Assignment, Equality, and Arithmetic¶
With some minor exceptions, Sage uses the Python programming language, so most introductory books on Python will help you to learn Sage.
= for assignment. It uses
sage: a = 5 sage: a 5 sage: 2 == 2 True sage: 2 == 3 False sage: 2 < 3 True sage: a == 5 True
Sage provides all of the basic mathematical operations:
sage: 2**3 # ** means exponent 8 sage: 2^3 # ^ is a synonym for ** (unlike in Python) 8 sage: 10 % 3 # for integer arguments, % means mod, i.e., remainder 1 sage: 10/4 5/2 sage: 10//4 # for integer arguments, // returns the integer quotient 2 sage: 4 * (10 // 4) + 10 % 4 == 10 True sage: 3^2*4 + 2%5 38
The computation of an expression like
3^2*4 + 2%5 depends on
the order in which the operations are applied; this is specified in
the “operator precedence table” in Arithmetical binary operator precedence.
Sage also provides many familiar mathematical functions; here are just a few examples:
sage: sqrt(3.4) 1.84390889145858 sage: sin(5.135) -0.912021158525540 sage: sin(pi/3) 1/2*sqrt(3)
As the last example shows, some mathematical expressions return
‘exact’ values, rather than numerical approximations. To get a
numerical approximation, use either the function
N or the method
n (and both of these have a longer name,
N is the same as
n)). These take optional
prec, which is the requested number of bits of
digits, which is the requested number of decimal
digits of precision; the default is 53 bits of precision.
sage: exp(2) e^2 sage: n(exp(2)) 7.38905609893065 sage: sqrt(pi).numerical_approx() 1.77245385090552 sage: sin(10).n(digits=5) -0.54402 sage: N(sin(10),digits=10) -0.5440211109 sage: numerical_approx(pi, prec=200) 3.1415926535897932384626433832795028841971693993751058209749
Python is dynamically typed, so the value referred to by each variable has a type associated with it, but a given variable may hold values of any Python type within a given scope:
sage: a = 5 # a is an integer sage: type(a) <type 'sage.rings.integer.Integer'> sage: a = 5/3 # now a is a rational number sage: type(a) <type 'sage.rings.rational.Rational'> sage: a = 'hello' # now a is a string sage: type(a) <... 'str'>
The C programming language, which is statically typed, is much different; a variable declared to hold an int can only hold an int in its scope.