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.”)
...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?
Or a rational number whose decimal representation doesn't repeat?
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.
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.
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.
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.
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.
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.
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.
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)
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.
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>.
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.
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.
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.
I've heard there's some pushback against the Chinese policy in that some people in the west keep an unofficial local time which is widely understood and quoted (though presumably not for things that are sufficiently official or relevant to other regions). Apparently there's currently an ethnic conflict over the time zone status in Xinjiang:
https://en.wikipedia.org/wiki/Xinjiang_Time
Maybe this conflict has now been pushed underground by force?
> In 2018, according to Human Rights Watch, a Uyghur man was arrested and sent to a detention center because he set his watch to Xinjiang Time.
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.
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.
e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional
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.
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.
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 or https://en.wikipedia.org/wiki/P-adic_number or the complex numbers.
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.
Come to think of it, this is one way of thinking about the relationship between symbols and geometry.
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.
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....
Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians!
Yeah representation is not reality.
Now, says the ratio of those two measuring sticks ...
Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions.
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?
The Circus Animal's Desertion, W. B. Yeats