I don't quite understand how to get the last line in this problem.
I'm assuming the bd = db kind of multiplication is already assumed to be commutative, or else you'd have yourself a circular argument. Multiply out the last line (foil it), and you'll get the second-to-last line. So, from the second-to-last line to the last line is a matter of factoring. Since foiling it out gets you the desired result, the method of factoring, whatever it was, was valid.
2nd line to the last line....
Split 2 into $\displaystyle 2=\sqrt{2}\sqrt{2}$ first
$\displaystyle ac+\left(\sqrt{2}\right)b\left(\sqrt{2}\right)d+\l eft(\sqrt{2}\right)ad+\left(\sqrt{2}\right)bc$
$\displaystyle =a(c+\sqrt{d})+\left(\sqrt{2}\right)b(c+\sqrt{d})$
However, there is no proof of commutativity in any of that example.
Hello, coopsterdude!
I don't understand the proof . . . It's very sloppy!
$\displaystyle \begin{array}{cccc}(c+d\sqrt{2})(a+b\sqrt{2}) &= & ca + cb\sqrt{2} + da\sqrt{2} + 2bd & [?]\\ \\
&=& (ac+2bd) + (ad+bc)\sqrt{2} & [?]\\
&& \!\!\!\!\uparrow \\ \\[-3mm]
&=& (a+b\sqrt{2})(c+d\sqrt{2}) \end{array}$
We are supposed to prove that multiplication is commutive.
But in the proof, they already assumed that mult'n is commutative
. . . as Ackbeet already pointed out.
The first step should look like this:
. . $\displaystyle (c+d\sqrt{2})(a+b\sqrt{2}) \;=\; (c)(a) + (c)(b\sqrt{2}) + (d\sqrt{2})(a) + (d\sqrt{2})(b\sqrt{2})$
They simplified the third term like this:
. . $\displaystyle \begin{array}{ccccccc}
(d\!\cdot\!\sqrt{2})\!\cdot\!(a) &=& d\!\cdot\!(\sqrt{2}\!\cdot\!a) & \text{assoc.} \\
& =& d\!\cdot\!(a\!\cdot\!\sqrt{2}) & \text{comm.} \\
& = & (d\!\cdot\!a)\!\cdot\!\sqrt{2} & \text{assoc.} \\
& = & da\sqrt{2}\end{array}$
They simplified the fourth term like this:
. . $\displaystyle \begin{array}{ccccc}
(d\!\cdot\!\sqrt{2})\!\cdot\!(b\!\cdot\!\sqrt{2}) &=& d\!\cdot\!(\sqrt{2}\!\cdot\!b)\!\cdot\!\sqrt{2} & \text{ assoc.} \\
& = & d\!\cdot\!(b\!\cdot\!\sqrt{2})\!\cdot\!\sqrt{2} & \text{comm.} \\
& = & (d\!\cdot\!b)\!\cdot\!(\sqrt{2}\!\cdot\sqrt{2}) & \text{assoc.} \\
& = & (b\!\cdot\!d)\!\cdot\!(2) & \text{comm.} \\
& = & 2bd & \text{comm.}\end{array}$
In the next step, they assume that: .$\displaystyle ca \,=\,ac,\;\;da \,=\,ad.$
The second line should be:
. . $\displaystyle ac + bc\sqrt{2} + ad\sqrt{2} + 2bd $
. . . .$\displaystyle =\;(ac + bc\sqrt{2}) + (ad\sqrt{2} + 2bd) $
Factor: .$\displaystyle c(a+b\sqrt{2}) + d\sqrt{2}(a + b\sqrt{2}) $
Factor: .$\displaystyle (a + b\sqrt{2})(c + d\sqrt{2}) $
Like I said, very sloppy!
Maybe we're trying to prove multiplication is commutative in the field $\displaystyle \mathbb{Q}(\sqrt{2})$? Probably easier than factoring would be to expand separately $\displaystyle (a+b\sqrt{2})(c+d\sqrt{2})$ and $\displaystyle (c+d\sqrt{2})(a+b\sqrt{2})$ (like Ackbeet suggested) and see that they are equal term-for-term.