document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120076>Question 120076</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a question. How can you determine if a number is rational or irrational?\r<BR>\n" );
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document.write( "1.) Is 64 squared rational or irrational?\r<BR>\n" );
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document.write( "2.) Is -1.2 repeating decimal rational or irrational?\r<BR>\n" );
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document.write( "Can some please explain different ways to determine if a number is irrational or rational? Thanks!!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88016 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=88016\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A rational number is one that can be expressed as the ratio (hence the name) of two integers.  So if you can find any two integers p and q such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=p%2Fq ALIGN=MIDDLE  ALT=\"x=p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then x is rational.  Otherwise, x is irrational.\r<BR>\n" );
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document.write( "A couple of rules:  Any integer is rational.  That's because you can express any integer as that same integer divided by 1.\r<BR>\n" );
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document.write( "Any repeating decimal is rational, conversely, all irrational numbers are non-repeating decimals.\r<BR>\n" );
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document.write( "Problem 1) Squaring an integer results in an integer.  64 is an integer, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64%5E2 ALIGN=MIDDLE  ALT=\"64%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer.  All integers are rational, therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64%5E2 ALIGN=MIDDLE  ALT=\"64%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is rational.\r<BR>\n" );
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document.write( "Problem 2) All repeating decimals are rational, so -1.2... is rational.  But what are the integers p and q?  Hint:  Any time the repeating part of the decimal is the same number, try 9 as a denominator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3=3%2F9 ALIGN=MIDDLE  ALT=\"1%2F3=3%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the rule holds).  Since the absolute value of the given number is slightly greater than 1, you need a numerator that is slightly greater than the denominator.  Put the minus sign on either the numerator or the denominator.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11%29%2F9 ALIGN=MIDDLE  ALT=\"%28-11%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > does quite nicely.\r<BR>\n" );
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document.write( "Here's a couple more rules for the other side of the question:\r<BR>\n" );
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document.write( "The square root of any number that is not a perfect square is irrational.  The cube root of any number that is not a perfect cube is irrational.  And so on...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the ratio of a circle's circumference to its diameter, is irrational.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the base of the natural logarithms, is irrational\r<BR>\n" );
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document.write( " </SPAN>\n" );
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