Wikipedia:Reference desk/Archives/Mathematics/2014 August 30

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August 30

Integers/whole numbers vs decimals

The advantage of using integers instead of decimals would seem obvious to most (9 mm instead of 0.09 cm, 1500 metres instead of 1.5 kilometres). But is preference for integers/whole numbers over decimals when using SI units an established principal?--Gibson Flying V (talk) 03:22, 30 August 2014 (UTC)

It is more that people like to use a system where their measurements have a whole number part but not be too big and to use the largest unit like that they can. 1500 meters is an example where one tries as far as possible to use the same scale for all ones measurements. In athletics one would say 1500 meters but in a car one might say 1.5 kilometers. Dmcq (talk) 07:31, 30 August 2014 (UTC)

Right, but for whatever reason 9mm and 1500m were chosen. Similarly, drinks are in 700ml bottles, not 0.7l bottles, snacks are in 200g packs, not 0.2kg packs, films are 90 minutes, not 1.5 hours. It seems that where integers can be used, they are, and I was curious to know from those knowledgeable in mathematics if this apparent preference has ever been acknowledged anywhere (or does it just go without saying).--Gibson Flying V (talk) 07:40, 30 August 2014 (UTC)

Note that 9 mm = 0.9 cm, (not = 0.09 cm). Integers are more elementary and were historically used before fractions, and so an integer number of subunits were preferred to fractions of larger units. The prefix c = 0.01 is usually considered part of the unit, cm = 0.01 m, rather than part of the number, 0.9c = 0.009 . Of course 0.9 cm = 0.9c m. Bo Jacoby (talk) 20:25, 30 August 2014 (UTC).

• Medical professionals are taught to avoid working with decimals, particularly when measuring dosages.[1][2][3][4][5]
• The UK Metric Association's Measurement units style guide says, "Use whole numbers and avoid decimal points if possible - e.g. write 25 mm rather than 2.5 cm."
• In his book entitled The Fear of Maths: How to Overcome It Steve Chinn opens the chapter entitled "Measuring" with I am sure that most people would rather avoid decimals and fractions. This is the reason we have "pence" rather than "one hundredths of a pound". The metric system allows us to avoid decimals by using a prefix instead of a decimal point. If £1 is the basic unit of money, then 1 metre is now the basic unit of length. The metre is too long for some measurements, so we use prefixes, as in "millemetre" as a way of dealing with fractions of a metre.
• This article cites the Australian construction industry's standardisation on millemetres for all measurements in 1970 as having saved it 10-15% in construction costs due to the eilimination of errors associated with decimals.

That's all I could find so far.--Gibson Flying V (talk) 01:12, 31 August 2014 (UTC)

Absurd or meaningless rate

I couldn't decide what desk to post this question to. It's kind of a logical/mathematical question but it's also a semantic/linguistic question, so if this is the wrong place to ask this question, please forgive.

Consider the following statements: 1) "I can run fast, up to 10 miles an hour" 2) "I can run at least one mile in at least an hour"

The first statement refers to a maximum possible speed or rate or ratio. But the second statement appears to be absurd or meaningless (I think). Can someone explain to me in a quasi-systematic way *why* the second statement is meaningless.--Jerk of Thrones (talk) 06:51, 30 August 2014 (UTC)

The Humanities reference desk would probably have been the right place for a question like this.

The first asserts that you can run at that speed for a short distance at least. The second is not meaningless, it says you can run one whole mile but sets no limit on the speed. The meaningless bit is because of the very reasonable expectation that the speaker actually meant something more otherwise they wouldn't have said so many unnecessary words, that implies they made a mistake in what they said. In English that sort of sentence can easily be the result of a common habit of duplicating a superlative and one would suppose they just made a mistake and meant "I can run at least one mile in an hour", but there may be some other explanation depending on the circumstances. Dmcq (talk) 07:21, 30 August 2014 (UTC)

It is absurd because it seems as if it should be a statement about how fast someone can run, but isn't. It could be paraphrased as 'I can run for some unspecified distance of a mile or more - but it will take me an hour or more to do it.' It isn't actually meaningless, just less informative than it first appears. AndyTheGrump (talk) 07:24, 30 August 2014 (UTC)

I think the odd part is claiming one can move a mile in a period of time without any upper limit. Unless the person is infirm, that should be true of everyone. Of course, just what constitutes "running" is open for debate, but most wouldn't call a mile in an hour to be a run at all, only a slow walk. If you said it as "I can travel at least a mile in at least an hour", then that might be a reasonable statement from somebody with some type of injury, or carrying a heavy load. StuRat (talk) 02:35, 31 August 2014 (UTC)

Running vs. walking isn't defined by the speed, but by the gait. When walking you have 1-2 feet on the ground at any time, when running you have 0-1. -- Meni Rosenfeld (talk) 07:50, 2 September 2014 (UTC)

It's defined by both: [6]. There's not much point to using a running gait when moving that slowly. Even joggers move faster than that. StuRat (talk) 14:45, 2 September 2014 (UTC)

The meaninglessness comes with both the over-generalization of the sentence (mentioned above, effectively weakening the statement to "I can run 1 mile before I die") and the contrast with the listener's expectation ("...in at least one hour? That doesn't help one bit").

Advertisers do this a lot, throwing a heap of positive-sounding phrases which don't actually synergize at the audience. ("Save up to 50%, and more" is the textbook example. It could be 50%, 99%, or only 1%, and due to the illogical structure of their promises, they didn't really lie even if most customers save much less than 50%.

Some politicians use similar patterns, usually for similar reasons (to suggest, rather than actually make, promises).

Sometimes employed for comedy ("A messy death is the last thing that could happen to you" – literally) or by a "lawful" character who would never lie. TV Tropes calls it a "false reassurance" . - ¡Ouch! (^{hurt me} / _{more pain}) 10:43, 1 September 2014 (UTC)

That's a good one. Absolutely true but conveying no information. I like those in my speech, like 'If I don't go to sleep I'll never wake up in the morning'. I think there's a term for those but I've forgotten it. Dmcq (talk) 11:07, 1 September 2014 (UTC)

Sports produce a lot of those, like "The reason we lost is that they scored more points than us." StuRat (talk) 17:07, 5 September 2014 (UTC)

Coequalizer

I read the article Coequalizer, and feel a little bit stupid, because even after repeatedly thinking about it, it evades my grasp.

The article tells me:

In the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the smallest equivalence relation $~\sim$ such that for every $x\in X$, we have $f(x)\sim g(x)$. In particular, if R is an equivalence relation on a set Y, and r₁, r₂ are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r₁ and r₂ is the quotient set Y/R.

Firstly I have trouble understanding what the smallest equivalence relation is. I assume, it's the finest?

To make a simple example, assume X=Y is the set of real numbers and $f(x)=|x|$ and $g(x)=x$. What would be the coequalizer? 77.3.137.128 (talk) 13:08, 30 August 2014 (UTC)

Yes, smallest means finest. The term smallest is justified by thinking of an equivalence relation as a set of pairs. Then the smallest one with property X is the intersection of all equivalence relations with property X.

Another way to view it is to start with $f(x)\sim g(x)$ for all $x$, then make it reflexive and symmetric and close under transitivity.

Using your example, for every nonnegative $x$, $x=f(-x)\sim g(-x)=-x$, so we start with $x\sim -x$ for all $x$. Of course, we also add symmetry and reflexivity. Normally we'd need to close under transitivity, but this is already transitive. So now we take the quotient of the reals by this, which gets us a set which can be naturally identified with the nonnegative reals.--80.109.106.3 (talk) 14:38, 30 August 2014 (UTC)

Excuse me, it really looks like I have some extraordinarily mental block on that subject. Please tell me what the morphism of this coequalizer would be. 77.3.137.128 (talk) 14:57, 30 August 2014 (UTC)

I'm not sure what morphism you're asking for. The equivalence relation from your example is given by $x\sim y$ if $x=y$ or $x=-y$. We get the coequalizer by taking the quotient of the reals by this, so the coequalizer is the set $\{\{x,-x\}\ :\ x\in \mathbb {R} _{\geq 0}\}$. The natural identification I mentioned earlier is given by $\{x,-x\}\mapsto x$.--80.109.106.3 (talk) 17:04, 30 August 2014 (UTC)

Thank you so far. I guess my problem is some misunderstanding deep inside my head, probably mixing limits and colimits. At least I now have an example that is not tainted by this fault inside my brain. Thanks. 77.3.137.128 (talk) 20:18, 30 August 2014 (UTC)

Got it! I finally got my brain bug fixed. Having been trained on resolving equations, my mind was tied on thinking about the domain, but, as the name co-equalizer strongly suggests, we are rather forcing equality on the codomain. Nice koan. 95.112.216.113 (talk) 08:53, 31 August 2014 (UTC)

{{reflist-talk}} added here for clarity 71.20.250.51 (talk) 11:58, 31 August 2014 (UTC)

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