document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117632>Question 117632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hii. Well i was assighned a Essay to do that has to be about 1-2 pages of why do we get a positive number when deviding a negative number by a negative number? and a feww examples and add diagrams and stuff. And i didnt really find any answers online. do u think you could help me? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85648 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=85648\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The simplest explanation that I can give is this:\r<BR>\n" );
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document.write( "If you divide any number by itself, you always get 1.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fa=1 ALIGN=MIDDLE  ALT=\"a%2Fa=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all real numbers.\r<BR>\n" );
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document.write( "You can also express any negative number as the product of that number's opposite and -1.  In other words, you could write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%284%29 ALIGN=MIDDLE  ALT=\"%28-1%29%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let's say that x is some positive number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y is some positive number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E0 ALIGN=MIDDLE  ALT=\"y%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Then we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > are negative numbers.  (I hope you clearly understand why you can't just say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a negative number without qualifying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > as positive in the first place.)\r<BR>\n" );
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document.write( "So let's divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%29%2F%28-y%29 ALIGN=MIDDLE  ALT=\"%28-x%29%2F%28-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  But we already said that you can also express any negative number as the product of that number's opposite and -1, so we can write: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%29x%29%2F%28%28-1%29y%29 ALIGN=MIDDLE  ALT=\"%28%28-1%29x%29%2F%28%28-1%29y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   But from the first rule we talked about above <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2F%28-1%29=1 ALIGN=MIDDLE  ALT=\"%28-1%29%2F%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%29x%29%2F%28%28-1%29y%29=%281%29%28x%2Fy%29=%28x%2Fy%29 ALIGN=MIDDLE  ALT=\"%28%28-1%29x%29%2F%28%28-1%29y%29=%281%29%28x%2Fy%29=%28x%2Fy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now all you have to do is prove that the quotient of a positive number divided by a positive number is positive -- or just take that one on faith.\r<BR>\n" );
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document.write( "There is another way to do this.  Remember that division is nothing more than multiplication by the reciprocal.  A reciprocal is a number formed from an original number such that the product of the original and the reciprocal equal one.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%281%2Fa%29=1 ALIGN=MIDDLE  ALT=\"a%2A%281%2Fa%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  A reciprocal is also called the multiplicative inverse.\r<BR>\n" );
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document.write( "So, if you are dividing a by b <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb ALIGN=MIDDLE  ALT=\"a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is the same as multiplying a by the reciprocal of b <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%281%2Fb%29 ALIGN=MIDDLE  ALT=\"a%2A%281%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we can define some <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1%2Fb ALIGN=MIDDLE  ALT=\"c=1%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > and our division becomes a straight multiplication:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ac ALIGN=MIDDLE  ALT=\"a%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and our problem becomes one of proving that a negative number times a negative number yields a positive product.\r<BR>\n" );
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document.write( "Let a and b be any two real numbers.\r<BR>\n" );
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document.write( "Consider the number x defined by <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ab+%2B+%28-a%29%28b%29+%2B+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x+=+ab+%2B+%28-a%29%28b%29+%2B+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can write \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ab+%2B+%28-a%29%28+%28b%29+%2B+%28-b%29+%29 ALIGN=MIDDLE  ALT=\"x+=+ab+%2B+%28-a%29%28+%28b%29+%2B+%28-b%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >       (factor out -a)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x++=+ab+%2B+%28-a%29%280%29 ALIGN=MIDDLE  ALT=\"x++=+ab+%2B+%28-a%29%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ab+%2B+0 ALIGN=MIDDLE  ALT=\"x+=+ab+%2B+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x++=+ab ALIGN=MIDDLE  ALT=\"x++=+ab\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28+a+%2B+%28-a%29+%29b+%2B+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x+=+%28+a+%2B+%28-a%29+%29b+%2B+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >      (factor out b)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x++=+0+%2A+b+%2B+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x++=+0+%2A+b+%2B+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x++=+0+%2B+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x++=+0+%2B+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x++=+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x++=+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we have \r<BR>\n" );
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document.write( "      <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ab ALIGN=MIDDLE  ALT=\"x+=+ab\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "      <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"x+=+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, by the transitivity of equality, we have <BR>\n" );
document.write( "      <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+=+%28-a%29%28-b%29 ALIGN=MIDDLE  ALT=\"ab+=+%28-a%29%28-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hope that helps. </SPAN>\n" );
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