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Likewise, TIL. In your link however it states "Example 1: Numerical Context Let's consider two real numbers, a and b. If a is neither greater than nor less th
by namanyayg 3y ago
Likewise, TIL. In your link however it states
"Example 1: Numerical Context
Let's consider two real numbers, a
and b. If a is neither greater than nor less than b, but they aren't explicitly equal, the relationship is ≹"
How can that be possible?
- magicalhippo 3y agoPerhaps this: if they represent angles, then 1 and 361 represents the same absolute orientation, but they're not the same as 361 indicates you went one full revolution to get there. Contrived, but only thing I could think of.
- thaumasiotes 3y agoIt isn't. The real numbers are a totally ordered field. Any two real numbers are comparable to each other.
- shaunregenbaum 3y agoI would imagine trying to compare a purely imaginary number (3i) to a real number (3) would suffice.
- thaumasiotes 3y agoAn imaginary number wouldn't obey the stated constraint of being real.
- coldtea 3y agoNo, but if the parent's question goes beyond "how can this happen with reals" to "how can this happen with numbers in general", this answers his question.
- thaumasiotes 3y agoThe very next example on the page is "imagine two complex numbers with the same magnitude and different angles". For that to answer the parent's question, you'd have to assume he stopped reading immediately after seeing the part he quoted. The question is why the page says "imagine two real numbers that aren't comparable".
- rustybolt 3y agoI think they mean the case where a and b are variables for which you don't know the values.
- didntcheck 3y agoYeah, that's how I understood it. E.g one might write (a+b)^2 != a^2 + b^2 To mean that in general the equality doesn't hold. Despite exceptions like a=b=0 Strictly you should write something like ¬[∀ a,b (a+b)^2 != a^2 + b^2] But shorthand and abuse of notation are hardly rare
- didntcheck 3y agoNoticed a copy and paste error too late - the != in the second expression should of course be =
- coldtea 3y agoNo, that can be the case also for mathematical entities for which you can know the values, not just for "unknown variables".
- salmonellaeater 3y agoThat doesn't seem possible with the reals. An example from programming that comes to mind is NaN ≹ NaN (in some languages).
- bruce343434 3y agoThe floating point specification mandates that nan does not compare to nan in any way, so it should be all languages. If you want to know nam, use isnan()
- jampekka 3y agoIsn't this what happens with infs (in maths and in many programming languages)? Edit: Not in many programming languages. In IEEE-754 inf == inf. In SymPy too oo == oo, although it's a bit controversial. Feels sketchy.
- yummypaint 3y agoThis doesn't really pass the small test for me either, but to play devils advocate: Imagine you have 2 irrational numbers, and for some a priori reason you know they cannot be equal. You write a computer program to calculate them to arbitrary precision, but no matter how many digits you generate they are identical to that approximation. You know that there must be some point at which they diverge, with one being larger than the other, but you cannot determine when or by how much.
- MikeDelta 3y agoMaybe you will find the proof that the infinite series 0.9999... exactly equals 1 interesting: https://en.wikipedia.org/wiki/0.999 https://en.wikipedia.org/wiki/0.999...
- jcul 3y agoWow, can't believe I've never realised this. How counterintuitive. The 1/3 * 3 argument, I found the most intuitive.
- quchen 3y agoIt's a flawed psychological argument though, because it hinges on accepting that 0.333...=1/3, for which the proof is the same as for 0.999...=1. People have less of a problem with 1/3 so they gloss over this - for some reason, nobody ever says "but there is always a ...3 missing to 1/3" or something.
- Scarblac 3y agoThe problem is that there are two different ways to write the same number in infinite decimals notation. (0.999... and 1.000...). Thats what's counter intuitive to people, it's not an issue with 1/3. That has just one way to write it as decimals, 0.333...
- thaumasiotes 3y agoI like the argument that observes "if you subtract 0.99(9) from 1, you get a number in which every decimal place is zero". The geometric series proof is less fun but more straightforward. As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.
- omeid2 3y agoThink how any number on the z axis of complex plain isn't equal to the a number of same magnitude, on x and y axis. Now if you really think about, a number of a given magnitude on x axis also isn't exactly "equal" to a name of same magnitude on y axis or vice versa. Other wise, -5 and 5 should be equal, because they're the same magnitude from 0.
- bruce343434 3y agoBut |5|=|-5| so I don't exactly see your point. Edit: oh, I see what you mean. 1 is not larger or smaller than i, but it also doesn't equal i.
- ska 3y agoThis is just imposing an chosen (and in this case, pretty natural) order on a set without total order. But as you note, you have to give something up. In this case the order identifies each number on a circle in the complex plane with a single value, their magnitude. I've never really seen this notation used, but it could have some use in partially-ordered sets.
- omeid2 3y agoIt is formalisms all the way down.
- kevindamm 3y agofor the Reals, it is only hypothetical, the domain has a total order. \inf and $\inf + 1$ comes to mind but I don't think it really counts
- thaumasiotes 3y ago> \inf and $\inf + 1$ comes to mind but I don't think it really counts That just depends on the numeric structure you're working with. In the extended reals, +inf is equal to +inf + 1. In a structure with more infinite values than that, it would generally be less. But they wouldn't be incomparable; nothing says "comparable values" quite like the pair "x" and "x + 1".
- kaba0 3y agoI guess it depends on the exact definitions, but reals usually doesn’t include the infinities. At my uni we introduced infinities precisely as an extension of the reals with two values defined by `lim`.
- kaba0 3y agoIt is false - real numbers fulfill the trichotomy property, which is precisely the lack of such a relationship: every two real number is either less than, equal or greater than. But the numerical context can still be correct: (edit: ~~imaginary~~) complex numbers for example don’t have such a property.
- fsckboy 3y agoi'm learning about this right here as I read, but do you mean complex numbers rather than imaginary?
- kaba0 3y agoYep, that’s what I meant, sorry!
- TeMPOraL 3y agoSame question. Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V. Is this is what ≹ is talking about?
- shemii 3y agoAs I've written in another comment here, a great example of a number-y field which is not totally ordered is Games ⊂ Surreal Numbers ⊂ ℝ. There you have certain "numbers" which can can confused with (read incomparable) whole intervals of numbers. Games are really cool :)
- SetTheorist 3y agoYou've got the subset relationships backwards: Reals are a subset of the field of Surreal Numbers which is a subset of the group of Games. (Probably better phrased as embeddings, rather than subsets, but the point remains...) Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.
- runiq 3y agoHmm, what about +0 and -0?
- RedNifre 3y ago+0 = -0
- tpoacher 3y agoProbably does not apply for real numbers, but could totally apply to, e.g., fuzzy numbers, whose 'membership function' bleeds beyond the 'crisp' number into nearby numbers. You could imagine two fuzzy numbers with the same 'crisp' number having different membership profiles, and thus not being "equal", while at the same time being definitely not less and not greater at the same time. Having said that, this all depends on appropriate definitions for all those concepts. You could argue that having the same 'crisp' representation would make them 'equal' but not 'equivalent', if that was the definition you chose. So a lot of this comes down to how you define equality / comparisons in whichever domain you're dealing with.