document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=60831>Question 60831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the difference bdtween rational and irrational numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #41818 by </B><A HREF=/tutors/aboutme.mpl?userid=jai_kos><B>jai_kos(139)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jai_kos><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=41818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Before we talk about rational and irrational numbers, let's make clear <BR>\n" );
document.write( "one other definition.  An INTEGER is in the set: \r<BR>\n" );
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document.write( "{...-3, -2, -1, 0, 1, 2, 3, ...}\r<BR>\n" );
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document.write( "It is just a positive or negative whole number.  Thus 454564 is an <BR>\n" );
document.write( "integer, but 1/2 isn't.\r<BR>\n" );
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document.write( "Now, a rational number is any number that can be written as a ratio of <BR>\n" );
document.write( "two integers (hence the name!).  In other words, a number is rational if<BR>\n" );
document.write( "we can write it as a fraction where the numerator and denominator are <BR>\n" );
document.write( "both integers.  Now then, every integer is a rational number, since <BR>\n" );
document.write( "each integer n can be written in the form n/1.  For example 5 = 5/1 - <BR>\n" );
document.write( "thus 5 is a rational number.  However, numbers like 1/2, <BR>\n" );
document.write( "45454737/2424242, and -3/7 are also rational since they are fractions <BR>\n" );
document.write( "where the numerator and denominator are integers.  \r<BR>\n" );
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document.write( "An irrational number is any real number that is not rational.  By \"real\" <BR>\n" );
document.write( "number I mean, loosely, a number that we can conceive of in this world,<BR>\n" );
document.write( "one with no square roots of negative numbers (numbers where square roots <BR>\n" );
document.write( "of negative numbers are involved are called complex, and there is lots <BR>\n" );
document.write( "of neat stuff there, if you are curious).  A real number is a number <BR>\n" );
document.write( "that is somewhere on your number line.  So, any number on the number <BR>\n" );
document.write( "line that isn't a rational number is irrational.  For example, the <BR>\n" );
document.write( "square root of 2 is an irrational number because it can't be written as <BR>\n" );
document.write( "a ratio of two integers.  \r<BR>\n" );
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document.write( "How would you imagine we would show something like that?  The proof <BR>\n" );
document.write( "is a proof by contradiction.  We assume that the square root of 2 CAN <BR>\n" );
document.write( "be written as p/q for some integers, p and q, and we get a contradiction.  <BR>\n" );
document.write( "The proof has a little trick to it, but if you're curious about it, write back <BR>\n" );
document.write( "and I can tell you more!  \r<BR>\n" );
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document.write( "Other irrational numbers include:<BR>\n" );
document.write( "square root of 3, the square root of 5, pi, e, ....\r<BR>\n" );
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document.write( "I hope this answers your question.  There are lots of neat properties of <BR>\n" );
document.write( "rational numbers, irrational numbers and real numbers.  For instance, it<BR>\n" );
document.write( "turns out that if you were to try to gauge how many rational numbers, <BR>\n" );
document.write( "irrational numbers, and real numbers there are between 0 and 1, you <BR>\n" );
document.write( "would find that while there are infinitely many of each kind of number, <BR>\n" );
document.write( "there are many, many more irrational numbers than rational numbers.  The <BR>\n" );
document.write( "sizes of the infinities involved are somehow a little different.  Another <BR>\n" );
document.write( "property is that between any two rational numbers on the number line <BR>\n" );
document.write( "there is an irrational number; also, between any two irrational numbers <BR>\n" );
document.write( "there is a rational number.  \r<BR>\n" );
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document.write( " </SPAN>\n" );
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