Testwiki:Reference desk/Archives/Mathematics/2015 May 12

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < May 11 ! width="25%" align="center"|<< Apr | May | Jun >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

May 12

Butterfly Method

I would like to know more about why the "butterfly method" works when comparing fractions. The procedure works, but I would like to know the concept. For example, (and feel free to put this in that fancy wikipedia math font)

4/10 (>,<,=) 6/9

I can cross multiply and the side with the greater product is also the greater fraction.

So in this example the comparison becomes

4/10 (>,<,=) 6/9 ---> 4 x 9 (>,<,=) 10 x 6 ---> 36 (>,<,=) 60 ---> 36 < 60

Generally it's:

a/b (>,<,=) c/d ---> ad (>,<,=) bc

I found this very cryptic and ungrammatical answer, which I could not decipher. Any help is appreciated. Thank you! — Preceding unsigned comment added by 66.226.194.210 (talk) 12:44, 12 May 2015 (UTC)

This follows from Equality_(mathematics)#Some_basic_logical_properties_of_equality. Let's use '?' to mean either equality or inequality, which you've written as (>,<,=). So we have $\frac{a}{b}?\frac{c}{d}\to \frac{ad}{b}?\frac{cd}{d}\to \frac{ad}{b}?\frac{c}{1}\to \frac{adb}{b}?cb\to ad?cb$, which works because at each step, we multiplied both sides of the "equation" by the same thing, which preserves the (in)equality. Note that multiplication by (-1) reverses the inequality, so your "butterfly" method will only work for positive numbers a,b,c,d, unless you have a convention to switch the sign. Does that make sense? SemanticMantis (talk) 13:16, 12 May 2015 (UTC)

More properly, for positive numbers b and d (per Inequality (mathematics)#Multiplication and division). a and c can be any sign. -- ToE 14:11, 12 May 2015 (UTC)

As Template:Ping noted above, multiplying both sides is dangerous with negative numbers — so one should avoid it as long as possible. First let's recall that adding a number to both sides does not change the equality, and does not reverse the inequality direction. So for any of three operators $\cancel{op\in \{<,=,>\}}$ relations $rel\in \{<,=,>\}$ we can safely convert the (in)equality

$\frac{a}{b}rel\frac{c}{d}$

into equivalent

$(\frac{a}{b}-\frac{c}{d})rel0$

just by subtracting the RHS expression from both sides, and further into

$\frac{ad-bc}{bd}rel0$

For positive denominator $(bd)>0$ that corresponds to

$(ad-bc)rel0$

and

$(ad)rel(bc)$

which is the main part of the answer. However the red condition reveals the other way, where a negative sign of one of original denominators causes that answer false due to reversing the inequality direction. --CiaPan (talk) 14:26, 12 May 2015 (UTC)

Just for the heck of it, I'll remind us all that {=,<,>} are all relations, not operators, and equality is a common example of an equivalence relation. Of course it doesn't really matter for your explanation, but we might as well use the right terms :) SemanticMantis (talk) 14:42, 12 May 2015 (UTC)

All 'op' replaced with 'rel'. Thank you, Template:Ping, for pointing out my mistake! --CiaPan (talk) 18:45, 12 May 2015 (UTC)

Cheers, nice \cancel by the way -- I learned some new LaTeX in trade :) SemanticMantis (talk) 23:51, 12 May 2015 (UTC)

OP here, thanks all. It makes sense now. — Preceding unsigned comment added by 66.226.194.210 (talk) 14:45, 13 May 2015 (UTC)