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The way I was taught decimals in school (in Romania) always made 0.99... seem like an absurdity to me: we were always taught that fractions are the "real" repre

The way I was taught decimals in school (in Romania) always made 0.99... seem like an absurdity to me: we were always taught that fractions are the "real" representation of rational numbers, and decimal notation is just a shorthand. Doing arithmetic with decimal numbers was seen as suspect, and never allowed for decimals with infinite expansions. So, for example, if a test asked you to calculate 2 × 0.2222... [which we notated as 2 × 0,(2)], then the right solution was to expand it: 2 × 0.2222... = 2 × 2/9 = 4/9 = 0.444... Once you're taught that this is how the numbers work, it's easy(ish) to accept that 0.999... is just a notational trick. At the very least, you're "immune" to certain legit-looking operations, like 0.33... + 0.66... = 1/3 + 2/3 = 3/3 = 1 Instead of 0.33... + 0.66... = 0.99... So, in this view, 0.3 or 0.333... are not numbers in the proper sense, they're just a convenient notation for 3/10 and 1/3 respectively. And there simply is no number whose notation would be 0.999..., it's just an abuse of the decimal notation.

lmm 1y ago
> Doing arithmetic with decimal numbers was seen as suspect, and never allowed for decimals with infinite expansions. With that attitude how do you handle e.g. pi or sqrt(2), which it's perfectly legitimate to do arithmetic with?
- AndrewDucker 1y ago
  Once you're dealing with irrational numbers you have to understand that all results are approximations.
  - lmm 1y ago
    Well, sure, but you should still be able to ask and answer questions like "Is pi + sqrt(2) less than or greater than 4.553?"
    - AndrewDucker 1y ago
      In that case you know how many decimal places you want to expand them to, in order to compare.
      - tsimionescu 1y ago
        Note that "expanding them to some number of decimal places" gives a somewhat misleading idea about how this works. What you're actually doing is computing a good enough approximation of pi, and expressing that as a decimal. But this is not the same kind of simple process that naturally gives decimals as it is for a rational fraction. Instead, you have to find some series with rational elements which converges to pi, and then compute enough terms of that series that you have a good enough approximation of pi for your purpose. Ideally, since you're interested in an inequality, you'd pick a series which is monotoniclaly increasing or decreasing, so that you know that computing more terms can't put you below or above the target number after you've reached a conclusion. But there is no canonical answer, there are numerous series which converge to pi that you could use, and they would givw you different decimal expansions as you are computing them.
    - dagw 1y ago
      When you have those sorts of problems the best way is to approach them using inequalities. 3.1415<pi<3.1416 and 1.4142<sqrt(2)<1.4143, => 4.5557<pi + sqrt(2)<4.5559 => 4.553 < 4.5557 < pi + sqrt(2) => 4.553 < pi + sqrt(2)
      - lmm 1y ago
        Finite decimal representations being of course the most convenient notation for inequalities like this.
    - tsimionescu 1y ago
      It's important to understand that this was a non-trivial question for thousands of years. The ancient Babylonians would have probably believed this to be false (their best known approximation had pi ≈ 25/8, which is too small). The right way to approach this problem from first principles would be to construct some geometrical objects that have these lengths and try to compare them (for example by taking the perimeter of a square inscribing a unit circle and a square inscribed in a unit circle as the upper and lower bounds for pi, though that may not be good enough for this particular problem). When you're doing something like pi + sqrt(2) ≈ 3.14159 + 1.41421 = 4.5558, you're taking known good approximations of these two real numbers and adding them up. The heavy lifting was done over thousands of years to produce these good approximations. It's not the arithmetic on the decimal representations that's doing the heavyh lifting, it's the algorithms needed to produce these good approximations in the first place that are the magic here. And it would be just as easy to compute this if I told you that pi ≈ 314159/100000, and sqrt(2) ≈ 141421/100000, so that their sum is 455580/100000, which is clearly larger than 4553/1000.
      - mcphage 1y ago
        > their best known approximation had pi ≈ 25/8, which is too small I'm curious if they had a better one that we don't know of yet—their best known approximation of sqrt(2) is significantly more accurate.
  - qayxc 1y ago
    Not really. Like the sibling comment said - you simply keep the symbolic values. I.e. instead of 4.442882938158... you write π√2, just like you would ⅚ and not 0,8333... in both cases you preserve the exact values. Decimal (or any other numbering system, really) approximations are only useful when you never want to do any further arithmetic with the result.
    - thaumasiotes 1y ago
      > Decimal (or any other numbering system, really) approximations are only useful when you never want to do any further arithmetic with the result. What? The opposite is the case. Anything you want to do something with, you can only measure inaccurately; arithmetic doesn't have any use if you can't apply it to inaccurate measurements. That's what we use it for!
      - qayxc 1y ago
        So I take it you never wrote any numerical simulations or did symbolic calculations then? Catastrophic cancellation and other failures are serious issues to consider when doing numerical analysis and can often be avoided completely by using symbolic calculation instead. You can easily end up with wrong results, especially when composing calculations. This would make it difficult to, for example, match your theoretical model against actual measurement results; particularly if the model includes expressions that don't have closed-form solutions.
  - rob_c 1y ago
    Less approximations and more representations of complex things at times. (Just my opinion) I prefer comparing it to complex numbers where I can't have "i" apples but I can calculate the phase difference between 2 power supplies in a circuit using such notation. Nobody really cares about the 3rd decimal place when taking about a speeding car at a turn but they do when talking about electrons in an accelerator, so accuracy and precision always feel mucky to talk about when dealing with irrationals (again my opinion).
    - tsimionescu 1y ago
      Well, 3.141 is an approximation of pi, not a representation of it, insomuch as you use it in an arithmetic expression. Of course, you can write 3.141... to just represent pi, but you can't eaisly use that in an arithmetic expression. For example, I can't tell you from "mechanical" operations if 3.141... - 3.1417 > 0, I have to lookup how big pi actually is.
- qsort 1y ago
  You don't. You keep them in symbolic form until they simplify and you do arithmetic at the last possible moment.
  - lmm 1y ago
    Sure, but when you reach that "last possible moment", what then?
    - dagw 1y ago
      If it's a maths problem you just leave it as symbols. If it's a science or engineering problem you expand it to a decimal approximation with the precision needed for the specific problem you are dealing with.
      - tsimionescu 1y ago
        Note that even for an engineering problem, you don't necessarily use a decimal representation. You may well want to represent pi as 3 or 4 or 22/7 or any other approximation that is good for your particular use case. Or you may even have usecases where you do things the opposite way - you may want to approximate 1 as pi/3 or something like that for certain kinds of problems (e.g if you're going to take the sin of your result).
    - chongli 1y ago
      Then you calculate the decimal expansion to the desired number of decimal places. This avoids accumulation of roundoff errors in intermediate results. Note that writing sqrt(2) as 1.41 or 1.41421 or any other decimal expansion you might want to write is incorrect: you will always get some roundoff error. If you want to calculate that sqrt(2)*sqrt(2)=2 then you can’t do so by multiplying the decimal expansions.
    - rob_c 1y ago
      You never evaluate symbols until your giving a numerical equivalent. Sure if a question asks for the escape velocity from Jupiter this has an approximate numerical value, but you don't just start by throwing numbers at a wall, you get the simplest equation which represents the value you're interested in an then evaluate it once you have a single equation for that parameter. Yes sqrt(2)*pi has a numerical approximation but you don't want that right at the start of taking about something like spin orbitals or momenta of spinning disks. Doing the latter compounds errors. It's no different to keeping around "i"/"j" until you need to express a phase or angle as it's cleaner and avoids compounding accuracy errors.
- tsimionescu 1y ago
  This is a very strange question. With repeating decimals, it is technically possible, though very complicated, to do arithmetic directly on the representations. You have to remember a bunch of extra rules, but it can be done. However, with numbers that have non-repeating inifinite decimal expansions, it is completely imposible to do arithmetic in the decimal notation. I'm not exagerating: it's literally physically impossible to represent on paper the result of doing 3pi in decimal notation in an unambiguous form other than 3pi. It's also completely impossible to use the decimal expansion of pi to compute that pi / pi = 1. Here, I'll show you what it would be like to try: pi / pi = 3.141592653589793238462643383279502884197169399375105820949445923078164062862089986280348253421170679821480865132820664709384460955058223172.... Now, of course you can do arithmetic with certain approximations of pi. For example, I can do this: pi / pi ≈ 3.1415 / 3.1415 = 1 Or even 3 × pi ≈ 3 × 3 = 9 But this is not doing arithmetic with the decimal expansion of pi, this is doing arithmetic with rational numbers that are close enoigh to pi for some purpose (that has to be defined).
  - anthk 1y ago
    pi/pi would evaluate to 1 as most proper languages would deal with pi symbolically and not arithmetically.
    - tsimionescu 1y ago
      My point was only that even trivial arithmetic is impossible to do with the infinite decimal representations of irrational numbers.
    - librasteve 1y ago
      in raku say pi/pi; #1
      - lizmat 1y ago
        Sadly, that's just Num.gist showing 1.0 as "1" though. say (pi/pi).^name; # Num
- anthk 1y ago
  You would use rational approximations good enough for different scales and roundings.
- cwmma 1y ago
  in American math classes (as opposed to science classes) you almost never expand PI or sqrt(2), you either cancel them out or leave them in the answer until the end. Maybe if it's a word problem you sub them in the very last step but the problem itself is almost certainly going to be designed so it's not an issue.
  - Suppafly 1y ago
    >in American math classes (as opposed to science classes) you almost never expand PI Except we have some fascination with memorizing the digits of pi and having competitions for doing so for some reason.
    - BobaFloutist 1y ago
      Just for fun.
    - mrguyorama 1y ago
      There is no school math test that will require you to know the digits of Pi, except as a silly extra credit question. The fascination is just dick measuring. "I'm smarter than you", for memorizing a longer string? It's quite dumb, but American media loves to use the dumbest possible ways of demonstrating that a character is intelligent, because uh it's really really hard to demonstrate "This person is very intelligent" to a subset of the population that is mostly at a middle school reading level and barely comprehends basic arithmetic, let alone algebra.
    - anthk 1y ago
      And that's useless for actual math.
      - Suppafly 1y ago
        >And that's useless for actual math. Agreed. The schools always seem to have these learning adjacent things that are theoretically supposed to make subjects engaging, but in reality are so disconnected from the subject that they are meaningless.
lcrz 1y ago
Both ways are just notation. There’s nothing more real about 3/10 compared to 0.3. Telling you otherwise might have worked as a educational “shorthand”, but there are no mathematical difficulties as long as you use good definitions of what you mean when you write them down. The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits.
- tsimionescu 1y ago
  I agree that ultimately both are just notations. I do think the fractional notation has some definite advantages and few disadvantages, so I think it's better to regard it as more canonical. I disagree though that it's necessary or even useful to think of 0.99... or 0.33... as sequences or limits. It's of course possible, but it complicates a very simple concept, in my opinion, and muddies the waters of how we should be using notions such as equality or inifinity. For example, it's generally seen as a bad idea to declare that some infinite sum is actually equal to its limit, because that only applies when the series of partial sums converges. It's more rigorous to say that sigma(1/n) for n from 2 going to infinity converges to 1, not that it is equal to 1; or to say that lim(sigma(1/n)) for n from 2 to infinity = 1. So, to say that 0.xxx... = sigma(x/10^n) for n from 1 to infinity, and to show that this is equal to 1 for x = 9, muddies the waters a bit. It still gives this impression that you need to do an infinite addition to show that 0.999... is equal to 1, when it's in fact just a notation for 9/9 = 1. It's better in my opinion to show how to calculate the repeating decimal expansion of a fraction, and to show that there exists no fraction whose decimal expansion is 0.9... repeating.
- WorldMaker 1y ago
  > The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits. Also a possible third thing: not enjoying working in a Base that makes factors of 3 hard to write. Thirds seem like common enough fractions "naturally" but decimal (Base-10) makes them hard to write. It's one of the reasons there are a lot of proponents of Base-12 as a better base for people, especially children, because it has a factor of 3 and thirds have nice clean duodecimal representations. (Base-60 is another fun option; it's also Babylonian approved and how we got 60 minutes and 60 seconds as common unit sizes.)
olau 1y ago
I was taught something of the same. But I think it was misguided. I'll note that 1/3 is not a number, it's a calculation. So more complicated. And fractions are generally much more complicated than the decimal system. Beyond some simple fractions that you're bound to experience in your everyday life, I don't think it makes sense to drill fractions. In the end, when you actually need to know the answer to a computation as a number, you're more likely to make a mistake because you spend your time juggling fractions instead of handling numerical instability. Decimal notation used to be impractical because calculating with multiple digits was slow and error-prone. But that's no longer the case.
- tsimionescu 1y ago
  This is ultimately a matter of definitions, and neither defining the fractions nor the decimals as the "true" representation of rationals is ultimately more or less correct. But, operations on fractions are definitely easier than operations on decimals. And fractions have the nice property that they have finite representations for all rational numbers, whereas decimal representations always have infinite representations even for very simple numbers, such as 1/3. Also, if you are going to do arithmetic with infinite decimal representations, the you have to be aware that the rules are more complex then simply doing digit-by-digit operations. That is, 0.77... + 0.44... ≠ 1.11... even though 7+4 = 11. And it gets even more complex for more complicated repeating patterns, such as 0.123123123... + 0.454545... (that is, 123/999 + 45/99). I doubt there is any reason whatsoever to attempt to learn the rules for these things, given that the arithmetic of fractions is much simpler and follows from the rules for division. The fact that a handful of simple cases work in simple ways doesn't make it a good idea to try.
- volemo 1y ago
  > I'll note that 1/3 is not a number, it's a calculation. So more complicated. 1/3 is a calculation the same way 42 is a calculation (4*10^1 + 2*10^0). Nothing is real except sets containing sets! /j
  - DemocracyFTW2 1y ago
    Yes, true. *BUT* 1/3 is a fraction with denominator 3. 1/5 is a fraction with another denominator, and 1/7 has yet another. So how much is 1/3 + 1/5 + 1/7? You can't just add up, you first have to multiply to get to common ground. The decimal expansions of these use the same base and are readily comparable.
anthk 1y ago
Forth and Lisp users often try to use rational first and floats later. On Scheme Lisps, you have exact-inexact and inexact->exact functions which convert rationals to floats and viceversa.
Suppafly 1y ago
>So, in this view, 0.3 or 0.333... are not numbers in the proper sense, they're just a convenient notation for 3/10 and 1/3 respectively. And there simply is no number whose notation would be 0.999..., it's just an abuse of the decimal notation. I wonder if students from Romania are hamstrung in more advanced mathematics from being taught this way.
- leereeves 1y ago
  I don't think they would be; I think they might even have an advantage. They'd understand that the only numbers that have infinite repeating expansions are rationals, and that decimals are, in general, just approximations.
- tsimionescu 1y ago
  I don't think decimals, especially repeated decimals or other infinite decimal expansions, show up much if at all in any advanced math subjects beyond the study of themselves, of course). Higher math is almost exclusively symbolic. You're more likely to need to learn that "1" is just a notation for the set which contains the empty set then to learn that it's OK to add 0.22... + 0.44... = 0.66...
mcphage 1y ago
> we were always taught that fractions are the "real" representation of rational numbers There is no "real" representation of rational numbers, and fractions are no more real—or fake—than decimals. > And there simply is no number whose notation would be 0.999... There is, though. It's 1.

lmm 1y ago

> Doing arithmetic with decimal numbers was seen as suspect, and never allowed for decimals with infinite expansions. With that attitude how do you handle e.g. pi or sqrt(2), which it's perfectly legitimate to do arithmetic with?

AndrewDucker 1y ago
Once you're dealing with irrational numbers you have to understand that all results are approximations.
- lmm 1y ago
  Well, sure, but you should still be able to ask and answer questions like "Is pi + sqrt(2) less than or greater than 4.553?"
  - AndrewDucker 1y ago
    In that case you know how many decimal places you want to expand them to, in order to compare.
    - tsimionescu 1y ago
      Note that "expanding them to some number of decimal places" gives a somewhat misleading idea about how this works. What you're actually doing is computing a good enough approximation of pi, and expressing that as a decimal. But this is not the same kind of simple process that naturally gives decimals as it is for a rational fraction. Instead, you have to find some series with rational elements which converges to pi, and then compute enough terms of that series that you have a good enough approximation of pi for your purpose. Ideally, since you're interested in an inequality, you'd pick a series which is monotoniclaly increasing or decreasing, so that you know that computing more terms can't put you below or above the target number after you've reached a conclusion. But there is no canonical answer, there are numerous series which converge to pi that you could use, and they would givw you different decimal expansions as you are computing them.
  - dagw 1y ago
    When you have those sorts of problems the best way is to approach them using inequalities. 3.1415<pi<3.1416 and 1.4142<sqrt(2)<1.4143, => 4.5557<pi + sqrt(2)<4.5559 => 4.553 < 4.5557 < pi + sqrt(2) => 4.553 < pi + sqrt(2)
    - lmm 1y ago
      Finite decimal representations being of course the most convenient notation for inequalities like this.
  - tsimionescu 1y ago
    It's important to understand that this was a non-trivial question for thousands of years. The ancient Babylonians would have probably believed this to be false (their best known approximation had pi ≈ 25/8, which is too small). The right way to approach this problem from first principles would be to construct some geometrical objects that have these lengths and try to compare them (for example by taking the perimeter of a square inscribing a unit circle and a square inscribed in a unit circle as the upper and lower bounds for pi, though that may not be good enough for this particular problem). When you're doing something like pi + sqrt(2) ≈ 3.14159 + 1.41421 = 4.5558, you're taking known good approximations of these two real numbers and adding them up. The heavy lifting was done over thousands of years to produce these good approximations. It's not the arithmetic on the decimal representations that's doing the heavyh lifting, it's the algorithms needed to produce these good approximations in the first place that are the magic here. And it would be just as easy to compute this if I told you that pi ≈ 314159/100000, and sqrt(2) ≈ 141421/100000, so that their sum is 455580/100000, which is clearly larger than 4553/1000.
    - mcphage 1y ago
      > their best known approximation had pi ≈ 25/8, which is too small I'm curious if they had a better one that we don't know of yet—their best known approximation of sqrt(2) is significantly more accurate.
- qayxc 1y ago
  Not really. Like the sibling comment said - you simply keep the symbolic values. I.e. instead of 4.442882938158... you write π√2, just like you would ⅚ and not 0,8333... in both cases you preserve the exact values. Decimal (or any other numbering system, really) approximations are only useful when you never want to do any further arithmetic with the result.
  - thaumasiotes 1y ago
    > Decimal (or any other numbering system, really) approximations are only useful when you never want to do any further arithmetic with the result. What? The opposite is the case. Anything you want to do something with, you can only measure inaccurately; arithmetic doesn't have any use if you can't apply it to inaccurate measurements. That's what we use it for!
    - qayxc 1y ago
      So I take it you never wrote any numerical simulations or did symbolic calculations then? Catastrophic cancellation and other failures are serious issues to consider when doing numerical analysis and can often be avoided completely by using symbolic calculation instead. You can easily end up with wrong results, especially when composing calculations. This would make it difficult to, for example, match your theoretical model against actual measurement results; particularly if the model includes expressions that don't have closed-form solutions.
- rob_c 1y ago
  Less approximations and more representations of complex things at times. (Just my opinion) I prefer comparing it to complex numbers where I can't have "i" apples but I can calculate the phase difference between 2 power supplies in a circuit using such notation. Nobody really cares about the 3rd decimal place when taking about a speeding car at a turn but they do when talking about electrons in an accelerator, so accuracy and precision always feel mucky to talk about when dealing with irrationals (again my opinion).
  - tsimionescu 1y ago
    Well, 3.141 is an approximation of pi, not a representation of it, insomuch as you use it in an arithmetic expression. Of course, you can write 3.141... to just represent pi, but you can't eaisly use that in an arithmetic expression. For example, I can't tell you from "mechanical" operations if 3.141... - 3.1417 > 0, I have to lookup how big pi actually is.
qsort 1y ago
You don't. You keep them in symbolic form until they simplify and you do arithmetic at the last possible moment.
- lmm 1y ago
  Sure, but when you reach that "last possible moment", what then?
  - dagw 1y ago
    If it's a maths problem you just leave it as symbols. If it's a science or engineering problem you expand it to a decimal approximation with the precision needed for the specific problem you are dealing with.
    - tsimionescu 1y ago
      Note that even for an engineering problem, you don't necessarily use a decimal representation. You may well want to represent pi as 3 or 4 or 22/7 or any other approximation that is good for your particular use case. Or you may even have usecases where you do things the opposite way - you may want to approximate 1 as pi/3 or something like that for certain kinds of problems (e.g if you're going to take the sin of your result).
  - chongli 1y ago
    Then you calculate the decimal expansion to the desired number of decimal places. This avoids accumulation of roundoff errors in intermediate results. Note that writing sqrt(2) as 1.41 or 1.41421 or any other decimal expansion you might want to write is incorrect: you will always get some roundoff error. If you want to calculate that sqrt(2)*sqrt(2)=2 then you can’t do so by multiplying the decimal expansions.
  - rob_c 1y ago
    You never evaluate symbols until your giving a numerical equivalent. Sure if a question asks for the escape velocity from Jupiter this has an approximate numerical value, but you don't just start by throwing numbers at a wall, you get the simplest equation which represents the value you're interested in an then evaluate it once you have a single equation for that parameter. Yes sqrt(2)*pi has a numerical approximation but you don't want that right at the start of taking about something like spin orbitals or momenta of spinning disks. Doing the latter compounds errors. It's no different to keeping around "i"/"j" until you need to express a phase or angle as it's cleaner and avoids compounding accuracy errors.
tsimionescu 1y ago
This is a very strange question. With repeating decimals, it is technically possible, though very complicated, to do arithmetic directly on the representations. You have to remember a bunch of extra rules, but it can be done. However, with numbers that have non-repeating inifinite decimal expansions, it is completely imposible to do arithmetic in the decimal notation. I'm not exagerating: it's literally physically impossible to represent on paper the result of doing 3pi in decimal notation in an unambiguous form other than 3pi. It's also completely impossible to use the decimal expansion of pi to compute that pi / pi = 1. Here, I'll show you what it would be like to try: pi / pi = 3.141592653589793238462643383279502884197169399375105820949445923078164062862089986280348253421170679821480865132820664709384460955058223172.... Now, of course you can do arithmetic with certain approximations of pi. For example, I can do this: pi / pi ≈ 3.1415 / 3.1415 = 1 Or even 3 × pi ≈ 3 × 3 = 9 But this is not doing arithmetic with the decimal expansion of pi, this is doing arithmetic with rational numbers that are close enoigh to pi for some purpose (that has to be defined).
- anthk 1y ago
  pi/pi would evaluate to 1 as most proper languages would deal with pi symbolically and not arithmetically.
  - tsimionescu 1y ago
    My point was only that even trivial arithmetic is impossible to do with the infinite decimal representations of irrational numbers.
  - librasteve 1y ago
    in raku say pi/pi; #1
    - lizmat 1y ago
      Sadly, that's just Num.gist showing 1.0 as "1" though. say (pi/pi).^name; # Num
anthk 1y ago
You would use rational approximations good enough for different scales and roundings.
cwmma 1y ago
in American math classes (as opposed to science classes) you almost never expand PI or sqrt(2), you either cancel them out or leave them in the answer until the end. Maybe if it's a word problem you sub them in the very last step but the problem itself is almost certainly going to be designed so it's not an issue.
- Suppafly 1y ago
  >in American math classes (as opposed to science classes) you almost never expand PI Except we have some fascination with memorizing the digits of pi and having competitions for doing so for some reason.
  - BobaFloutist 1y ago
    Just for fun.
  - mrguyorama 1y ago
    There is no school math test that will require you to know the digits of Pi, except as a silly extra credit question. The fascination is just dick measuring. "I'm smarter than you", for memorizing a longer string? It's quite dumb, but American media loves to use the dumbest possible ways of demonstrating that a character is intelligent, because uh it's really really hard to demonstrate "This person is very intelligent" to a subset of the population that is mostly at a middle school reading level and barely comprehends basic arithmetic, let alone algebra.
  - anthk 1y ago
    And that's useless for actual math.
    - Suppafly 1y ago
      >And that's useless for actual math. Agreed. The schools always seem to have these learning adjacent things that are theoretically supposed to make subjects engaging, but in reality are so disconnected from the subject that they are meaningless.

lcrz 1y ago

Both ways are just notation. There’s nothing more real about 3/10 compared to 0.3. Telling you otherwise might have worked as a educational “shorthand”, but there are no mathematical difficulties as long as you use good definitions of what you mean when you write them down. The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits.

tsimionescu 1y ago
I agree that ultimately both are just notations. I do think the fractional notation has some definite advantages and few disadvantages, so I think it's better to regard it as more canonical. I disagree though that it's necessary or even useful to think of 0.99... or 0.33... as sequences or limits. It's of course possible, but it complicates a very simple concept, in my opinion, and muddies the waters of how we should be using notions such as equality or inifinity. For example, it's generally seen as a bad idea to declare that some infinite sum is actually equal to its limit, because that only applies when the series of partial sums converges. It's more rigorous to say that sigma(1/n) for n from 2 going to infinity converges to 1, not that it is equal to 1; or to say that lim(sigma(1/n)) for n from 2 to infinity = 1. So, to say that 0.xxx... = sigma(x/10^n) for n from 1 to infinity, and to show that this is equal to 1 for x = 9, muddies the waters a bit. It still gives this impression that you need to do an infinite addition to show that 0.999... is equal to 1, when it's in fact just a notation for 9/9 = 1. It's better in my opinion to show how to calculate the repeating decimal expansion of a fraction, and to show that there exists no fraction whose decimal expansion is 0.9... repeating.
WorldMaker 1y ago
> The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits. Also a possible third thing: not enjoying working in a Base that makes factors of 3 hard to write. Thirds seem like common enough fractions "naturally" but decimal (Base-10) makes them hard to write. It's one of the reasons there are a lot of proponents of Base-12 as a better base for people, especially children, because it has a factor of 3 and thirds have nice clean duodecimal representations. (Base-60 is another fun option; it's also Babylonian approved and how we got 60 minutes and 60 seconds as common unit sizes.)

olau 1y ago

I was taught something of the same. But I think it was misguided. I'll note that 1/3 is not a number, it's a calculation. So more complicated. And fractions are generally much more complicated than the decimal system. Beyond some simple fractions that you're bound to experience in your everyday life, I don't think it makes sense to drill fractions. In the end, when you actually need to know the answer to a computation as a number, you're more likely to make a mistake because you spend your time juggling fractions instead of handling numerical instability. Decimal notation used to be impractical because calculating with multiple digits was slow and error-prone. But that's no longer the case.

tsimionescu 1y ago
This is ultimately a matter of definitions, and neither defining the fractions nor the decimals as the "true" representation of rationals is ultimately more or less correct. But, operations on fractions are definitely easier than operations on decimals. And fractions have the nice property that they have finite representations for all rational numbers, whereas decimal representations always have infinite representations even for very simple numbers, such as 1/3. Also, if you are going to do arithmetic with infinite decimal representations, the you have to be aware that the rules are more complex then simply doing digit-by-digit operations. That is, 0.77... + 0.44... ≠ 1.11... even though 7+4 = 11. And it gets even more complex for more complicated repeating patterns, such as 0.123123123... + 0.454545... (that is, 123/999 + 45/99). I doubt there is any reason whatsoever to attempt to learn the rules for these things, given that the arithmetic of fractions is much simpler and follows from the rules for division. The fact that a handful of simple cases work in simple ways doesn't make it a good idea to try.
volemo 1y ago
> I'll note that 1/3 is not a number, it's a calculation. So more complicated. 1/3 is a calculation the same way 42 is a calculation (4*10^1 + 2*10^0). Nothing is real except sets containing sets! /j
- DemocracyFTW2 1y ago
  Yes, true. *BUT* 1/3 is a fraction with denominator 3. 1/5 is a fraction with another denominator, and 1/7 has yet another. So how much is 1/3 + 1/5 + 1/7? You can't just add up, you first have to multiply to get to common ground. The decimal expansions of these use the same base and are readily comparable.

anthk 1y ago

Forth and Lisp users often try to use rational first and floats later. On Scheme Lisps, you have exact-inexact and inexact->exact functions which convert rationals to floats and viceversa.

Suppafly 1y ago

>So, in this view, 0.3 or 0.333... are not numbers in the proper sense, they're just a convenient notation for 3/10 and 1/3 respectively. And there simply is no number whose notation would be 0.999..., it's just an abuse of the decimal notation. I wonder if students from Romania are hamstrung in more advanced mathematics from being taught this way.

leereeves 1y ago
I don't think they would be; I think they might even have an advantage. They'd understand that the only numbers that have infinite repeating expansions are rationals, and that decimals are, in general, just approximations.
tsimionescu 1y ago
I don't think decimals, especially repeated decimals or other infinite decimal expansions, show up much if at all in any advanced math subjects beyond the study of themselves, of course). Higher math is almost exclusively symbolic. You're more likely to need to learn that "1" is just a notation for the set which contains the empty set then to learn that it's OK to add 0.22... + 0.44... = 0.66...

mcphage 1y ago

> we were always taught that fractions are the "real" representation of rational numbers There is no "real" representation of rational numbers, and fractions are no more real—or fake—than decimals. > And there simply is no number whose notation would be 0.999... There is, though. It's 1.