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<head><title>CC -- the class of all complex numbers</title>
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<div><h1>CC -- the class of all complex numbers</h1>
<div class="single"><h2>Description</h2>
<div>In Macaulay2, complex numbers are represented as floating point numbers, and so are only approximate. The symbol <a href="_ii.html" title="the square root of -1">ii</a> represents the square root of -1 in many numeric contexts. A complex number is obtained by using the symbolic constant <a href="_ii.html" title="the square root of -1">ii</a> or the conversion functions <a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> and <a href="_numeric.html" title="convert to floating point">numeric</a>, in combination with real numbers (see <a href="___R__R.html" title="the class of all real numbers">RR</a>). It is stored internally as a pair of arbitrary precision floating point real numbers, using the <a href="___M__P__F__R.html" title="">MPFR</a> library.<table class="examples"><tr><td><pre>i1 : z = 3-4*ii
o1 = 3-4*ii
o1 : CC (of precision 53)</pre>
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<tr><td><pre>i2 : z^5
o2 = -237.000000000002+3116*ii
o2 : CC (of precision 53)</pre>
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<tr><td><pre>i3 : 1/z
o3 = .12+.16*ii
o3 : CC (of precision 53)</pre>
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<tr><td><pre>i4 : +ii
o4 = ii
o4 : CC (of precision 53)</pre>
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<tr><td><pre>i5 : numeric_200 ii
o5 = ii
o5 : CC (of precision 200)</pre>
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Complex numbers are ordered lexicographically, mingled with real numbers.<table class="examples"><tr><td><pre>i6 : sort {1+ii,2+ii,1-ii,2-ii,1/2,2.1,7/5}
1 7
o6 = {-, 1-ii, 1+ii, -, 2-ii, 2+ii, 2.1}
2 5
o6 : List</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>i7 : precision z
o7 = 53</pre>
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For complex 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>i8 : class z
o8 = CC
o8 : InexactFieldFamily</pre>
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<tr><td><pre>i9 : ring z
o9 = CC
53
o9 : ComplexField</pre>
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A computation involving numbers of different precisions has a result with the minimal precision occurring. 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>i10 : 3p100+2p90e3*ii
o10 = 3+2000*ii
o10 : CC (of precision 90)</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>i11 : printingPrecision
o11 = 6</pre>
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<tr><td><pre>i12 : x = {1/3.*ii,1/3p100*ii}
o12 = {.333333*ii, .333333*ii}
o12 : List</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>i13 : y = toExternalString x
o13 = {toCC(.0p53,.33333333333333331p53),toCC(.0p100
,.33333333333333333333333333333346p100)}</pre>
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<tr><td><pre>i14 : value y === x
o14 = true</pre>
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<div class="single"><h2>Caveat</h2>
<div>Currently, most transcendental functions are not implemented for complex arguments.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ii.html" title="the square root of -1">ii</a> -- the square root of -1</span></li>
<li><span><a href="_to__C__C.html" title="convert to high-precision complex number">toCC</a> -- convert to high-precision complex number</span></li>
<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>
</ul>
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<div class="waystouse"><h2>Functions and methods returning a complex number :</h2>
<ul><li><span>CC * 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 * 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 * 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>CC * 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>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>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>ZZ * 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, 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 + 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 + 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 + 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>CC + 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>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>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>ZZ + 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, 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 - 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 - 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 - 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>CC - 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>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>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>ZZ - 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 / 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 / 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 / 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>CC / 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>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>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>ZZ / 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>acos(CC), see <span><a href="_acos.html" title="arccosine">acos</a> -- arccosine</span></span></li>
<li><span>agm(CC,CC), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</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(CC,RR), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(CC,ZZ), 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(RR,CC), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>agm(ZZ,CC), see <span><a href="_agm.html" title="arithmetic-geometric mean">agm</a> -- arithmetic-geometric mean</span></span></li>
<li><span>asin(CC), see <span><a href="_asin.html" title="arcsine">asin</a> -- arcsine</span></span></li>
<li><span>atan(CC), 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>cos(CC), see <span><a href="_cos.html" title="compute the cosine">cos</a> -- compute the cosine</span></span></li>
<li><span>cosh(CC), see <span><a href="_cosh.html" title="compute the hyperbolic cosine">cosh</a> -- compute the hyperbolic cosine</span></span></li>
<li><span>cot(CC), see <span><a href="_cot.html" title="cotangent">cot</a> -- cotangent</span></span></li>
<li><span>coth(CC), see <span><a href="_coth.html" title="hyperbolic cotangent">coth</a> -- hyperbolic cotangent</span></span></li>
<li><span>csc(CC), see <span><a href="_csc.html" title="cosecant">csc</a> -- cosecant</span></span></li>
<li><span>csch(CC), see <span><a href="_csch.html" title="hyperbolic cosecant">csch</a> -- hyperbolic cosecant</span></span></li>
<li><span>exp(CC), see <span><a href="_exp.html" title="exponential function">exp</a> -- exponential function</span></span></li>
<li><span>log(CC), 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(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>log(ZZ,CC), see <span><a href="_log.html" title="logarithm function">log</a> -- logarithm function</span></span></li>
<li><span>sec(CC), see <span><a href="_sec.html" title="secant">sec</a> -- secant</span></span></li>
<li><span>sech(CC), see <span><a href="_sech.html" title="hyperbolic secant">sech</a> -- hyperbolic secant</span></span></li>
<li><span>sin(CC), see <span><a href="_sin.html" title="compute the sine">sin</a> -- compute the sine</span></span></li>
<li><span>sinh(CC), see <span><a href="_sinh.html" title="compute the hyperbolic sine">sinh</a> -- compute the hyperbolic sine</span></span></li>
<li><span>sqrt(CC), see <span><a href="_sqrt.html" title="square root function">sqrt</a> -- square root function</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>sqrt(RR), see <span><a href="_sqrt.html" title="square root function">sqrt</a> -- square root function</span></span></li>
<li><span>sqrt(ZZ), see <span><a href="_sqrt.html" title="square root function">sqrt</a> -- square root function</span></span></li>
<li><span>tan(CC), see <span><a href="_tan.html" title="compute the tangent">tan</a> -- compute the tangent</span></span></li>
<li><span>tanh(CC), see <span><a href="_tanh.html" title="compute the hyperbolic tangent">tanh</a> -- compute the hyperbolic tangent</span></span></li>
<li><span>toCC(CC), 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(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(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), 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,CC), 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), 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), 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>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>
<li><span>toCC(ZZ,ZZ,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>
</ul>
<h2>Methods that use a complex number :</h2>
<ul><li><span>CC % CC, 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="__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="__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 % 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 // CC, 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="__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="__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 // 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 == CC, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>CC == QQ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>CC == RR, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>CC == ZZ, 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>RR == CC, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == CC, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>CC ^ ZZ, see <span><a href="_^.html" title="a binary operator, usually used for powers">^</a> -- a binary operator, usually used for powers</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><a href="_conjugate_lp__C__C_rp.html" title="complex conjugate">conjugate(CC)</a> -- complex conjugate</span></li>
<li><span>floor(CC), see <span><a href="_floor.html" title="floor function">floor</a> -- floor function</span></span></li>
<li><span>isReal(CC), 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>CC << ZZ, see <span><a href="_left_spshift.html" title="">left shift</a></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(CC,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>numeric(CC), see <span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</span></span></li>
<li><span>numeric(ZZ,CC), see <span><a href="_numeric.html" title="convert to floating point">numeric</a> -- convert to floating point</span></span></li>
<li><span>CC >> ZZ, see <span><a href="_right_spshift.html" title="">right shift</a></span></span></li>
<li><span>ring(CC), 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(CC), see <span><a href="_round.html" title="round a number">round</a> -- round a number</span></span></li>
<li><span>size2(CC), 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>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___C__C.html" title="the class of all complex numbers">CC</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>
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