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

= 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). 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)). a and c can be any sign. -- ToE 14:11, 12 May 2015 (UTC)


 * As 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
 * $$\frac ab\ rel\ \frac cd$$
 * into equivalent
 * $$\left( \frac ab - \frac cd\right)\ rel\ 0$$
 * 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,, 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)