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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_basic_springs_spof_spnumbers.html" title="">basic rings of numbers</a></div>
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<div><h1>basic rings of numbers</h1>
<div>The following rings are initially present in every session with Macaulay2.<ul><li><span><a href="___Z__Z.html" title="the class of all integers">ZZ</a> -- the class of all integers</span></li>
<li><span><a href="___Q__Q.html" title="the class of all rational numbers">QQ</a> -- the class of all rational numbers</span></li>
<li><span><a href="___R__R.html" title="the class of all real numbers">RR</a> -- the class of all real numbers</span></li>
<li><span><a href="___C__C.html" title="the class of all complex numbers">CC</a> -- the class of all complex numbers</span></li>
</ul>
The names of some of these rings are double letters so the corresponding symbols with single letters are preserved for use as variables.<p/>
Numbers in these rings are constructed as follows.<table class="examples"><tr><td><pre>i1 : 1234
o1 = 1234</pre>
</td></tr>
<tr><td><pre>i2 : 123/4
123
o2 = ---
4
o2 : QQ</pre>
</td></tr>
<tr><td><pre>i3 : 123.4
o3 = 123.4
o3 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i4 : 1.234e-20
o4 = 1.234e-20
o4 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i5 : 123+4*ii
o5 = 123+4*ii
o5 : CC (of precision 53)</pre>
</td></tr>
</table>
The usual arithmetic operations are available.<table class="examples"><tr><td><pre>i6 : 4/5 + 2/3
22
o6 = --
15
o6 : QQ</pre>
</td></tr>
<tr><td><pre>i7 : 10^20
o7 = 100000000000000000000</pre>
</td></tr>
<tr><td><pre>i8 : 3*5*7
o8 = 105</pre>
</td></tr>
<tr><td><pre>i9 : 3.1^2.1
o9 = 10.7611716060997
o9 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i10 : sqrt 3.
o10 = 1.73205080756888
o10 : RR (of precision 53)</pre>
</td></tr>
</table>
An additional pair of division operations that produce integer quotients and remainders is available.<table class="examples"><tr><td><pre>i11 : 1234//100
o11 = 12</pre>
</td></tr>
<tr><td><pre>i12 : 1234%100
o12 = 34</pre>
</td></tr>
</table>
Numbers can be promoted to larger rings as follows, see <a href="_promote.html" title="promote to another ring">RingElement _ Ring</a>.<table class="examples"><tr><td><pre>i13 : 1_QQ
o13 = 1
o13 : QQ</pre>
</td></tr>
<tr><td><pre>i14 : (2/3)_CC
o14 = .666666666666667
o14 : CC (of precision 53)</pre>
</td></tr>
</table>
One way to enter real and complex numbers with more precision is to insert the desired number of bits of precision after the letter p at the end of the number, but before the possible e that indicates the exponent of 10.<table class="examples"><tr><td><pre>i15 : 1p300
o15 = 1
o15 : RR (of precision 300)</pre>
</td></tr>
<tr><td><pre>i16 : 1p300e-30
o16 = 1e-30
o16 : RR (of precision 300)</pre>
</td></tr>
</table>
Numbers can be lifted to smaller rings as follows, see <a href="_lift.html" title="lift to another ring">lift</a>.<table class="examples"><tr><td><pre>i17 : x = 2/3*ii/ii
o17 = .666666666666667
o17 : CC (of precision 53)</pre>
</td></tr>
<tr><td><pre>i18 : lift(x,RR)
o18 = .666666666666667
o18 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i19 : lift(x,QQ)
2
o19 = -
3
o19 : QQ</pre>
</td></tr>
</table>
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