5 ms·

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way t

by mjd 6y ago

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat. I've met many people who were just boggled by the idea that “10” might mean ●●, or ●●●● ●●●● ●●●● ●●●●, rather than ●●●●● ●●●●●. A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right? (https://math.stackexchange.com/q/709657/25554 https://math.stackexchange.com/q/709657/25554) Computer programmers (and historians!) have a similar problem with dates, and in particular with issues like daylight saving time and time zones. I think a lot of the problem is that again there's no way to talk about a particular instant of time without adopting some necessarily arbitrary and relative nomenclature like “January 17, 1706 at 09:37 local time in Boston”. But when was this _really_? Unfortunately there is no “really”. (“Oh, you mean Ramadan 1117 AH, now I understand.”)

blauditore 6y ago
>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat ...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?
- tom-thistime 6y ago
  Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?
  - barrkel 6y ago
    I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.
    - samatman 6y ago
      Thank you! This thread is full of people insisting on something wrong because they were taught incorrectly; an irrational number is defined in terms of integer ratios for a reason. It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.
      - deleted 6y ago
        [deleted]
    - mjd 6y ago
      That is not what I meant at all. (My phrasing was unclear. Sorry for the confusion.) Jordi understood what I meant though.
      - barrkel 6y ago
        It was unclear; I understand rational numbers to be ratios, and irrationals to be inexpressible as fractions, and I see an almost direct connection between that and digital representation in rational bases, so it seemed deeply confusing to see a consequence of the definition of rationals being refuted. The only way out was an irrational base.
  - mjd 6y ago
    My phrasing was bad. I should have said "the mistaken belief that an irrational number is * defined to be * one whose decimal representation doesn't repeat”. Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational. It's much easier to start from the intrinsic properties, and use those to prove things about the representation, than the other way around. But if you don't distinguish the representation from the thing itself, you can't tell which way you are going.
    - btilly 6y ago
      I am still not in agreement. The proof that the usual definition is equivalent to the representation is fairly straightforward and easy, no matter which side you picked as the definition. And once the equivalence is established, all other proofs proceed naturally. It therefore matters a lot that we pick one as a definition and know which one we picked, but not so much which one we picked. Now in fact the quotient definition is by far more interesting mathematically. There is also a clear foundational reason to prefer it, namely that you can easily construct and prove things about the rational numbers long before you construct the real numbers. However it is unlikely that anyone who is confused about the definition of a rational number has a clear understanding of how the reals are constructed, so that is not a particularly important consideration for them. Furthermore the fact that foundational considerations argue for one construction over another has little bearing on what is pedagogically preferable. As a famous example, the easiest way to rigorously define logarithms is through the integral of 1/x. However explaining logarithms that way to someone who doesn't know them is a pedagogical disaster.
- jordigh 6y ago
  I think the problem is that the definition, while valid, makes you hyperfocus on irrational numbers this way. Seldom do we prove that a number is irrational by inspecting its decimal expansion. This would be in most cases a very unnatural proof. Since irrationality is a negative property (meaning, one arising out of a negation: the number is not a ratio), most of the time you prove it by contradiction. But people who just know the "digits don't repeat" definition expect us to somehow be able to list all of the digits of an irrational number and show that this infinite list doesn't repeat, which is, of course, an impossible task.
- mjd 6y ago
  It's _true_, but it's not good as a definition, because it's hard to reason about. It drags in all sorts of contingent facts about base-10 representations that are not usually of interest. The equivalent property, that a number is irrational if it's not equal to m÷n for any integers m and n, is much simpler. So we use that as the definition, and from that simple and intrinsic definition, we prove the _theorem_ that the decimal representation of an irrational number never repeats.
throwaway894345 6y ago
Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since Unix epoch (or some similarly arbitrary point in time), then writing programs about time would be much simpler. I think the numerals and numbers issue is more complex because numerals are fundamentally hard to reason about. Even the question "what is a number?" is deceivingly deep.
- barrkel 6y ago
  No, dates are harder than that. Humans use time to coordinate; the representation of time is fundamentally about communication. Only timestamps of events in the physical world are easy (ish). But that's not always, or perhaps mostly, what people are interested in. When people receive a time, they may (usually) want it in their own time zone, but they might instead want it in the time zone of the entity they're getting the time from, if they're subsequently going to talk to that entity about the time. When they talk about meeting someone else, when they convey the time of the meeting, they usually mean whatever that time means in the place where they meet, which might be different from the current location of either. It might even be different due to political changes around time zones and daylight saving if the meeting is far enough in the future.
  - throwaway894345 6y ago
    You say "no, dates are harder than that", but it sounds like you agree with me exactly based on the rest of your post. Time is hard because human representations of it are varied, complicated, and often arbitrary--not because there's anything fundamentally hard about the math of time (again, relativity notwithstanding). Contrast that with numbers which are inherently difficult to intuit about apart from issues of representation.
- deleted 6y ago
  [deleted]
foobuzzHN 6y ago
> A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right? I don't think there are many areas of math where ∞ is a number. In my experience people have a whole other problem with ∞, thinking that it is some sort of huge concept defined globally in math, where it is just a notation shared by various non-mystical definitions across subjects (e.g. bijection-based definition of infinite set, epsilon-based definition of convergence, etc)
btilly 6y ago
For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat. How is this a mistaken belief? Every rational number winds up in a repeating decimal representation and every number with a repeating decimal representation is a rational number. We learn algorithms to go back and forth between the two in elementary school. Therefore irrational numbers cannot have repeating decimal representations. Conversely numbers with decimal representations that don't wind up repeating cannot be rational and so must be irrational.
- mjd 6y ago
  My phrasing was poor. See https://news.ycombinator.com/item?id=23013090 https://news.ycombinator.com/item?id=23013090
schoen 6y ago
Apparently the New Math (https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math) tried to address this kind of issue quite explicitly, by drawing a consistent distinction between numbers and numerals (where a numeral is a symbol that names a number). Reportedly most American math students found this kind of distinction extremely hard to grasp when they were presented with this kind of issue in elementary school. Maybe it would have worked better when they were a bit older. I wonder if there's a way of teaching this kind of distinction and issue well in a way that would make sense for most students. I think Feynman said somewhere that the New Math explicitly taught base representation and base conversions, probably as a way of trying to underscore the idea that "123" is a representation of a number rather than a number. Feynman found this to be of questionable value and thought that most students didn't manage to get the point. Edit: there's a similar issue in linguistics because you have words, phonemes, phones, graphemes, and glyphs. You could say that "dog" isn't a word, but is rather the standard way of writing a particular word in the standard writing system for English (which would sometimes be indicated by <dog> in linguistic contexts). This idea lets you refer to <alright> and <all right> as ways of writing the same word, or <color> and <colour>, or in the case of languages with multiple writing systems <हिन्दुस्तानी> and <ہندوستانی>, or <אַ שפּראַך איז אַ דיאַלעקט מיט אַן אַרמיי און פֿלאָט> and <a shprakh iz a dialekt mit an armey un flot>.
austincheney 6y ago
Proof: bring up removing time zones, such that there is only one time zone. The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.
- bmn__ 6y ago
  > Let’s not forget that time zones are a human invention less than 200 years old. What's the point of that statement? Before the introduction of formal time zones, we had thousands of informal ones, one for each settlement, calibrating noon to the zenith of the sun.
  - austincheney 6y ago
    The point is to indicate that humans knew how to wake up, eat food, communicate, and conduct business (even over long distances) before they had clocks much less the relatively recently invented time zones.
- ISO-morphism 6y ago
  200 years ago we didn't need timezones because we couldn't communicate or travel fast enough or often enough for them to matter. With railroads and telegraphs the level of granularity required for "some time 300 miles away" went from days to hours and minutes. No one in the U.S. west wanted to be told "The sun is at its zenith around 8 a.m. for you." For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day with people near them than to communicate an absolute moment in time and do the math to figure out how bright it is outside.
  - austincheney 6y ago
    > 200 years ago we didn't need timezones because... We still do not. China and India are examples of large geographies spanning across a vast amount of longitude and yet each are a single time zone. Time zones are a political entity only, an unnecessary complexity. The absence of time zones will not halt business or communication over long distances. > For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day People have done this for thousands of years without modern chronometers. Examples: dusk, dawn, morning, midday, afternoon, evening, twilight.

by mjd 6y ago

blauditore 6y ago

>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat ...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?

tom-thistime 6y ago
Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?
- barrkel 6y ago
  I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.
  - samatman 6y ago
    Thank you! This thread is full of people insisting on something wrong because they were taught incorrectly; an irrational number is defined in terms of integer ratios for a reason. It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.
    - deleted 6y ago
      [deleted]
  - mjd 6y ago
    That is not what I meant at all. (My phrasing was unclear. Sorry for the confusion.) Jordi understood what I meant though.
    - barrkel 6y ago
      It was unclear; I understand rational numbers to be ratios, and irrationals to be inexpressible as fractions, and I see an almost direct connection between that and digital representation in rational bases, so it seemed deeply confusing to see a consequence of the definition of rationals being refuted. The only way out was an irrational base.
- mjd 6y ago
  My phrasing was bad. I should have said "the mistaken belief that an irrational number is * defined to be * one whose decimal representation doesn't repeat”. Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational. It's much easier to start from the intrinsic properties, and use those to prove things about the representation, than the other way around. But if you don't distinguish the representation from the thing itself, you can't tell which way you are going.
  - btilly 6y ago
    I am still not in agreement. The proof that the usual definition is equivalent to the representation is fairly straightforward and easy, no matter which side you picked as the definition. And once the equivalence is established, all other proofs proceed naturally. It therefore matters a lot that we pick one as a definition and know which one we picked, but not so much which one we picked. Now in fact the quotient definition is by far more interesting mathematically. There is also a clear foundational reason to prefer it, namely that you can easily construct and prove things about the rational numbers long before you construct the real numbers. However it is unlikely that anyone who is confused about the definition of a rational number has a clear understanding of how the reals are constructed, so that is not a particularly important consideration for them. Furthermore the fact that foundational considerations argue for one construction over another has little bearing on what is pedagogically preferable. As a famous example, the easiest way to rigorously define logarithms is through the integral of 1/x. However explaining logarithms that way to someone who doesn't know them is a pedagogical disaster.
jordigh 6y ago
I think the problem is that the definition, while valid, makes you hyperfocus on irrational numbers this way. Seldom do we prove that a number is irrational by inspecting its decimal expansion. This would be in most cases a very unnatural proof. Since irrationality is a negative property (meaning, one arising out of a negation: the number is not a ratio), most of the time you prove it by contradiction. But people who just know the "digits don't repeat" definition expect us to somehow be able to list all of the digits of an irrational number and show that this infinite list doesn't repeat, which is, of course, an impossible task.
mjd 6y ago
It's _true_, but it's not good as a definition, because it's hard to reason about. It drags in all sorts of contingent facts about base-10 representations that are not usually of interest. The equivalent property, that a number is irrational if it's not equal to m÷n for any integers m and n, is much simpler. So we use that as the definition, and from that simple and intrinsic definition, we prove the _theorem_ that the decimal representation of an irrational number never repeats.

throwaway894345 6y ago

Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since Unix epoch (or some similarly arbitrary point in time), then writing programs about time would be much simpler. I think the numerals and numbers issue is more complex because numerals are fundamentally hard to reason about. Even the question "what is a number?" is deceivingly deep.

barrkel 6y ago
No, dates are harder than that. Humans use time to coordinate; the representation of time is fundamentally about communication. Only timestamps of events in the physical world are easy (ish). But that's not always, or perhaps mostly, what people are interested in. When people receive a time, they may (usually) want it in their own time zone, but they might instead want it in the time zone of the entity they're getting the time from, if they're subsequently going to talk to that entity about the time. When they talk about meeting someone else, when they convey the time of the meeting, they usually mean whatever that time means in the place where they meet, which might be different from the current location of either. It might even be different due to political changes around time zones and daylight saving if the meeting is far enough in the future.
- throwaway894345 6y ago
  You say "no, dates are harder than that", but it sounds like you agree with me exactly based on the rest of your post. Time is hard because human representations of it are varied, complicated, and often arbitrary--not because there's anything fundamentally hard about the math of time (again, relativity notwithstanding). Contrast that with numbers which are inherently difficult to intuit about apart from issues of representation.
deleted 6y ago
[deleted]

foobuzzHN 6y ago

> A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right? I don't think there are many areas of math where ∞ is a number. In my experience people have a whole other problem with ∞, thinking that it is some sort of huge concept defined globally in math, where it is just a notation shared by various non-mystical definitions across subjects (e.g. bijection-based definition of infinite set, epsilon-based definition of convergence, etc)

btilly 6y ago

For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat. How is this a mistaken belief? Every rational number winds up in a repeating decimal representation and every number with a repeating decimal representation is a rational number. We learn algorithms to go back and forth between the two in elementary school. Therefore irrational numbers cannot have repeating decimal representations. Conversely numbers with decimal representations that don't wind up repeating cannot be rational and so must be irrational.

mjd 6y ago
My phrasing was poor. See https://news.ycombinator.com/item?id=23013090 https://news.ycombinator.com/item?id=23013090

schoen 6y ago

Apparently the New Math (https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math) tried to address this kind of issue quite explicitly, by drawing a consistent distinction between numbers and numerals (where a numeral is a symbol that names a number). Reportedly most American math students found this kind of distinction extremely hard to grasp when they were presented with this kind of issue in elementary school. Maybe it would have worked better when they were a bit older. I wonder if there's a way of teaching this kind of distinction and issue well in a way that would make sense for most students. I think Feynman said somewhere that the New Math explicitly taught base representation and base conversions, probably as a way of trying to underscore the idea that "123" is a representation of a number rather than a number. Feynman found this to be of questionable value and thought that most students didn't manage to get the point. Edit: there's a similar issue in linguistics because you have words, phonemes, phones, graphemes, and glyphs. You could say that "dog" isn't a word, but is rather the standard way of writing a particular word in the standard writing system for English (which would sometimes be indicated by <dog> in linguistic contexts). This idea lets you refer to <alright> and <all right> as ways of writing the same word, or <color> and <colour>, or in the case of languages with multiple writing systems <हिन्दुस्तानी> and <ہندوستانی>, or <אַ שפּראַך איז אַ דיאַלעקט מיט אַן אַרמיי און פֿלאָט> and <a shprakh iz a dialekt mit an armey un flot>.

austincheney 6y ago

Proof: bring up removing time zones, such that there is only one time zone. The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.

bmn__ 6y ago
> Let’s not forget that time zones are a human invention less than 200 years old. What's the point of that statement? Before the introduction of formal time zones, we had thousands of informal ones, one for each settlement, calibrating noon to the zenith of the sun.
- austincheney 6y ago
  The point is to indicate that humans knew how to wake up, eat food, communicate, and conduct business (even over long distances) before they had clocks much less the relatively recently invented time zones.
ISO-morphism 6y ago
200 years ago we didn't need timezones because we couldn't communicate or travel fast enough or often enough for them to matter. With railroads and telegraphs the level of granularity required for "some time 300 miles away" went from days to hours and minutes. No one in the U.S. west wanted to be told "The sun is at its zenith around 8 a.m. for you." For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day with people near them than to communicate an absolute moment in time and do the math to figure out how bright it is outside.
- austincheney 6y ago
  > 200 years ago we didn't need timezones because... We still do not. China and India are examples of large geographies spanning across a vast amount of longitude and yet each are a single time zone. Time zones are a political entity only, an unnecessary complexity. The absence of time zones will not halt business or communication over long distances. > For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day People have done this for thousands of years without modern chronometers. Examples: dusk, dawn, morning, midday, afternoon, evening, twilight.