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# 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
- RRi -- the class of all real intervals
- 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)

Integers may be entered in bases 2, 8, or 16 using particular prefixes.

    i6 : 0b10011010010 -- binary

    o6 = 1234

    i7 : 0o2322 -- octal

    o7 = 1234

    i8 : 0x4d2 -- hexadecimal

    o8 = 1234

The usual arithmetic operations are available.

    i9 : 4/5 + 2/3

         22
    o9 = --
         15

    o9 : QQ

    i10 : 10^20

    o10 = 100000000000000000000

    i11 : 3*5*7

    o11 = 105

    i12 : 3.1^2.1

    o12 = 10.76117160609969

    o12 : RR (of precision 53)

    i13 : sqrt 3.

    o13 = 1.732050807568877

    o13 : RR (of precision 53)

An additional pair of division operations that produce integer quotients and remainders is available.

    i14 : 1234//100

    o14 = 12

    i15 : 1234%100

    o15 = 34

Numbers can be promoted to larger rings as follows, see RingElement _ Ring.

    i16 : 1_QQ

    o16 = 1

    o16 : QQ

    i17 : (2/3)_CC

    o17 = .6666666666666666

    o17 : 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.

    i18 : 1p300

    o18 = 1

    o18 : RR (of precision 300)

    i19 : 1p300e-30

    o19 = 1e-30

    o19 : RR (of precision 300)

Numbers can be lifted to smaller rings as follows, see lift.

    i20 : x = 2/3*ii/ii

    o20 = .6666666666666666

    o20 : CC (of precision 53)

    i21 : lift(x,RR)

    o21 = .6666666666666666

    o21 : RR (of precision 53)

    i22 : lift(x,QQ)

          2
    o22 = -
          3

    o22 : QQ

### Menu

- ZZ -- the class of all integers
- QQ -- the class of all rational numbers
- RR -- the class of all real numbers
- RRi -- the class of all real intervals
- CC -- the class of all complex numbers

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_rings.m2:111:0.