Timeline for The concept of infinity for a 5 year old
Current License: CC BY-SA 4.0
28 events
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| Oct 1, 2021 at 14:30 | comment | added | user14805 | What's the difference between "infinity is what you get if you keep on going forever" and "infinity is the biggest number"? IMO, your explanation, "infinity is what you get if you keep on going forever", also give a child the impression that "infinity is the biggest number". | |
| S Feb 18, 2019 at 15:04 | history | suggested | CommunityBot | CC BY-SA 4.0 |
Fix typo: "begin told" is supposed to say "being told"
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| Feb 18, 2019 at 15:03 | review | Suggested edits | |||
| S Feb 18, 2019 at 15:04 | |||||
| Feb 17, 2019 at 22:03 | comment | added | Leonid Dworzanski | And your answer is compliant with the axiom of infinity. So, mathlogic could be used as a source for explanations. en.wikipedia.org/wiki/Axiom_of_infinity | |
| Feb 17, 2019 at 21:23 | comment | added | Qwertiy | Sometime I've named 999 999 999 999 as the biggest number I know. And it was the biggest just as I haven't known the name f the next number, so this one is the biggest and I can't say what will be the next number. I knew how to write any number, but this one is the biggest I knew. | |
| Feb 17, 2019 at 9:51 | comment | added | Evpok | This seems a bit circular : “the infinity if the number you'd get if you counted for an infinite amount of time” does not sound very satisfying, but whatever works, I guess | |
| Feb 14, 2019 at 16:56 | comment | added | orion2112 | @CortAmmon If a child decides that the biggest number (so "infinity") is $100$ and they realize that $101$ is bigger, then $101$ is "the new infinity". All the numbers have the property of being themselves ($1$ is $1$, $2$ is $2$, etc.), but as you keep on adding $1$, you realize that $\infty$ is neither $100$, $101$, $102$, and so on, so it must be different than a number; you can't put your finger on it (while $1000$ is definitely $1000$, and not $1001$). The approach is not meant to be intuitive, it's meant to be "constructive". (You build the idea that $\infty$ doesn't work like numbers). | |
| Feb 14, 2019 at 16:40 | comment | added | Cort Ammon | The one thing I might add is that this intuitive exploration process can hit a snag if the child already has decided that infinity is a number. Whether infinity is a number or not is actually a rather complicated question (see cardinals vs. ordinals to study how different infinities can be used in arithmetic operations). That certainty that infinity is actually a number that supports arithmetic as we know it (a natural number or a real number) may need to be gently dislodged before the intuitive approaches can do their job. | |
| Feb 14, 2019 at 16:30 | history | edited | orion2112 | CC BY-SA 4.0 |
Added explanation on personal experience
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| Feb 14, 2019 at 13:59 | comment | added | Carl Witthoft | This is the method provided in the book "The Phantom Tollbooth" | |
| Feb 14, 2019 at 10:31 | comment | added | Tommi | @orion2112 The answer would be improved by being explicit that this worked with teens but has not been tested with 5-year olds. There might be differences in thinking between the groups. Questions are supposed to rely on personal experience or reliable sources, so mentioning the experience, if only tangentially related, is good and improves the credibity of the answer. | |
| Feb 14, 2019 at 0:35 | vote | accept | QMC | ||
| May 11, 2023 at 1:44 | |||||
| Feb 13, 2019 at 18:01 | comment | added | rrauenza | "The largest number may be around 45 billion" : vimeo.com/13497928#t=1m18s | |
| Feb 13, 2019 at 15:33 | comment | added | Thorbjørn Ravn Andersen | @PeterA.Schneider Many things about infinity are not making sense. You know that some infinities are more infinite than others? | |
| Feb 13, 2019 at 15:30 | comment | added | Peter Cordes | @PeterA.Schneider: one last link about IEEE FP: Bruce Dawson wrote an interesting series of articles with some neat FP fun facts and useful real facts and real implementations especially on x86 and Windows/Linux. randomascii.wordpress.com/2012/02/25/… has an index of the articles. It's nice and somewhat skimmable to find a fun part to jump into, highly recommend it for anyone who wants an intro or deeper understanding of FP, especially in C and C++. | |
| Feb 13, 2019 at 15:23 | comment | added | orion2112 | I appreciate the comments relative to the programming point of view; it hadn't come to mind when I wrote the post. I would just like to reiterate that the presentation I proposed does not take into account everything about infinity. (There are many types of infinities, and ways to think about it; a 5 year-old does not need to know them all. I chose one I thought was simple enough). I may actually have presented the induction principle, rather than infinity, but I think it can give a glimpse of what infinity "looks like". | |
| Feb 13, 2019 at 15:05 | comment | added | Peter Cordes | @PeterA.Schneider: IEEE floating point is rather beautifully engineered in a lot of ways. Fun fact: the exponent bias and field layout of IEEE binary32 / binary64 formats (en.wikipedia.org/wiki/Single-precision_floating-point_format) means that an integer increment of the bit-pattern gives you the next representable float, and you can compare floats using a plain integer compare of the bit-patterns as integers (if they have the same sign). | |
| Feb 13, 2019 at 14:56 | comment | added | Peter - Reinstate Monica | @PeterCordes Thanks for the comments. You guessed right, I do not much work with floating point, so I read your information with interest. | |
| Feb 13, 2019 at 14:36 | comment | added | Peter Cordes |
(sorry for the long comments. I guess when you said "as a programmer", you were still talking about pure math infinity, which has different rules than IEEE floating-point infinity in programs. I'm more familiar with IEEE floating point, but some answers here do say that inf+1 is still inf. And even compares equal with ==, so IEEE math assumes all infinities (of the same sign) are equal to each other.)
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| Feb 13, 2019 at 14:22 | comment | added | Peter Cordes | In mathematics, though, 1/inf is undefined (One divided by infinity is not zero?), so IEEE math defining that result as 0 (and raising an underflow exception) instead of NaN + invalid exception is more of an engineering hack. But IEEE Infinity usually represents an overflowed finite value, not a true infinity. So I guess @PeterA.Schneider it does make some sense to consider true infinity not a number. But you can add 1 to it and get infinity again, if you're using a type that can represent infinity in the first place, for cardinal numbers anyway. | |
| Feb 13, 2019 at 13:53 | comment | added | Peter Cordes |
TL:DR: IEEE floating point has overflow that saturates to +- infinity (unlike integer math where overflow typically wraps or is undefined). Similarly, IEEE also has underflow to 0 (gradual with denormals) if you keep doing x *= 0.75 or something. Yes infinity is special, but overflow exceptions are separate from invalid operation exceptions (like trying to take the real sqrt a negative number). e.g. gnu.org/software/libc/manual/html_node/FP-Exceptions.html
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| Feb 13, 2019 at 13:48 | comment | added | Peter Cordes |
@PeterA.Schneider: I guess you don't work with floating-point very often? std::numeric_limits<double>::infinity() is a perfectly valid double in C++, or C +INFINITY. You only get a NaN if you do inf - inf, or inf / inf. inf + 1, inf + inf, inf - 1 are all not errors and give you inf. 1.0/inf evaluates to 0. (godbolt.org/z/oPWwzc) This is somewhat questionable, but it's considered useful to make stuff like 1/(1/x + 1/y) "work" even for x=0 or overflow in the sum.
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| Feb 13, 2019 at 13:18 | comment | added | JRN | In IEEE floating-point arithmetic, $\infty+1=\infty$. :) | |
| Feb 13, 2019 at 12:24 | comment | added | Alma Do | One important thing is also that ∞+1=∞ is only valid for cardinal numbers and is not true for ordinal ones. So I'm really in doubt if there is an "easy" way to explain all of that to 5 y.o. child without resorting to teaching the "simple, but wrong" thing | |
| Feb 13, 2019 at 11:59 | comment | added | Dave L Renfro | @Peter A. Schneider: Exactly what I think also. Thus, the issue is not "what is $\infty + 1,$" but rather (if the questioner persists along this line of reasoning) "what might be a reasonable way to define what we mean by adding $1$ and $\infty$". For example, we can talk about mixing colors, such as mixing red paint and blue paint, and thus think of this as adding "red" and "blue", but what might we mean by adding "red" and $1$"? Is this even a useful path of inquiry? FYI, the notion of addition of cardinal numbers is simply one way of going about this, not THE way (as useful as it is). | |
| Feb 13, 2019 at 11:26 | comment | added | Peter - Reinstate Monica |
To me as a programmer $\infty+1=\infty$ results in a type error because + is not defined for a left hand operand of $\infty$ and a right hand operand of a natural number; or if it is defined, it is overloaded for these types and does something different than the other overloads. The important thing is that $\infty$ is, from my programmer's perspective, "not a number" (literally a NaN ;-) ), and you cannot naively use it like a number.
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| Feb 13, 2019 at 9:46 | comment | added | Nelson | Infinity is an abstract concept. It's actually not a number, but a concept. Hence ∞+1=∞ makes sense because +1 doesn't do anything to the concept of infinity. | |
| Feb 13, 2019 at 5:32 | history | answered | orion2112 | CC BY-SA 4.0 |