7 ms·

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number. I don't know if it's because decimals resemble

by lmkg 7y ago

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number. I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

Sharlin 7y ago
Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.
- tom-thistime 7y ago
  Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.
  - Sharlin 7y ago
    Every number is a limit, yes, but people think of 0.999… as a series. Not rigorously, of course, but that's a common argument even by people (perhaps especially by people) who have a highschool or even math-minor level understanding of series and limits.
tom-thistime 7y ago
Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element." You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.
- remexre 7y ago
  I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true" by the rules of the system. e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional
  - tom-thistime 7y ago
    Yeah forcing myself to keep making sure I'm defining "equality" seems to be a really useful way of "breaking my brain" until I understand what an algebraic structure "really is."
- pasquinelli 7y ago
  I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.
  - karatestomp 7y ago
    I think a lot of people see “find me a number between 0.9999... and 1” as no more or less valid (plus, perhaps, no more or less pedantic and dickish) than “ok smarty pants, why don’t you just keep adding 9s until you reach 1, then we’ll talk.”
  - tom-thistime 7y ago
    I imagine that's true sometimes. On the other hand, hypothetically, if it were the case that we were missing a key definition that's needed for some proof, and someone didn't believe that proof, then maybe we would't quite know enough yet to decide that the person is in mental kindergarten. A better first step for us might be to supply the missing definition.
L_226 7y ago
Reminds me of Terry Pratchett's description of the mathematical reasoning of camels in Pyramids: Lack of fingers was another big spur to the development of camel intellect. Human mathematical development had always been held back by everyone’s instinctive tendency, when faced with something really complex in the way of triform polynomials or parametric differentials, to count fingers. Camels started from the word go by counting numbers.
mjd 7y 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 7y 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 7y 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 7y ago
      I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.
Bodell 7y ago
"Players and painted stage took all my love, And not those things that they were emblems of." The Circus Animal's Desertion, W. B. Yeats
theincredulousk 7y ago
great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in base-10 decimal.
- scrollaway 7y ago
  Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
  - jacobparker 7y ago
    Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).
  - Koshkin 7y ago
    The Babylonians though it was 60. (But imagine having to remember additional 50 numeric characters.)
    - bmn__ 7y ago
      > remember additional 50 numeric characters The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember. https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg https://commons.wikimedia.org/wiki/File:Babylonian_numerals....
- throwaway894345 7y ago
  I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).
obastani 7y ago
This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard. The simplest definition is: a finite decimal ak ... a1.b1 ... bh is defined to be a fraction and an infinite decimal is defined to be a limit. You'd still have to define what a limit is, but that is somewhat more intuitive. [1] https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut [2] https://en.wikipedia.org/wiki/Hyperreal_number https://en.wikipedia.org/wiki/Hyperreal_number
- btilly 7y ago
  Sorry, no. There are TWO standard definitions of the real numbers. Namely Dedekind cuts and Cauchy sequences. (They are completely equivalent.) The usual decimal representation of a number turns out to be a Cauchy sequence. The "simplest definition" that you provide turns out to be rather non-simple in practice. Try proving that multiplication is commutative to see the difficulty. There are plenty of other number systems out there. Try https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number or https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number or the complex numbers. https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number
jordigh 7y ago
Decimals are easier to compare than fractions. I can't easily work out in my head which one is bigger, 457/790 or 580/924, but I can easily see that (approximately) 0.57848 is smaller than (approximately) 0.62771. Since the fundamental thing most people want to do with numbers is see which one is bigger, they favour decimal expansions. And since decimals worked so well for fractions, why not use them for everything else?
BurningFrog 7y ago
Similarly, perhaps, many take for granted that the written version of a word, in current standard spelling is the word.

by lmkg 7y ago

Sharlin 7y ago

Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.

tom-thistime 7y ago
Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.
- Sharlin 7y ago
  Every number is a limit, yes, but people think of 0.999… as a series. Not rigorously, of course, but that's a common argument even by people (perhaps especially by people) who have a highschool or even math-minor level understanding of series and limits.

tom-thistime 7y ago

Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element." You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.

remexre 7y ago
I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true" by the rules of the system. e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional
- tom-thistime 7y ago
  Yeah forcing myself to keep making sure I'm defining "equality" seems to be a really useful way of "breaking my brain" until I understand what an algebraic structure "really is."
pasquinelli 7y ago
I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.
- karatestomp 7y ago
  I think a lot of people see “find me a number between 0.9999... and 1” as no more or less valid (plus, perhaps, no more or less pedantic and dickish) than “ok smarty pants, why don’t you just keep adding 9s until you reach 1, then we’ll talk.”
- tom-thistime 7y ago
  I imagine that's true sometimes. On the other hand, hypothetically, if it were the case that we were missing a key definition that's needed for some proof, and someone didn't believe that proof, then maybe we would't quite know enough yet to decide that the person is in mental kindergarten. A better first step for us might be to supply the missing definition.

L_226 7y ago

Reminds me of Terry Pratchett's description of the mathematical reasoning of camels in Pyramids: Lack of fingers was another big spur to the development of camel intellect. Human mathematical development had always been held back by everyone’s instinctive tendency, when faced with something really complex in the way of triform polynomials or parametric differentials, to count fingers. Camels started from the word go by counting numbers.

mjd 7y 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 7y 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 7y 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 7y ago
    I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.

Bodell 7y ago

"Players and painted stage took all my love, And not those things that they were emblems of." The Circus Animal's Desertion, W. B. Yeats

theincredulousk 7y ago

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in base-10 decimal.

scrollaway 7y ago
Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
- jacobparker 7y ago
  Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).
- Koshkin 7y ago
  The Babylonians though it was 60. (But imagine having to remember additional 50 numeric characters.)
  - bmn__ 7y ago
    > remember additional 50 numeric characters The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember. https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg https://commons.wikimedia.org/wiki/File:Babylonian_numerals....
throwaway894345 7y ago
I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).

obastani 7y ago

This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard. The simplest definition is: a finite decimal ak ... a1.b1 ... bh is defined to be a fraction and an infinite decimal is defined to be a limit. You'd still have to define what a limit is, but that is somewhat more intuitive. [1] https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut [2] https://en.wikipedia.org/wiki/Hyperreal_number https://en.wikipedia.org/wiki/Hyperreal_number

btilly 7y ago
Sorry, no. There are TWO standard definitions of the real numbers. Namely Dedekind cuts and Cauchy sequences. (They are completely equivalent.) The usual decimal representation of a number turns out to be a Cauchy sequence. The "simplest definition" that you provide turns out to be rather non-simple in practice. Try proving that multiplication is commutative to see the difficulty. There are plenty of other number systems out there. Try https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number or https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number or the complex numbers. https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number

jordigh 7y ago

Decimals are easier to compare than fractions. I can't easily work out in my head which one is bigger, 457/790 or 580/924, but I can easily see that (approximately) 0.57848 is smaller than (approximately) 0.62771. Since the fundamental thing most people want to do with numbers is see which one is bigger, they favour decimal expansions. And since decimals worked so well for fractions, why not use them for everything else?

BurningFrog 7y ago

Similarly, perhaps, many take for granted that the written version of a word, in current standard spelling is the word.