _It 's a kind of relative parity._
|
| What is relative parity? The original question was whether
| infinity is even or odd... I don't know what you mean by
| _relative_ parity.
| adverbly wrote:
| > we can...prove infinity is even....and prove infinity is
| odd...
|
| > maybe I'm missing something
|
| The answer said:
|
| > the usual definition is that an ordinal number is even if...
| Otherwise, it is odd.
|
| In other words, if a number could be proved to be even, it is
| even. If not, it is odd.
|
| Using their definition, there is no such thing as "proving a
| number is odd". You'd have to do it by failing to prove it's
| evenness. In the case of infinity, because we can successfully
| prove evenness, it's even and not odd.
| smallnamespace wrote:
| The definition given was 'if there is another ordinal such
| that 2[?]=' [1], but the intuition is better explained by the
| post below:
|
| > A set S has even cardinality if it can be written as the
| disjoint union of two subsets A,B which have the same
| cardinality. [2]
|
| In other words, a set is even if it can be paired up, by
| finding one grouping where it pairs. Finding alternative
| groupings that do not pair does not matter.
|
| [1] https://math.stackexchange.com/a/49046
|
| [2] https://math.stackexchange.com/a/49045
| crazygringo wrote:
| OK, so I guess I'm just understanding that mathematicians
| arbitrarily decided to prioritize "even" over "odd"?
|
| Because as I stated in another comment, you could just as
| easily say odd cardinality exists if you can find two subsets
| with the same cardinality and there's one element left over,
| and otherwise we call it even.
|
| So at the end of the day, what you're saying is that
| ultimately infinity would be even just because mathematicians
| arbitrarily defined 'even' that way -- not because there's
| any intuitive logic behind it, any deeper justification, or
| any necessary consistency with parity for finite sets.
| skulk wrote:
| well, if you claim omega is odd, are you willing to claim
| omega + 1 is even? There is no ordinal B such that 2 * B is
| omega + 1, so it fails that definition. So you have to say
| omega is odd and omega + 1 is also odd, which is... odd.
| crazygringo wrote:
| But that "oddness" is precisely my whole point.
|
| I'm arguing that because it's just as easy to say that
| omega is odd as to say that it's even, that the whole
| concept breaks down and loses and all meaning.
|
| Because if you want to divide omega + 1 in half to show
| that it's even, go ahead. If we denote the set element
| inside of the "1" of "+ 1" by the symbol "a", then we can
| have: [1, 3, 5, 7, ...] [a, 2, 4,
| 6, ...]
|
| You'll see that it's easy to infinitely extend a 1-1
| correspondence between these two disjoint subsets, so
| they're the same size. Voila, omega + 1 is evenly
| divisible if you want it to be.
|
| But again, I'm not saying that this is useful or
| interesting. My whole point is that it's _not_ because
| even /odd is _not_ relevant for transfinite numbers,
| because you can make them anything you want.
| smallnamespace wrote:
| > arbitrarily decided
|
| Modern mathematics is all about coming up with definitions
| and rules that give rise to interesting (to a
| mathematician!) properties when further investigated.
|
| The definition given naturally lets the ordinal numbers
| continue the odd/even/odd... pattern. Choosing the
| alternative definition would not.
|
| In one sense that's 'arbitrary' because we decided on one
| definition over another. But another sense, we picked the
| parity rule that lets us extend the same pattern from the
| natural numbers, so it's a 'better' parity rule. And the
| fact that one rule gives this pattern while the other does
| not, did not come from humans, but is a 'metamathematical
| fact' from the universe of possible ways to define things.
|
| So I would say this definition is not fully arbitrary, it's
| an interaction between what mathematicians find interesting
| and the Platonic realm of possible mathematical constructs.
|
| Anyway, I'm not a mathematician but it seems this is how
| the game of math is played: to continually discover new
| rules that give rise to more interesting math.
| crazygringo wrote:
| Thanks, but you may have misunderstood the definition I
| have for defining odd numbers, because that corresponds
| equally to the natural numbers as well.
|
| So there is no better parity rule as you say, it is
| entirely arbitrary. It's not extending the same pattern,
| it's seeing that there are two ways of extending it and
| picking one arbitrarily that happens to prioritize even.
|
| In that case, if it were me, I'd call it something else.
| Not parity or even/odd. Because it's not the natural
| obvious extension. It's arbitrary.
| [deleted]
| jetunsaure wrote:
| Evenness is a more natural condition, so to speak, in that
| it has a simple definition and is easy to generalize.
| Having defined an even number, if an integer isn't even,
| it's odd.
|
| To get a feel for why this is convenient, consider that you
| can generalize by replacing "multiples of 2" with
| "multiples of n". Then, instead of splitting everything
| into two sets (even/odd), we can naturally split the
| integers into n sets called equivalence classes modulo n.
| For n=10, these would be "multiples of 10", "numbers whose
| remainder after dividing by 10 is 1", "numbers whose
| remainder after dividing by 10 is 2", and so on. Seen this
| way, you may find it less arbitrary now.
| crazygringo wrote:
| I understand what you're saying, so thank you, but I
| still find myself disagreeing.
|
| There are just as many odd numbers as even, so there's
| nothing more natural about it. They alternate. Yes you
| can extend to higher multiples, but there's still nothing
| more natural about multiples of 7 vs. multiples of 7 with
| remainder 3.
|
| And it's just as easy to say that infinity is divisible
| by 7, as it is to say that it divisible by 7 with
| remainder 3: [1, 2, 3, 4, 5, 6, 7], [8,
| 9, 10, 11, 12, 13, 14], ... 1, 2, 3, [4, 5, 6, 7,
| 8, 9, 10], [11, 12, 13, 14, 15 16, 17], ...
|
| So the entire idea I'm arguing against is that there's
| anything more natural, more default, more basic about the
| concept of "evenness" next to "oddness". The very first
| natural number, 1, is odd -- not even.
| [deleted]
| emrah wrote:
| Infinity is a concept not an actual specific number. How could it
| be even or odd?
| zac23or wrote:
| I see infinity and perfection as a direction, not a place. That's
| how I explain it to kids.
| [deleted]
| ioslipstream wrote:
| Yes
| gigel82 wrote:
| Is the last digit of Pi odd or even?
| paulddraper wrote:
| Trick question. Pi is irrational and does not have a last
| digit.
| UncleOxidant wrote:
| I guess I would have thought the answer to this question would be
| "yes"?
| w0mbat wrote:
| Infinity is not a number, it's the absence of a limit.
| [deleted]
| mkup wrote:
| Absense of limit is not always an infinity, e.g. for sequence
| (-1)^n.
|
| Even if sequence is unbounded, it does not always converge to
| infinity, e.g. n^((1+(-1)^n)/2): 1, 2, 1, 4, 1, 6, 1, 8, 1, 10,
| 1, 12, 1, 14, 1, 16, 1, 18, 1, 20 ...
|
| Convergence of sequence x(n) to infinity by definition is: for
| each real number e>0 there exists a natural number N(e) such
| that for every number n>=N(e) we have |x(n)|>e.
| cubefox wrote:
| Yeah. Though the defenders of transfinite (ordinal and
| cardinal) numbers do in fact assert that there are many
| infinite numbers, such as aleph zero or omega. They are just
| usually somewhat embarrassed about this and therefore only talk
| about "ordinals" or "cardinals". It's like trying to hide that
| you drink beer by saying you merely drink lagers and ales.
| MetaWhirledPeas wrote:
| NaN
| jedberg wrote:
| I think the best answer is the first comment to the question:
|
| Infinity is neither even nor odd, just like 1.4 isn't even or
| odd.
| fsckboy wrote:
| i think making the odd/even distinction is a mistake. It makes
| people say things like "2 is the only even prime" as if that's
| somehow different than "3 is the only prime _even_ ly divisible
| by 3".
|
| Even is a quality of division when the remainder is 0; even is a
| quality of making a rectangle with discrete integral sides.
| sampo wrote:
| > In the context of transfinite ordinals
|
| I don't really find that transfinite ordinals match my childlike
| intuitive concept of infinity. Transfinite cardinals match
| better.
| mnunez wrote:
| Well, I asked ChatGPT:
|
| > Infinity is not a number, odd or even, but rather a concept or
| a mathematical idea that represents an unbounded or limitless
| quantity. Infinity is not a real number that can be used in
| ordinary arithmetic operations, but it is used to describe a
| quantity that is larger than any finite number. Therefore, the
| concept of odd or even does not apply to infinity.
| awestroke wrote:
| Much more reasonable answer than the others in the thread
| seiferteric wrote:
| Imagine if we had ChatGPT a couple hundred years ago: "What is
| the square root of -1?": "The square root of -1 is not defined
| because you cannot take the square root of a negative number."
| JdeBP wrote:
| So you are proposing that a couple of hundred years ago, in
| 1823, a hypothetical ChatGPT would not have been trained on
| the works of Leonhard Euler, from 80 years before that.
|
| That actually sounds about right. (-:
| irrational wrote:
| >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.
| SideburnsOfDoom wrote:
| > 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.
|
| This is true,
|
| but the same is true of (infinity - 1)
|
| Therefor infinity must also be odd.
| kccqzy wrote:
| The concept of "infinity - 1" doesn't exist. Subtraction
| isn't defined for ordinals. Furthermore even if you try to
| define it, it doesn't work for limit ordinals.
|
| If you are thinking about the difference between
| [0,1,2,3,...]
|
| and 0, [1,2,3,4,...]
|
| Then I regret to inform you the former is omega and the
| latter is 1+omega which is the same as omega. In other words
| attempting to subtract one from infinity by removing from the
| front results in infinity.
| SideburnsOfDoom wrote:
| > In other words attempting to subtract one from infinity
| by removing from the front results in infinity.
|
| And I regret to inform you that if you read more carefully,
| you will find that my comment above makes use of that very
| same property of infinity. Not only do I already know it;
| that's the joke.
|
| Specifically, that statements about omega are also
| statements about 1 + omega. The parent post saying "I think
| that infinity must be even" is such a statement. Regardless
| of if it's true or not, well-defined or not, coherent or
| not, it's equally all that about (infinity - 1).
|
| Should I also spell out that an argument that "n - 1 is
| even" is also an argument that "n is odd" ?
| [deleted]
| tlarkworthy wrote:
| But zero is in the middle, so it must be odd
| amanj41 wrote:
| 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?
| dkarl wrote:
| In mathematics, you can define things in different ways to get
| different answers. Ways of defining things tend to be
| highlighted as true (in at least some context) if they are
| interesting and useful, and ignored if not. I don't think the
| definition based on "dividing into pairs" is particularly
| interesting or useful in the context of the child's
| understanding of numbers, because it's too vague to be useful,
| and it doesn't lead to any insights.
|
| The definition based on transfinite ordinals explained in the
| same answer does seem interesting, and I wouldn't be surprised
| if it were useful. I think this is a case of simplification
| gone wrong, where everything interesting was lost in the
| translation to more accessible terminology.
|
| A more honest thing to say to a child would be that the way
| even and odd are defined only make sense for finite numbers.
| It's true for the definition they know, and it introduces them
| to the important insight that logical rules that are created
| for one kind of thing might not work when applied to something
| else. I think this would be more accessible and stimulating for
| a six-year-old than giving them a half-baked verbal imitation
| of a result from transfinite mathematics.
|
| They'll be thrilled later if they study math and discover that
| there are definitions of "infinity" and "even" that yield an
| answer to their childhood question.
| amelius wrote:
| An even more honest thing to say is that infinity when used
| as a number is a hack introduced by mathematicians to make
| notation and reasoning simple in some cases, but that it can
| be dangerous in other cases, like any other hack. If you want
| to use infinity in a safe way, then use limits around your
| expressions.
|
| (And this quickly resolves the case of this article, since
| lim x->inf x-2*floor(x/2) does not exist).
| chowells wrote:
| It's not a hack to create a new set and work out rules for
| how to use it which are both internally consistent and
| support easy morphisms with more familiar sets.
|
| It may not be _easy_ , but it's hardly a hack. It's one of
| the big ways math works, really. Are negative numbers a
| hack? Rational numbers? Algebraic numbers? Well then
| neither is the two-point compactification of the reals or
| extending the natural numbers into the ordinal numbers.
|
| These are things with very precise models and
| interpretations. No hacks at all.
| amelius wrote:
| But nobody said that hacks cannot have precise
| interpretations. It's the unreasonable cognitive load
| that is the problem.
| dkarl wrote:
| That's true in calculus, and probably a lot of other
| applied mathematics contexts where rigor tends to get swept
| under the rug, but it's not completely fair since there are
| versions of "infinity" that are defined and used
| rigorously. (The transfinite ordinals and cardinals
| mentioned in the Stack Overflow article are the example I'm
| familiar with.)
| taneq wrote:
| Infinities aren't comparable for equality... are they?
| kccqzy wrote:
| In the general case, the comparability of cardinals relies on
| the axiom of choice. In other words, they are comparable, but
| they require a slightly unintuitive foundation to establish
| that they are always comparable.
| [deleted]
| bombolo wrote:
| Not if you aim to pass your exam.
| FabHK wrote:
| Sure they are. You can define a one-to-one mapping, they're
| equal.
| orbital223 wrote:
| You can define a one-to-one mapping between the sets {1
| 2} and {3 4}, but I don't think anyone would say they are
| equal.
| srcreigh wrote:
| You're thinking of isomorphic, not equal.
| chowells wrote:
| They meant "their cardinalities are equal". It's honestly
| an easy mistake to make, especially if typing on a small
| screen. Or especially if having a discussion where sizes
| of infinity are already being discussed.
| eternalban wrote:
| Infinity is the twin of zero [?] | [?] .
| mkup wrote:
| Infinity is out of domain of integer numbers where notions of
| even and odd are defined and make sense.
|
| Notion of infinity is applicable when we are discussing sequences
| and their behaviour, such as convergence.
|
| Convergence of sequence x(n) to infinity by definition is: for
| each real number e>0 there exists a natural number N(e) such that
| for every number n>=N(e) we have |x(n)|>e.
|
| Convergence of sequence x(n) to plus infinity by definition is:
| for each real number e>0 there exists a natural number N(e) such
| that for every number n>=N(e) we have x(n)>e.
|
| Convergence of sequence x(n) to minus infinity by definition is:
| for each real number e>0 there exists a natural number N(e) such
| that for every number n>=N(e) we have x(n)<-e.
|
| For example sequence of natural numbers 1, 2, 3... converges to
| plus infinity and to infinity; sequence of negated natural
| numbers -1, -2, -3... converges to minus infinity and to
| infinity; and sequence of sign-alternating numbers (-1)^n * n:
| -1, 2, -3, 4, -5, 6, -7, 8, -9, 10... converges to infinity.
|
| So notion of infinity applies to _behaviour_ of sequences, whose
| elements remain finite nevertheless. If we consider other
| mathematical objects, e.g. integer numbers, then notion of
| infinity does not apply. If we consider convergence of sequences
| where notion of infinity is applicable, then notion of even /odd
| is not applicable.
|
| While discussing sequences converging to an infinity with a
| child, it may be useful to consider some interesting
| counterexamples: sequences which are unbounded, but still do not
| converge to infinity, e.g. 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12,
| 1, 14, 1, 16, 1, 18, 1, 20... (formula is n^((1+(-1)^n)/2)).
| pk-protect-ai wrote:
| -- the first comment here is the correct one, sorry for posting
| it without reading the comments. Infinity is not a number.
| dathinab wrote:
| which infinity? ;=)
| remram wrote:
| Infinity is not a number. If you want to extend even/odd to it,
| you can pick whatever you want.
| Sharlin wrote:
| There's always someone who sees a question in a submission
| title and feels the need to comment simply to answer said
| question in the most boring, banal, and least insightful way
| possible. Most people realize that if an article that poses a
| seemingly-simple question makes it to HN frontpage, there's
| almost certainly some unexpected, interesting, and/or
| insightful discussion there that reveals that the question
| wasn't so simple after all.
|
| Almost everything interesting in mathematics stems from a
| simple question: "How could we extend a concept to be more
| generally applicable?" Saying that infinity is not even or odd
| because it's not a number is like claiming that matrices can't
| be multiplied because they are not numbers.
| remram wrote:
| I am not preventing the discussion or claiming that you
| shouldn't extend this concept, just objecting to the way to
| question is formulated ("is infinity an odd or even _number_
| "). The stackexchange comments agree with me, and do this
| extension by pointing out reasons it would be useful to pick
| one or the other, but I found it important to point out this
| caveat that there is no logical answer (in terms of _numbers_
| ) and whatever you pick would be an extension, not a
| conclusion.
|
| Nowhere in my comment do I say that this is a silly
| submission or suggest that it shouldn't be in the front page.
| sukilot wrote:
| I suggest you reread the OP and ask questions, because you seem
| to have overlooked some of the ideas explained therein, such as
| transfinite ordinals.
| remram wrote:
| You can build those and then pick if you want them even or
| odd, which are regular-number concepts. That is exactly what
| they did, and what I described. You go and re-read it.
| cushpush wrote:
| Depends on where you start counting =).
| shadowgovt wrote:
| The key insight when dealing with infinities is that the tools we
| use to deal with finite numbers extrapolate to infinite sets by
| talking about relationships between numbers, not individual
| numbers.
|
| This is also how we get to the notion of infinities larger than
| other infinities.
| paulddraper wrote:
| npm i is-even const isEven = require("is-even");
| console.log(isEven(Infinity)); TypeError: is-odd
| expects a number.
| dist-epoch wrote:
| Nice find. I will file an issue, this is a very used package,
| it's important for it to be accurate.
| Bilal_io wrote:
| I read the title and I sarcastically thought "ask JavaScript!"
| Your comment didn't disappoint. Importing the is-even package
| is the cherry on top.
| sltkr wrote:
| Even with plain Javascript, `Infinity % 2` evaluates to
| `NaN`, as it should.
|
| And if you implement, e.g.: function
| IsEven(x) { return x % 2 == 0; } function IsOdd(x) {
| return x % 2 == 1; }
|
| Then IsEven(Infinity) == false and IsOdd(Infinity) == false,
| as expected.
| paulddraper wrote:
| The amazing thing is that the is-even package depends on the
| is-odd package.
|
| I would have thought the reverse, but -\\_(tsu)_/-
| layer8 wrote:
| It's an odd choice for sure.
| sawyna wrote:
| The is-odd package depends on is-number!
| nologic01 wrote:
| Even if its odd, infinity is not a number
| gus_massa wrote:
| The problem with transfinite is that you lose commutatively.
|
| Flowing the standard notation, where the usual infinite in the
| integer or the real line is "o = [?] = 1,2,3,..."
|
| o+1 = o+1 , i.e. "the next thing after infinity"
|
| 1+o = o , i.e. "the same infinity as before"
|
| 2o = o , i.e. "the same infinity as before", so it's even
|
| 1+2o = o , i.e. "the same infinity as before", so it looks odd,
| but don't fall in that trap
|
| o2 = o2 , i.e. "two infinities chained together", that is weird
|
| Two more weird example from
| https://en.wikipedia.org/wiki/Even_and_odd_ordinals
|
| > _Unlike the case of even integers, one cannot go on to
| characterize even ordinals as ordinal numbers of the form b2 = b
| + b. Ordinal multiplication is not commutative, so in general 2b
| [?] b2. In fact, the even ordinal o + 4 cannot be expressed as b
| + b, and the ordinal number_
|
| > _(o + 3)2 = (o + 3) + (o + 3) = o + (3 + o) + 3 = o + o + 3 =
| o2 + 3_
|
| > _is not even._
|
| For a six year old, I'd tell that infinite is not a number so
| it's not even or odd. If s/he even get's a Ph.D. in math, s/he
| will understand.
|
| Moreover, I remember when I was a graduate T.A. that one day
| before lunch I went to a class to learn about the
| https://en.wikipedia.org/wiki/Alexandroff_extension in the
| morning. (The idea is that you add one [?] to a set of numbers to
| get a compact set. And in the new set [?] is (almost) a number as
| good as the other numbers.) After lunch, I went to teach limits
| to first years students, and with a total straight face I told
| them that [?] is not a number.
| squidsoup wrote:
| > o+1 = o+1 , i.e. "the next thing after infinity"
|
| I find this concept perplexing. To me this implies that
| "infinity" has a value. How can you add 1 to a thing that by
| definition has no value?
| NegativeK wrote:
| I'd say that the problem with transfinites is that you lose
| intuitive understanding of what's going on, and one of those
| intuitions is commutativity.
|
| People seem to assume that they know a couple of tricks about
| infinity (adding, multiplying) and don't stop to think that
| there should be a much more rigorous definition. Which, they
| shouldn't -- the average person will never _actually_ care
| about transfinites.
| SAI_Peregrinus wrote:
| > 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.
| C-x_C-f wrote:
| > After lunch, I went to teach limits to first years students,
| and with a total straight face I told them that [?] is not a
| number.
|
| When you apply Alexandroff extension to add the point at
| infinity to, say, the real numbers, what you're left with is
| not a set of numbers (i.e. a field) anymore. So it makes sense
| to say that [?] is not a number. Moreover, the way [?] is used
| in analysis is different from Alexandroff compactification, in
| that you usually use _two_ infinities (+-[?]) as a shorthand
| for quantification over increasing or decreasing sequences of
| real numbers (this can be formalized using extended real
| numbers [0] or other gadgets but doing so has no advantages in
| a first-year analysis class, and might in fact make matters
| worse).
|
| [0] https://en.wikipedia.org/wiki/Extended_real_number_line
| gus_massa wrote:
| It was a long time ago, something like an optional course in
| Advanced Functional Analysis. It was about the algebras of
| functions with and without unity, and how to complete the
| ones without unity using the compactification (i.e. including
| a [?]) and a few variants.
|
| > _two infinities (+-[?])_
|
| It depends. In the real numbers it depends, but in most cases
| I agree that it's better to use two. In complex analysis it's
| much better to have only one infinity. And there are more
| weird case like the projective plane where you have one
| infinity in each direction.
|
| > _So it makes sense to say that [?] is not a number._
|
| I agree, it's not longer a field and the operation lose many
| properties if you try to extend them. So I said " _(almost) a
| number_ ". Anyway, the weird part is that in some cases you
| can write f([?]) in an advanced math course, but you can
| never write f([?]) in a fist year math course.
| omgomgomgomg wrote:
| Well its not a natural number or real, its a concept entailing
| all odd and even numbers.
|
| Unless this is some sort of trick question or I am missong
| something.
| corn13read2 wrote:
| Definitely even, no chance half of infinity is a decimal.
| osigurdson wrote:
| I've always thought infinity is not a number but a concept that
| represents something which is ever increasing.
| paulddraper wrote:
| It is that as well.
| kccqzy wrote:
| That's the definition of infinity in calculus and analysis.
| Most of the comments in this HN discussion are talking about
| infinity as a set theoretical concept, i.e. cardinals and
| ordinals.
| JdeBP wrote:
| Ah, StackExchange!
|
| The answer that says "Here is a simple example that has some hope
| of being comprehensible to a 6-year-old." and then begins
| "Consider the ring of polynomial functions with integer
| coefficients, ..." gets upvoted tens of times.
|
| Even the answer that uses "numerocity", "refined cardinality",
| and "logarithm" as the explanation to a 6-year-old gets upvoted.
|
| The answer, https://math.stackexchange.com/a/49065/13638, that
| says as the answer-to-a-6-year-old _the same thing that several
| commenters have actually posted here_ (e.g.
| https://news.ycombinator.com/item?id=35790064 for one of many),
| on Hacker News in just the past hour or so, and that explains in
| terms that a 6-year-old has at least a chance of having
| encountered, gets 5 votes in 12 years and the submitter is banned
| from the site.
| Spivak wrote:
| The answer is using big words but the concept is simple. Like
| talking about fractions as a quotient ring over a field.
|
| A six year old can absolutely grasp that even means "being able
| to be split into two equally sized piles" where equally sized
| means each thing in the left pile can be matched to something
| in the right. 6 apples is even because you can split them into
| 3 and 3.
|
| Then for infinity you separate them into the even and odd
| numbers, boom. Infinity is even.
|
| Saying "infinity isn't a number", to me, is so much worse an
| answer because it's not satisfying. Because both you and the 6
| year old know that isn't right. The 6 year old is grasping at a
| bigger concept but doesn't have the words.
| caturopath wrote:
| Infinity, 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.
| lisper wrote:
| To explain to a six-year-old I would start by telling them that
| there are many different kinds of infinity, not just one. Some
| infinities are odd, others are even, and others are neither. It
| matters whether you are asking "how many" (cardinals) or "in what
| position" (ordinals). For regular finite numbers, cardinals and
| ordinals are (more or less) the same, but for infinities they
| behave differently. Then, if they want to get into the weeds, you
| can introduce them to transfinite ordinals, diagonalization, and
| all that fun stuff.
| Georgelemental wrote:
| For a child, I think the simplest kind of infinity to explain
| is the cardinality of integers--"how many numbers there are."
| lisper wrote:
| OK, but then where do you go from there? There are infinity
| numbers. Then what?
| kccqzy wrote:
| Right. The problem with teaching infinity by starting with
| cardinal numbers is that it's either too trivial or too
| hard. You can establish that several other sets of numbers
| are identified by the same infinity but there's not much
| you can do.
|
| An old HN comment echoes the same sentiment:
| https://news.ycombinator.com/item?id=17677010
|
| If we really are teaching kids, teach ordinals not
| cardinals.
| lisper wrote:
| > You can establish that several other sets of numbers
| are identified by the same infinity but there's not much
| you can do.
|
| Well, you can introduce them to the diagonal argument and
| the idea of a one-to-one correspondence. That's nothing
| to sneeze at.
|
| But I think the real trick here is to teach them that
| numbers can stand for different kinds of ideas, and in
| particular, they can stand for "how many" or "what
| position", and that these are _different_. I would start,
| not with infinity, but with negative numbers. You can 't
| have "one less than zero" because you can't take away
| anything from zero. That is the _definition_ of zero. But
| you can have "the thing before zero", or, to be more
| precise, "the thing before the zeroth thing (where the
| zeroth thing is the thing before the first thing)", which
| we call -1.
|
| Likewise you can't have "one more than infinity" because
| that's just infinity. That's the _definition_ of
| infinity. But you can have "the thing _after_ infinity "
| (or, to be more precise, "the thing after all the things
| that are the nth thing for all finite values of n", which
| we call o.
| Georgelemental wrote:
| Then you explain the properties of that infinity, like how
| infinity + infinity = infinity, or (as per OP) that it's
| even.
| lisper wrote:
| But if infinity is even then infinity + 1 must be odd.
| But infinity +1 = infinity, so infinity must be odd as
| well as even.
| feoren wrote:
| > It matters whether you are asking "how many" (cardinals) or
| "in what position" (ordinals)
|
| But "even" and "odd" are all about whether you can partition
| something into an equal number of pairs or not. If you're
| asking "in what position" (ordinals), you've explicitly said
| you're not in the realm of counting sets of things. I would
| argue division makes no sense in the realm of ordinals!
| Everyone is saying the transfinite ordinals alternate even-odd,
| but those are exactly the numbers where we've stated we're
| _only_ interested in position, not counting. It 's not clear to
| me why "dividing" an ordinal number into equal pairs makes any
| sense. (Whereas it makes perfect sense for cardinal numbers.)
| lisper wrote:
| > But "even" and "odd" are all about whether you can
| partition something into an equal number of pairs or not.
|
| Sez you. I can just as easily define even and odd in terms of
| whether or not I can arrive at a given position in a
| (potentially infinite) sequence taking by taking two steps at
| a time.
| hypertexthero wrote:
| It's an even number that fell on its side: [?]
| billpg wrote:
| No
| sukilot wrote:
| [dead]
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