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<div><h1>QQ -- the class of all rational numbers</h1>
<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : 1/2 + 3/5
11
o1 = --
10
o1 : QQ</pre>
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<div class="waystouse"><h2>Functions and methods returning a rational number :</h2>
<ul><li><span>QQ * QQ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>QQ * ZZ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>ZZ * QQ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>+ QQ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>QQ + QQ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>QQ + ZZ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>ZZ + QQ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>- QQ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>QQ - QQ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>QQ - ZZ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>ZZ - QQ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>QQ / QQ, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>QQ / ZZ, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>ZZ / QQ, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>gcd(QQ,QQ), see <span><a href="_gcd.html" title="greatest common divisor">gcd</a> -- greatest common divisor</span></span></li>
<li><span>gcd(QQ,ZZ), see <span><a href="_gcd.html" title="greatest common divisor">gcd</a> -- greatest common divisor</span></span></li>
<li><span>gcd(ZZ,QQ), see <span><a href="_gcd.html" title="greatest common divisor">gcd</a> -- greatest common divisor</span></span></li>
</ul>
<h2>Methods that use a rational number :</h2>
<ul><li><span>QQ !, see <span><a href="_!.html" title="factorial">!</a> -- factorial</span></span></li>
<li><span>CC % QQ, see <span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li>
<li><span>QQ % QQ, see <span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li>
<li><span>QQ % ZZ, see <span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li>
<li><span>RR % QQ, see <span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li>
<li><span>CC * QQ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>QQ * CC, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>QQ * RR, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>RR * QQ, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>CC + QQ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>QQ + CC, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>QQ + RR, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>RR + QQ, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>CC - QQ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>QQ - CC, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>QQ - RR, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>RR - QQ, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>CC / QQ, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>QQ / CC, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>QQ / RR, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>RR / QQ, see <span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></span></li>
<li><span>CC // QQ, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>QQ // QQ, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>QQ // ZZ, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>RR // QQ, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>CC == QQ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>QQ == CC, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>QQ == QQ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>QQ == RR, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>QQ == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>RR == QQ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == QQ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>abs(QQ), see <span><a href="_abs.html" title="absolute value function">abs</a> -- absolute value function</span></span></li>
<li><span>acos(QQ), see <span><a href="_acos.html" title="arccosine">acos</a> -- arccosine</span></span></li>
<li><span>agm(CC,QQ), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(QQ,CC), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(QQ,QQ), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(QQ,RR), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(QQ,ZZ), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(RR,QQ), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(ZZ,QQ), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>asin(QQ), see <span><a href="_asin.html" title="arcsine">asin</a> -- arcsine</span></span></li>
<li><span>atan(QQ), see <span><a href="_atan_lp__R__R_rp.html" title="compute the arctangent of a number ">atan(RR)</a> -- compute the arctangent of a number </span></span></li>
<li><span>atan2(QQ,QQ), see <span><a href="_atan2_lp__R__R_cm__R__R_rp.html" title="compute an angle of a certain triangle">atan2(RR,RR)</a> -- compute an angle of a certain triangle</span></span></li>
<li><span>atan2(QQ,RR), see <span><a href="_atan2_lp__R__R_cm__R__R_rp.html" title="compute an angle of a certain triangle">atan2(RR,RR)</a> -- compute an angle of a certain triangle</span></span></li>
<li><span>atan2(QQ,ZZ), see <span><a href="_atan2_lp__R__R_cm__R__R_rp.html" title="compute an angle of a certain triangle">atan2(RR,RR)</a> -- compute an angle of a certain triangle</span></span></li>
<li><span>atan2(RR,QQ), see <span><a href="_atan2_lp__R__R_cm__R__R_rp.html" title="compute an angle of a certain triangle">atan2(RR,RR)</a> -- compute an angle of a certain triangle</span></span></li>
<li><span>atan2(ZZ,QQ), see <span><a href="_atan2_lp__R__R_cm__R__R_rp.html" title="compute an angle of a certain triangle">atan2(RR,RR)</a> -- compute an angle of a certain triangle</span></span></li>
<li><span>BesselJ(ZZ,QQ), see <span><a href="___Bessel__J.html" title="Bessel function of the first kind">BesselJ</a> -- Bessel function of the first kind</span></span></li>
<li><span>BesselY(ZZ,QQ), see <span><a href="___Bessel__Y.html" title="Bessel function of the second kind">BesselY</a> -- Bessel function of the second kind</span></span></li>
<li><span>QQ * BettiTally, see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li><span>cos(QQ), see <span><a href="_cos.html" title="compute the cosine">cos</a> -- compute the cosine</span></span></li>
<li><span>cosh(QQ), see <span><a href="_cosh.html" title="compute the hyperbolic cosine">cosh</a> -- compute the hyperbolic cosine</span></span></li>
<li><span>cot(QQ), see <span><a href="_cot.html" title="cotangent">cot</a> -- cotangent</span></span></li>
<li><span>coth(QQ), see <span><a href="_coth.html" title="hyperbolic cotangent">coth</a> -- hyperbolic cotangent</span></span></li>
<li><span>csc(QQ), see <span><a href="_csc.html" title="cosecant">csc</a> -- cosecant</span></span></li>
<li><span>csch(QQ), see <span><a href="_csch.html" title="hyperbolic cosecant">csch</a> -- hyperbolic cosecant</span></span></li>
<li><span>eint(QQ), see <span><a href="_eint.html" title="exponential integral">eint</a> -- exponential integral</span></span></li>
<li><span>erf(QQ), see <span><a href="_erf.html" title="error function">erf</a> -- error function</span></span></li>
<li><span>erfc(QQ), see <span><a href="_erfc.html" title="complementary error function">erfc</a> -- complementary error function</span></span></li>
<li><span>exp(QQ), see <span><a href="_exp.html" title="exponential function">exp</a> -- exponential function</span></span></li>
<li><span>expm1(QQ), see <span><a href="_expm1.html" title="exponential minus 1">expm1</a> -- exponential minus 1</span></span></li>
<li><span>factor(QQ), see <span><a href="_factor_lp__Ring__Element_rp.html" title="factor a ring element">factor(RingElement)</a> -- factor a ring element</span></span></li>
<li><span>floor(QQ), see <span><a href="_floor.html" title="floor function">floor</a> -- floor function</span></span></li>
<li><span>Gamma(QQ), see <span><a href="___Gamma.html" title="Gamma function">Gamma</a> -- Gamma function</span></span></li>
<li><span>isReal(QQ), see <span><a href="_is__Real.html" title="whether a number is real">isReal</a> -- whether a number is real</span></span></li>
<li><span>lcm(QQ,QQ), see <span><a href="_lcm.html" title="least common multiple">lcm</a> -- least common multiple</span></span></li>
<li><span>lcm(QQ,ZZ), see <span><a href="_lcm.html" title="least common multiple">lcm</a> -- least common multiple</span></span></li>
<li><span>lcm(ZZ,QQ), see <span><a href="_lcm.html" title="least common multiple">lcm</a> -- least common multiple</span></span></li>
<li><span>lift(CC,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Ideal,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Matrix,type of QQ,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(Matrix,type of QQ,type of ZZ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(QQ,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(QQ,type of ZZ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lift(RR,type of QQ), see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>lngamma(QQ), see <span><a href="_lngamma.html" title="logarithm of the Gamma function">lngamma</a> -- logarithm of the Gamma function</span></span></li>
<li><span>log(QQ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(QQ,CC), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(QQ,QQ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(QQ,RR), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(QQ,ZZ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(RR,QQ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(ZZ,QQ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log1p(QQ), see <span><a href="_log1p.html" title="logarithm of 1+x">log1p</a> -- logarithm of 1+x</span></span></li>
<li><span>promote(RR,type of QQ), see <span><a href="_promote.html" title="promote to another ring">promote</a> -- promote to another ring</span></span></li>
<li><span>round(QQ), see <span><a href="_round.html" title="round a number">round</a> -- round a number</span></span></li>
<li><span>sec(QQ), see <span><a href="_sec.html" title="secant">sec</a> -- secant</span></span></li>
<li><span>sech(QQ), see <span><a href="_sech.html" title="hyperbolic secant">sech</a> -- hyperbolic secant</span></span></li>
<li><span>sin(QQ), see <span><a href="_sin.html" title="compute the sine">sin</a> -- compute the sine</span></span></li>
<li><span>sinh(QQ), see <span><a href="_sinh.html" title="compute the hyperbolic sine">sinh</a> -- compute the hyperbolic sine</span></span></li>
<li><span>sqrt(QQ), see <span><a href="_sqrt.html" title="square root function">sqrt</a> -- square root function</span></span></li>
<li><span>tan(QQ), see <span><a href="_tan.html" title="compute the tangent">tan</a> -- compute the tangent</span></span></li>
<li><span>tanh(QQ), see <span><a href="_tanh.html" title="compute the hyperbolic tangent">tanh</a> -- compute the hyperbolic tangent</span></span></li>
<li><span>toCC(QQ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,QQ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,QQ,QQ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,QQ,RR), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,QQ,ZZ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,RR,QQ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>toCC(ZZ,ZZ,QQ), see <span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></span></li>
<li><span>zeta(QQ), see <span><a href="_zeta.html" title="Riemann zeta function">zeta</a> -- Riemann zeta function</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Q__Q.html" title="the class of all rational numbers">QQ</a> is <span>a <a href="___Ring.html">ring</a></span>, with ancestor classes <a href="___Number.html" title="the class of all numbers">Number</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
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