181–190 of 398 posts
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I think you need to spend more time around six year olds ;)
My 6 year old nephew can do primes and I taught him to count and add in binary.
I 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.
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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 do…
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).
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No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
Six-year-olds know multiplication?
My 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...
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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)
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I really don't think that block quoting ChatGPT is a good contribution.
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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 ...
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Six-year-olds know multiplication?
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).
Making a subject "fun" is alright, but making it entertaining (IME) makes for more productive engagement.
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 "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 =…
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.