.. _section-rings: 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 coefficents and real coefficients, respectively. The sets are named "ratpoly" and "realpoly", but these aren't important here; however, note that the strings "." and "." name the *variables* used in the two cases. :: sage: ratpoly. = PolynomialRing(QQ) sage: realpoly. = PolynomialRing(RR) Now we illustrate the point about factoring x^2-2: .. link :: 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 :ref:section-poly, and for more about matrices, see :ref:section-linalg. The symbol I represents the square root of :math:-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 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 -1, but it is a *formal* or *symbolic* square root of -1. Calling CC(i) or CC.0 returns the *complex* square root of -1. Arithmetic involving different kinds of numbers is possible by so-called coercion, see :ref:section-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 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 (see :ref:section-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 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