Basic Rings

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:

  • the integers \(\{..., -1, 0, 1, 2, ...\}\), called ZZ in Sage.

  • the rational numbers – i.e., fractions, or ratios, of integers – called QQ in Sage.

  • the real numbers, called RR in Sage.

  • the complex numbers, called CC 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 \(x^2-2\) has two roots, \(\pm \sqrt{2}\). 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 coefficients 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)

>>> from sage.all import *
>>> ratpoly = PolynomialRing(QQ, names=('t',)); (t,) = ratpoly._first_ngens(1)
>>> realpoly = PolynomialRing(RR, names=('z',)); (z,) = realpoly._first_ngens(1)

ratpoly.<t> = PolynomialRing(QQ)
realpoly.<z> = PolynomialRing(RR)

Now we illustrate the point about factoring \(x^2-2\):

sage: factor(t^2-2)
t^2 - 2
sage: factor(z^2-2)
(z - 1.41421356237310) * (z + 1.41421356237310)

>>> from sage.all import *
>>> factor(t**Integer(2)-Integer(2))
t^2 - 2
>>> factor(z**Integer(2)-Integer(2))
(z - 1.41421356237310) * (z + 1.41421356237310)

factor(t^2-2)
factor(z^2-2)

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 \(-1\); 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

>>> from sage.all import *
>>> i  # square root of -1
I
>>> i in QQ
False

i  # square root of -1
i in QQ

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')

>>> from sage.all import *
>>> reset('i')

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 \(-1\), but it is a formal square root of \(-1\) as an algebraic number. Calling CC(i) or CC.0 or CC.gen(0) returns the complex square root of \(-1\). Arithmetic involving different kinds of numbers is possible by so-called coercion, see Parents, Conversion and Coercion.

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

>>> from sage.all import *
>>> i = CC(i)       # floating point complex number
>>> i == CC.gen(0)
True
>>> a, b = Integer(4)/Integer(3), Integer(2)/Integer(3)
>>> z = a + b*i
>>> z
1.33333333333333 + 0.666666666666667*I
>>> z.imag()        # imaginary part
0.666666666666667
>>> z.real() == a   # automatic coercion before comparison
True
>>> a + b
2
>>> Integer(2)*b == a
True
>>> parent(Integer(2)/Integer(3))
Rational Field
>>> parent(Integer(4)/Integer(2))
Rational Field
>>> Integer(2)/Integer(3) + RealNumber('0.1')       # automatic coercion before addition
0.766666666666667
>>> RealNumber('0.1') + Integer(2)/Integer(3)       # coercion rules are symmetric in Sage
0.766666666666667

i = CC(i)       # floating point complex number
i == CC.0
a, b = 4/3, 2/3
z = a + b*i
z
z.imag()        # imaginary part
z.real() == a   # automatic coercion before comparison
a + b
2*b == a
parent(2/3)
parent(4/2)
2/3 + 0.1       # automatic coercion before addition
0.1 + 2/3       # coercion rules are symmetric in Sage

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

>>> from sage.all import *
>>> RationalField()
Rational Field
>>> QQ
Rational Field
>>> Integer(1)/Integer(2) in QQ
True

RationalField()
QQ
1/2 in QQ

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 (see Parents, Conversion and Coercion). The numbers \(\pi\) and \(\sqrt{2}\) 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

>>> from sage.all import *
>>> RealNumber('1.2') in QQ
True
>>> pi in QQ
False
>>> pi in RR
True
>>> sqrt(Integer(2)) in QQ
False
>>> sqrt(Integer(2)) in CC
True

1.2 in QQ
pi in QQ
pi in RR
sqrt(2) in QQ
sqrt(2) in CC

For use in higher mathematics, Sage also knows about other rings, such as finite fields, \(p\)-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

>>> from sage.all import *
>>> GF(Integer(3))
Finite Field of size 3
>>> GF(Integer(27), 'a')  # need to name the generator if not a prime field
Finite Field in a of size 3^3
>>> Zp(Integer(5))
5-adic Ring with capped relative precision 20
>>> sqrt(Integer(3)) in QQbar # algebraic closure of QQ
True

GF(3)
GF(27, 'a')  # need to name the generator if not a prime field
Zp(5)
sqrt(3) in QQbar # algebraic closure of QQ