Wikipedia:Reference desk/Archives/Mathematics/2014 April 26

= April 26 =

1/2÷1/2
In mathematics, 1/2÷1/2=1, but if you take an apple, and divide it in half, you get half an apple. Then if you take that half and divide it half again, you get 1/4 an apple. So then why in mathematics is 1/2÷1/2=1? ScienceApe (talk) 17:08, 26 April 2014 (UTC)


 * Dividing in half is not the same as dividing by one half. If you divide one half in half, you get (1/2)/2=1/4. 1/2 divided by 1/2 gives you the answer of how many times 1/2 is 1/2, which is 1. —Kusma (t·c) 17:44, 26 April 2014 (UTC)


 * Dividing in half is the same as dividing by 2 (or multiplying by a half). This is the multiplicative inverse of dividing by a half (or multiplying by 2).    D b f i r s   18:17, 26 April 2014 (UTC)


 * "dividing in half" is x divide 2.
 * "dividing by half" is x divide (1/2) or x times 2
 * 24.192.240.37 (talk) 20:09, 26 April 2014 (UTC)


 * Could you say that in English, please?   D b f i r s   20:44, 26 April 2014 (UTC)


 * You don't speak Reverse Polish? --ColinFine (talk) 21:54, 26 April 2014 (UTC)


 * There's a similar English problem with percentages:


 * $10 increased by 150% = $25.


 * $10 increased to 150% = $15. StuRat (talk) 16:42, 28 April 2014 (UTC)


 * Now to give a real world example. Say you have half a pizza, and ask "how many times can that half a pizza be divided up into half a pizza ?".  The answer, of course, is one time. StuRat (talk) 12:53, 27 April 2014 (UTC)


 * Yes, I see the problem now: dividing an apple "in two" and "by two" are both the same as dividing it "in half", but dividing "by half" is the inverse. It's just a quirk of the language, but it must confuse many people, including some native speakers of English.    D b f i r s   08:42, 28 April 2014 (UTC)


 * The truth is that there is no reason that it has to make any sense. If you are paying your friend a debt of 4 (you have -4) then when you are done paying half, you have -2 since (-4 * (1/2) = -2).  If you are dividing a pizza in 2, (1 ÷ 2 ) then if you only do a quarter of the division you should have 2 pizzas (1 ÷ (2/4)) since 2/4 is a quarter of 2.  But that means that when you are a quarter of the way dividing a pizza in half, you temporarily have two whole pizzas.  When you are halfway through dividing a pizza in half, you have a full pizza.  These are obviously absurd results that don't make sense.


 * We therefore realize that the symbolism is arbitrary. What do you think of this reasoning, guys?  91.120.14.30 (talk) 14:32, 28 April 2014 (UTC)


 * I don't see what's so absurd about the first one, if you take negative money as representing a debt (which you did). Then you owe your friend 2 instead of 4 after your partial repayment, and hence have −2.
 * Could you explain what you meant by doing only a quarter of the division? I'm not sure if I understand you. Double sharp (talk) 14:52, 28 April 2014 (UTC)


 * While advanced math often no longer has any relation to the physical world, I don't see a problem with dividing by fractions. It takes a bit more thought, but it's quite doable.  In your (1 ÷ (2/4)) pizza case, the way to phrase that in English is "Divide one pizza up into fourths.  How many groups of two/fourths do you now have ?". StuRat (talk) 16:38, 28 April 2014 (UTC)


 * I'm rather surprised that someone who asks intelligent questions elsewhere is so confused by fractions. May I politely ask 91.120.14.30, are you just trolling us for fun, or would you like us to find a website that explains division of fractions for you?  The concept of a division only a quarter completed seems nonsense to me.     D b f i r s   17:24, 28 April 2014 (UTC)


 * You're right - I just wasn't thinking about it clearly. There is nothing suprising.  "How many quarters of a pizza are there in a pizza" is a division problem: 1 ÷ (1/4) and clearly has the answer 4 - just not "4 pizzas" but "4 pieces-of-a-pizza." So is "How many quarters of a pizza are there in 2 pizzas" which is 2 ÷ (1/4).  Neither result is surprising.  The only mistake I made is in calling the latter result "8 pizzas" rather than "8 pieces of pizza". 91.120.14.30 (talk) 09:08, 29 April 2014 (UTC)


 * This is a very common type of set problem about fractions, and probably as many people get it wrong as the Monty Hall problem and need it explained. Luckily quite a few eventually agree with the result rather than still disputing it like that famous problem! Anyway the first question to ask so one knows there is a potential problem with reasoning is if dividing a half by two gives a quarter, then is it likely that dividing it by a half which is a totally different number will give the same result? Dmcq (talk) 19:11, 28 April 2014 (UTC)


 * If you want to talk about doing a fraction ("a quarter") of a multiplication ("of the division"), you need to work in logarithmic space to get the natural symmetries. Then doing a quarter of halving is multiplying by $$2^{-1/4}\approx0.841$$.  Do that twice and you will have multiplied by $$\sqrt{1/2}\approx0.707$$; do it four times and you will have halved the original.  --Tardis (talk) 02:07, 29 April 2014 (UTC)


 * True, but in "real space", that's just multiplying by the square root of the square root of a half. ( Are we not confused enough already? )    D b f i r s   07:18, 30 April 2014 (UTC)

Variant of the modulo operation
Is there a name for this function?

$$f(a,n) = \begin{cases} a\,\bmod\,n, & n \nmid a \\ n, & n \mid a\end{cases}$$

Jackmcbarn (talk) 20:59, 26 April 2014 (UTC)


 * Not this one per se, but if you fix n, then the function $$f(a) := f(a,n)$$ is a lift of the canonical projection map $$\mathbb Z\to\mathbb Z_n$$ that associates to each integer in $$\mathbb Z$$ the corresponding element of the residue class group $$\mathbb Z_n$$. There are many lifts.  The modulo map is one, and the one you wrote down is.  All lifts are of the form
 * $$f(a)= (a\,\bmod\,n) + k(a)n$$
 * for some (arbitrary) function $$k:\mathbb Z_n\to\mathbb Z$$.  Sławomir Biały  (talk) 21:31, 26 April 2014 (UTC)
 * You can get rid of the cases by writing $$f(a,n)=(a-1\,\bmod\,n)+1$$. —Kusma (t·c) 12:52, 27 April 2014 (UTC)
 * I hadn't seen that particular notation before, with $$\nmid$$ and $$\mid$$, but Kusma's simplification makes everything much clearer. Is this notation common? -- The Anome (talk) 15:25, 27 April 2014 (UTC)
 * That's not the same function. In my function, f(10.5, 10) = 0.5, but in yours, f(10.5, 10) = 10.5. (Perhaps I should have specified that I wanted this to work on non-integers, since it seems that's not always the case with mod). I guess $$f(a,n)=\lim_{x \to 0^+}(((a-x)\,\bmod\,n)+x)$$ would work, but at that point it's just as complicated as the piecewise definition. Jackmcbarn (talk) 17:31, 27 April 2014 (UTC)
 * (ec) I think people here were probably assuming you were working over the integers in which case Kusma is correct. If your working with reals you probably want something like
 * $$r = a - n \left\lceil {a \over n} \right\rceil + n.$$
 * using the ceiling function. Although you want to check you get the right answer for negative values.--Salix alba (talk): 18:12, 27 April 2014 (UTC)
 * I had assumed this, since the notation $$\mid,\nmid$$ certainly suggests that the arguments are integers. But my response only requires a small change.  Then the residue class group is replaced by the quotient group $$\mathbb R/n\mathbb Z$$ of the reals by the lattice $$n\mathbb Z$$.  There is a natural projection $$\mathbb R\to\mathbb R/n\mathbb Z$$ and the function f is a lift that is continuous at every point except the identity.   Sławomir Biały  (talk) 00:28, 28 April 2014 (UTC)