When defining matrices, vectors, or polynomials, it is sometimes useful and sometimes necessary to specify the “ring” over which it is defined. A ring is a mathematical construction in which there are well-behaved notions of addition and multiplication; if you’ve never heard of them before, you probably just need to know about these four commonly used rings:
, called ZZ in Sage.You may need to know about these distinctions because the same
polynomial, for example, can be treated differently depending on the
ring over which it is defined. For instance, the polynomial 
has two roots, 
. Those roots are not rational, so if
you are working with polynomials with rational coefficients, the
polynomial won’t factor. With real coefficients, it will. Therefore
you may want to specify the ring to insure that you are getting the
information you expect. The following two commands defines the sets
of polynomials with rational coefficents and real coefficients,
respectively. The sets are named “ratpoly” and “realpoly”, but these
aren’t important here; however, note that the strings “.<t>” and
“.<z>” name the variables used in the two cases.
sage: ratpoly.<t> = PolynomialRing(QQ)
sage: realpoly.<z> = PolynomialRing(RR)
Now we illustrate the point about factoring :
sage: factor(t^2-2)
t^2 - 2
sage: factor(z^2-2)
(z - 1.41421356237310) * (z + 1.41421356237310)
Similar comments apply to matrices: the row-reduced form of a matrix can depend on the ring over which it is defined, as can its eigenvalues and eigenvectors. For more about constructing polynomials, see Polynomials, and for more about matrices, see Linear Algebra.
The symbol I represents the square root of 
; i is a
synonym for I. Of course, this is not a rational number:
sage: i # square root of -1
I
sage: i in QQ
False
Note: The above code may not work as expected if the variable i has been assigned a different value, for example, if it was used as a loop variable. If this is the case, type
sage: reset('i')
to get the original complex value of i.
There is one subtlety in defining complex numbers: as mentioned above,
the symbol i represents a square root of , but it is a
formal or symbolic square root of . Calling CC(i) or
CC.0 returns the complex square root of .
sage: i = CC(i) # floating point complex number
sage: i == CC.0
True
sage: a, b = 4/3, 2/3
sage: z = a + b*i
sage: z
1.33333333333333 + 0.666666666666667*I
sage: z.imag() # imaginary part
0.666666666666667
sage: z.real() == a # automatic coercion before comparison
True
sage: a + b
2
sage: 2*b == a
True
sage: parent(2/3)
Rational Field
sage: parent(4/2)
Rational Field
sage: 2/3 + 0.1 # automatic coercion before addition
0.766666666666667
sage: 0.1 + 2/3 # coercion rules are symmetric in SAGE
0.766666666666667
Here are more examples of basic rings in Sage. As noted above, the ring of rational numbers may be referred to using QQ, or also RationalField() (a field is a ring in which the multiplication is commutative and in which every nonzero element has a reciprocal in that ring, so the rationals form a field, but the integers don’t):
sage: RationalField()
Rational Field
sage: QQ
Rational Field
sage: 1/2 in QQ
True
The decimal number 1.2 is considered to be in QQ: decimal numbers
which happen to also be rational can be “coerced” into the rational
numbers. The numbers 
and 
are not rational, though:
sage: 1.2 in QQ
True
sage: pi in QQ
False
sage: pi in RR
True
sage: sqrt(2) in QQ
False
sage: sqrt(2) in CC
True
For use in higher mathematics, Sage also knows about other rings, such
as finite fields, 
-adic integers, the ring of algebraic numbers,
polynomial rings, and matrix rings. Here are constructions of some of
these:
sage: GF(3)
Finite Field of size 3
sage: GF(27, 'a') # need to name the generator if not a prime field
Finite Field in a of size 3^3
sage: Zp(5)
5-adic Ring with capped relative precision 20
sage: sqrt(3) in QQbar # algebraic closure of QQ
True