document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=453160>Question 453160</A>: <I><SPAN STYLE=\"word-break: break-all;\"> can you explain how multiplying or dividing a negative by a negative equals a positive </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #311368 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=311368\">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( "Imagine that you have a number line large enough for you to walk on, and like one of the X-men, you have the ability to stretch your limbs as far as you like so that you can take as long of steps as you like.\r<BR>\n" );
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document.write( "Now, let's define the operation of multiplication of two numbers, call them <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE N\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE M\">, as follows.  To multiply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE N\ *\ M\"> you take <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE N\"> steps each of which is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE M\"> units long.\r<BR>\n" );
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document.write( "For example, let's say that you want to multiply 2 times 3.  You would take 2 steps each of which was 3 units long.  If you started at zero, you would end up at six.\r<BR>\n" );
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document.write( "So far, so good, right?  Now let's add a little more to the definition of multiplication to take care of signed numbers.\r<BR>\n" );
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document.write( "Lets say that the sign on the first number is the direction you start out facing.  So if you have a positive <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE N\">, then, you would start standing on the zero and facing toward the positive numbers.  Then, the sign on the second number tells you whether you should walk forward, that is in the direction you are facing, when you take <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE M\"> steps, or if you should walk backwards, away from the direction you are facing.\r<BR>\n" );
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document.write( "So if you multiply +2 times +3, you face toward the positive and walk forward, making you end up on +6.  (Plus times Plus is Plus)\r<BR>\n" );
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document.write( "If you multiply -2 times +3, you face toward the negative and walk forward, making you end up on -6.  (Minus times Plus is Minus)\r<BR>\n" );
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document.write( "If you multiply +2 times -3, you face toward the positive but you walk backwards, making you end up on -6.  (Plus times Minus is Minus)\r<BR>\n" );
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document.write( "Finally, if you multiply -2 times -3, you face the negative and walk backwards making you end up on +6.  (Minus times Minus is Plus)\r<BR>\n" );
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document.write( "Consider the double negative analogy from English:\r<BR>\n" );
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document.write( "I have a pencil.  (Positive that you have, and positive existence of a pencil)\r<BR>\n" );
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document.write( "I don't have a pencil.  (Negative that you have, and positive existence of a pencil)\r<BR>\n" );
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document.write( "I have no pencil.  (Positive that you have, but negative existence of a pencil)\r<BR>\n" );
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document.write( "I don't have no pencil.  (Negative that you have, and negative existence, hence you must have at least one pencil -- or all of the pencils in existence.  If you do not have NO pencil, then you must have SOME pencil)\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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