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

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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 @SemanticMantis: 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

$\left({\frac {a}{b}}-{\frac {c}{d}}\right)\ rel\ 0$

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

${\frac {ad-bc}{bd}}\ rel\ 0$

For positive denominator $\color {Red}(bd)>0$ that corresponds to

$(ad-bc)\ rel\ 0$

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, @SemanticMantis:, 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)