document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187446>Question 187446</A>: <I><SPAN STYLE=\"word-break: break-all;\"> is the commutative property true for the operation of subtraction? multiplication n division? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140512 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=140512\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The commutative property only applies to addition and multiplication.\r<BR>\n" );
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document.write( "Commutative Property of Addition:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  a + b = b + a\ \forall\ a,\,b\,\in\,\R\">\r<BR>\n" );
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document.write( "Commutative Property of Multiplication:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  ab = ba\ \forall\ a,\,b\,\in\,\R\">\r<BR>\n" );
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document.write( "Of course, this causes us no problems whatsoever because, for those of us who understand the concepts of signed numbers and reciprocals, there is no such thing as subtraction and division.\r<BR>\n" );
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document.write( "Subtraction is actually addition of the additive inverse, in other words:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 7 - 2\">\r<BR>\n" );
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document.write( "Which operation is <i><b>not</b></i> commutative because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 7 - 2 \neq 2 - 7\">\r<BR>\n" );
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document.write( "But the operation is really:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 7 +(- 2)\">\r<BR>\n" );
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document.write( "Which is commutative because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 7 +(- 2) = -2 + 7\">\r<BR>\n" );
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document.write( "Likewise, division is simply multiplication by the reciprocal.\r<BR>\n" );
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document.write( "Division is not commutative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4 \div 2 \neq 2 \div 4\">\r<BR>\n" );
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document.write( "But\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4 \div 2 \">\r<BR>\n" );
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document.write( "is really:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4 \times \frac{1}{2}\">\r<BR>\n" );
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document.write( "and this operation <i><b>is</b></i> commutative because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4 \times \frac{1}{2} = \frac{1}{2} \times 4\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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