# basic rings of numbers

The following rings are initially present in every session with Macaulay2.
• ZZ -- the class of all integers
• QQ -- the class of all rational numbers
• RR -- the class of all real numbers
• CC -- the class of all complex numbers
The names of some of these rings are double letters so the corresponding symbols with single letters are preserved for use as variables.

Numbers in these rings are constructed as follows.

 i1 : 1234 o1 = 1234 i2 : 123/4 123 o2 = --- 4 o2 : QQ i3 : 123.4 o3 = 123.4 o3 : RR (of precision 53) i4 : 1.234e-20 o4 = 1.234e-20 o4 : RR (of precision 53) i5 : 123+4*ii o5 = 123+4*ii o5 : CC (of precision 53)
The usual arithmetic operations are available.
 i6 : 4/5 + 2/3 22 o6 = -- 15 o6 : QQ i7 : 10^20 o7 = 100000000000000000000 i8 : 3*5*7 o8 = 105 i9 : 3.1^2.1 o9 = 10.7611716060997 o9 : RR (of precision 53) i10 : sqrt 3. o10 = 1.73205080756888 o10 : RR (of precision 53)
An additional pair of division operations that produce integer quotients and remainders is available.
 i11 : 1234//100 o11 = 12 i12 : 1234%100 o12 = 34
Numbers can be promoted to larger rings as follows, see RingElement _ Ring.
 i13 : 1_QQ o13 = 1 o13 : QQ i14 : (2/3)_CC o14 = .666666666666667 o14 : CC (of precision 53)
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.
 i15 : 1p300 o15 = 1 o15 : RR (of precision 300) i16 : 1p300e-30 o16 = 1e-30 o16 : RR (of precision 300)
Numbers can be lifted to smaller rings as follows, see lift.
 i17 : x = 2/3*ii/ii o17 = .666666666666667 o17 : CC (of precision 53) i18 : lift(x,RR) o18 = .666666666666667 o18 : RR (of precision 53) i19 : lift(x,QQ) 2 o19 = - 3 o19 : QQ