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.
Sage uses ``=`` for assignment. It uses ``==``, ``<=``, ``>=``, ``<`` and ``>`` for
comparison:
::
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 :ref:`section-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, ``numerical_approx``, and
the function ``N`` is the same as ``n``)). These take optional
arguments ``prec``, which is the requested number of bits of
precision, and ``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)
sage: a = 5/3 # now a is a rational number
sage: type(a)
sage: a = 'hello' # now a is a string
sage: type(a)
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.
A potential source of confusion in Python is that an integer
literal that begins with a zero is treated as an octal number,
i.e., a number in base 8.
::
sage: 011
9
sage: 8 + 1
9
sage: n = 011
sage: n.str(8) # string representation of n in base 8
'11'
This is consistent with the C programming language.