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<head><title>RR -- the class of all real numbers</title>
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<div><h1>RR -- the class of all real numbers</h1>
<div class="single"><h2>Description</h2>
<div>A real number is entered as a sequence of decimal digits with a point. It is stored internally as an arbitrary precision floating point number, using the <a href="___M__P__F__R.html" title="">MPFR</a> library.<table class="examples"><tr><td><pre>i1 : 3.14159
o1 = 3.14159
o1 : RR (of precision 53)</pre>
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The precision is measured in bits, is visible in the ring displayed on the second of each pair of output lines, and can be recovered using <a href="_precision.html" title="">precision</a>.<table class="examples"><tr><td><pre>i2 : precision 3.14159
o2 = 53</pre>
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For real numbers, the functions <a href="_class.html" title="class of an object">class</a> and <a href="_ring.html" title="get the associated ring of an object">ring</a> yield different results. That allows numbers of various precisions to be used without creating a new ring for each precision.<table class="examples"><tr><td><pre>i3 : class 3.1
o3 = RR
o3 : InexactFieldFamily</pre>
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<tr><td><pre>i4 : ring 3.1
o4 = RR
53
o4 : RealField</pre>
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The precision can be specified on input by appending the letter <tt>p</tt> and a positive number.<table class="examples"><tr><td><pre>i5 : 3p300
o5 = 3
o5 : RR (of precision 300)</pre>
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An optional exponent (for the power of ten to multiply by) can be specified on input by appending the letter <tt>e</tt> and a number.<table class="examples"><tr><td><pre>i6 : 3e3
o6 = 3000
o6 : RR (of precision 53)</pre>
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<tr><td><pre>i7 : -3e-3
o7 = -.003
o7 : RR (of precision 53)</pre>
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<tr><td><pre>i8 : -3p111e-3
o8 = -.003
o8 : RR (of precision 111)</pre>
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Numbers that appear alone on an output line are displayed with all their meaningful digits. (Specifying 100 bits of precision yields about 30 decimal digits of precision.)<table class="examples"><tr><td><pre>i9 : 1/3.
o9 = .333333333333333
o9 : RR (of precision 53)</pre>
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<tr><td><pre>i10 : 1/3p100
o10 = .333333333333333333333333333333
o10 : RR (of precision 100)</pre>
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<tr><td><pre>i11 : 100 * log(10,2)
o11 = 30.1029995663981
o11 : RR (of precision 53)</pre>
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Numbers displayed inside more complicated objects are printed with the number of digits specified by <a href="_printing__Precision.html" title="current precision for printing numbers">printingPrecision</a>.<table class="examples"><tr><td><pre>i12 : printingPrecision
o12 = 6</pre>
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<tr><td><pre>i13 : {1/3.,1/3p100}
o13 = {.333333, .333333}
o13 : List</pre>
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The notion of equality tested by <a href="__eq_eq.html" title="equality">==</a> amounts to equality of the internal binary digits.<table class="examples"><tr><td><pre>i14 : .5p100 == .5p30
o14 = true</pre>
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<tr><td><pre>i15 : .2p100 == .2p30
o15 = false</pre>
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The notion of (strict) equality tested by <a href="___Thing_sp_eq_eq_eq_sp__Thing.html" title="strict equality">===</a> also takes the precision into account.<table class="examples"><tr><td><pre>i16 : .5p100 === .5p30
o16 = false</pre>
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<tr><td><pre>i17 : .2p100 === .2p30
o17 = false</pre>
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Perhaps surprisingly, the IEEE floating point standard also specifies that every number, including 0, has a sign bit, and strict equality testing takes it into account, as it must do, because some arithmetic and transcendental functions take it into account.<table class="examples"><tr><td><pre>i18 : 0.
o18 = 0
o18 : RR (of precision 53)</pre>
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<tr><td><pre>i19 : -0.
o19 = -0
o19 : RR (of precision 53)</pre>
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<tr><td><pre>i20 : 1/0.
o20 = infinity
o20 : RR (of precision 53)</pre>
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<tr><td><pre>i21 : 1/-0.
o21 = -infinity
o21 : RR (of precision 53)</pre>
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<tr><td><pre>i22 : log 0
o22 = -infinity
o22 : RR (of precision 53)</pre>
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<tr><td><pre>i23 : csc (0.)
o23 = infinity
o23 : RR (of precision 53)</pre>
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<tr><td><pre>i24 : csc (-0.)
o24 = -infinity
o24 : RR (of precision 53)</pre>
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Use <a href="_to__External__String.html" title="convert to a readable string">toExternalString</a> to produce something that, when encountered as input, will reproduce exactly what you had before.<table class="examples"><tr><td><pre>i25 : x = {1/3.,1/3p100}
o25 = {.333333, .333333}
o25 : List</pre>
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<tr><td><pre>i26 : x == {.333333, .333333}
o26 = false</pre>
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<tr><td><pre>i27 : y = toExternalString x
o27 = {.33333333333333331p53,.33333333333333333333333333333346p100}</pre>
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<tr><td><pre>i28 : x === value y
o28 = true</pre>
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Transcendental constants and functions are available to high precision, with <a href="_numeric.html" title="convert to floating point">numeric</a>.<table class="examples"><tr><td><pre>i29 : numeric pi
o29 = 3.14159265358979
o29 : RR (of precision 53)</pre>
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<tr><td><pre>i30 : numeric_200 pi
o30 = 3.14159265358979323846264338327950288419716939937510582097494
o30 : RR (of precision 200)</pre>
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<tr><td><pre>i31 : Gamma oo
o31 = 2.28803779534003241795958890906023392288968815335622244119938
o31 : RR (of precision 200)</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_to__R__R.html" title="convert to high-precision real number">toRR</a> -- convert to high-precision real number</span></li>
<li><span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</span></li>
<li><span><a href="_precision.html" title="">precision</a></span></li>
<li><span><a href="_format.html" title="format a string or a real number">format</a> -- format a string or a real number</span></li>
<li><span><a href="_printing__Precision.html" title="current precision for printing numbers">printingPrecision</a> -- current precision for printing numbers</span></li>
<li><span><a href="_printing__Accuracy.html" title="current accuracy for printing numbers">printingAccuracy</a> -- current accuracy for printing numbers</span></li>
<li><span><a href="_printing__Lead__Limit.html" title="maximum number of leading zeroes to use when printing real numbers">printingLeadLimit</a> -- maximum number of leading zeroes to use when printing real numbers</span></li>
<li><span><a href="_printing__Trail__Limit.html" title="maximum number of additional trailing digits to use when printing real numbers">printingTrailLimit</a> -- maximum number of additional trailing digits to use when printing real numbers</span></li>
<li><span><a href="_printing__Separator.html" title="string used to separate mantissa from exponent when printing real numbers">printingSeparator</a> -- string used to separate mantissa from exponent when printing real numbers</span></li>
<li><span><a href="_max__Exponent.html" title="">maxExponent</a></span></li>
<li><span><a href="_min__Exponent.html" title="">minExponent</a></span></li>
</ul>
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<div class="waystouse"><h2>Functions and methods returning a real number :</h2>
<ul><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>RR * 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 * 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 * 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, 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>RR + 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 + 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 + 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, 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>RR - 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 - 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 - 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>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>RR / 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 / 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 / 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>abs(CC), see <span><a href="_abs.html" title="absolute value function">abs</a> -- absolute value function</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>abs(RR), 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>acos(RR), see <span><a href="_acos.html" title="arccosine">acos</a> -- arccosine</span></span></li>
<li><span>acos(ZZ), see <span><a href="_acos.html" title="arccosine">acos</a> -- arccosine</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(RR,RR), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(RR,ZZ), 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>agm(ZZ,RR), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(ZZ,ZZ), 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>asin(RR), see <span><a href="_asin.html" title="arcsine">asin</a> -- arcsine</span></span></li>
<li><span>asin(ZZ), 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><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></li>
<li><span>atan(ZZ), 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><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></li>
<li><span>atan2(RR,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(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>atan2(ZZ,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(ZZ,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>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>BesselJ(ZZ,RR), 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>BesselJ(ZZ,ZZ), 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>BesselY(ZZ,RR), 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>BesselY(ZZ,ZZ), 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>cos(QQ), see <span><a href="_cos.html" title="compute the cosine">cos</a> -- compute the cosine</span></span></li>
<li><span>cos(RR), see <span><a href="_cos.html" title="compute the cosine">cos</a> -- compute the cosine</span></span></li>
<li><span>cos(ZZ), 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>cosh(RR), see <span><a href="_cosh.html" title="compute the hyperbolic cosine">cosh</a> -- compute the hyperbolic cosine</span></span></li>
<li><span>cosh(ZZ), 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>cot(RR), see <span><a href="_cot.html" title="cotangent">cot</a> -- cotangent</span></span></li>
<li><span>cot(ZZ), 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>coth(RR), see <span><a href="_coth.html" title="hyperbolic cotangent">coth</a> -- hyperbolic cotangent</span></span></li>
<li><span>coth(ZZ), 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>csc(RR), see <span><a href="_csc.html" title="cosecant">csc</a> -- cosecant</span></span></li>
<li><span>csc(ZZ), 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>csch(RR), see <span><a href="_csch.html" title="hyperbolic cosecant">csch</a> -- hyperbolic cosecant</span></span></li>
<li><span>csch(ZZ), 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>eint(RR), see <span><a href="_eint.html" title="exponential integral">eint</a> -- exponential integral</span></span></li>
<li><span>eint(ZZ), 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>erf(RR), see <span><a href="_erf.html" title="error function">erf</a> -- error function</span></span></li>
<li><span>erf(ZZ), 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>erfc(RR), see <span><a href="_erfc.html" title="complementary error function">erfc</a> -- complementary error function</span></span></li>
<li><span>erfc(ZZ), 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>exp(RR), see <span><a href="_exp.html" title="exponential function">exp</a> -- exponential function</span></span></li>
<li><span>exp(ZZ), 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>expm1(RR), see <span><a href="_expm1.html" title="exponential minus 1">expm1</a> -- exponential minus 1</span></span></li>
<li><span>expm1(ZZ), see <span><a href="_expm1.html" title="exponential minus 1">expm1</a> -- exponential minus 1</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>Gamma(RR), see <span><a href="___Gamma.html" title="Gamma function">Gamma</a> -- Gamma function</span></span></li>
<li><span>Gamma(ZZ), see <span><a href="___Gamma.html" title="Gamma function">Gamma</a> -- 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,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), 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(RR,ZZ), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(ZZ), 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>log(ZZ,RR), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(ZZ,ZZ), 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>log1p(RR), see <span><a href="_log1p.html" title="logarithm of 1+x">log1p</a> -- logarithm of 1+x</span></span></li>
<li><span>log1p(ZZ), see <span><a href="_log1p.html" title="logarithm of 1+x">log1p</a> -- logarithm of 1+x</span></span></li>
<li><span><a href="_random_lp__R__R_rp.html" title="random real number">random(RR)</a> -- random real number</span></li>
<li><span>sec(QQ), see <span><a href="_sec.html" title="secant">sec</a> -- secant</span></span></li>
<li><span>sec(RR), see <span><a href="_sec.html" title="secant">sec</a> -- secant</span></span></li>
<li><span>sec(ZZ), 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>sech(RR), see <span><a href="_sech.html" title="hyperbolic secant">sech</a> -- hyperbolic secant</span></span></li>
<li><span>sech(ZZ), 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>sin(RR), see <span><a href="_sin.html" title="compute the sine">sin</a> -- compute the sine</span></span></li>
<li><span>sin(ZZ), 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>sinh(RR), see <span><a href="_sinh.html" title="compute the hyperbolic sine">sinh</a> -- compute the hyperbolic sine</span></span></li>
<li><span>sinh(ZZ), see <span><a href="_sinh.html" title="compute the hyperbolic sine">sinh</a> -- compute the hyperbolic sine</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>tan(RR), see <span><a href="_tan.html" title="compute the tangent">tan</a> -- compute the tangent</span></span></li>
<li><span>tan(ZZ), 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>tanh(RR), see <span><a href="_tanh.html" title="compute the hyperbolic tangent">tanh</a> -- compute the hyperbolic tangent</span></span></li>
<li><span>tanh(ZZ), see <span><a href="_tanh.html" title="compute the hyperbolic tangent">tanh</a> -- compute the hyperbolic tangent</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>
<li><span>zeta(RR), see <span><a href="_zeta.html" title="Riemann zeta function">zeta</a> -- Riemann zeta function</span></span></li>
<li><span>zeta(ZZ), see <span><a href="_zeta.html" title="Riemann zeta function">zeta</a> -- Riemann zeta function</span></span></li>
</ul>
<h2>Methods that use a real number :</h2>
<ul><li><span>RR !, see <span><a href="_!.html" title="factorial">!</a> -- factorial</span></span></li>
<li><span>CC % RR, 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>RR % RR, 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 % 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>CC * 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 * 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>CC + 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 + 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>CC - 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 - 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>CC / 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 / 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>CC // RR, 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>RR // RR, 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 // 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>CC == RR, 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>RR == CC, 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>RR == RR, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>RR == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == RR, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>agm(CC,RR), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(RR,CC), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>clean(RR,Matrix), see <span><a href="_clean.html" title="Set to zero elements that are approximately zero">clean</a> -- Set to zero elements that are approximately zero</span></span></li>
<li><span>clean(RR,MutableMatrix), see <span><a href="_clean.html" title="Set to zero elements that are approximately zero">clean</a> -- Set to zero elements that are approximately zero</span></span></li>
<li><span>clean(RR,Number), see <span><a href="_clean.html" title="Set to zero elements that are approximately zero">clean</a> -- Set to zero elements that are approximately zero</span></span></li>
<li><span>clean(RR,RingElement), see <span><a href="_clean.html" title="Set to zero elements that are approximately zero">clean</a> -- Set to zero elements that are approximately zero</span></span></li>
<li><span>floor(RR), see <span><a href="_floor.html" title="floor function">floor</a> -- floor function</span></span></li>
<li><span>isReal(RR), 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>RR << ZZ, see <span><a href="_left_spshift.html" title="">left shift</a></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>lift(RR,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>lngamma(RR), 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(RR,CC), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>log(RR,RR), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>norm(RR,Matrix), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>norm(RR,MutableMatrix), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>norm(RR,Number), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>norm(RR,RingElement), see <span><a href="_norm.html" title="">norm</a></span></span></li>
<li><span>numeric(RR), see <span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</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><a href="_random_lp__R__R_cm__R__R_rp.html" title="random real number">random(RR,RR)</a> -- random real number</span></li>
<li><span><a href="_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.html" title="a random mutable matrix of integers">randomMutableMatrix(ZZ,ZZ,RR,ZZ)</a> -- a random mutable matrix of integers</span></li>
<li><span>RR >> ZZ, see <span><a href="_right_spshift.html" title="">right shift</a></span></span></li>
<li><span>ring(RR), see <span><a href="_ring.html" title="get the associated ring of an object">ring</a> -- get the associated ring of an object</span></span></li>
<li><span>round(RR), see <span><a href="_round.html" title="round a number">round</a> -- round a number</span></span></li>
<li><span>round(ZZ,RR), see <span><a href="_round.html" title="round a number">round</a> -- round a number</span></span></li>
<li><span>size2(RR), see <span><a href="_size2.html" title="number of binary digits to the left of the point">size2</a> -- number of binary digits to the left of the point</span></span></li>
<li><span>sqrt(RR), see <span><a href="_sqrt.html" title="square root function">sqrt</a> -- square root function</span></span></li>
<li><span>toCC(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(RR,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,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,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,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,RR,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,RR,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,ZZ,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>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___R__R.html" title="the class of all real numbers">RR</a> is <span>an <a href="___Inexact__Field__Family.html">inexact field family</a></span>, with ancestor classes <a href="___Inexact__Number.html" title="">InexactNumber</a> < <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>
</div>
</div>
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