Assignment, Equality, and Arithmetic - Tutorial

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# 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

    sage: a = 5
    sage: a
    5
    sage: 2 == 2
    True
    sage: 2 == 3
    False
    sage: 2 < 3
    True
    sage: a == 5
    True

Python

    >>> from sage.all import *
    >>> a = Integer(5)
    >>> a
    5
    >>> Integer(2) == Integer(2)
    True
    >>> Integer(2) == Integer(3)
    False
    >>> Integer(2) < Integer(3)
    True
    >>> a == Integer(5)
    True

Sage provides all of the basic mathematical operations:

Sage

    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

[Python]

    >>> from sage.all import *
    >>> Integer(2)**Integer(3)    #  ** means exponent
    8
    >>> Integer(2)**Integer(3)     #  ^ is a synonym for ** (unlike in Python)
    8
    >>> Integer(10) % Integer(3)  #  for integer arguments, % means mod, i.e., remainder
    1
    >>> Integer(10)/Integer(4)
    5/2
    >>> Integer(10)//Integer(4)   #  for integer arguments, // returns the integer quotient
    2
    >>> Integer(4) * (Integer(10) // Integer(4)) + Integer(10) % Integer(4) == Integer(10)
    True
    >>> Integer(3)**Integer(2)*Integer(4) + Integer(2)%Integer(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

    sage: sqrt(3.4)
    1.84390889145858
    sage: sin(5.135)
    -0.912021158525540
    sage: sin(pi/3)
    1/2*sqrt(3)

Python

    >>> from sage.all import *
    >>> sqrt(RealNumber('3.4'))
    1.84390889145858
    >>> sin(RealNumber('5.135'))
    -0.912021158525540
    >>> sin(pi/Integer(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

    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]

    >>> from sage.all import *
    >>> exp(Integer(2))
    e^2
    >>> n(exp(Integer(2)))
    7.38905609893065
    >>> sqrt(pi).numerical_approx()
    1.77245385090552
    >>> sin(Integer(10)).n(digits=Integer(5))
    -0.54402
    >>> N(sin(Integer(10)),digits=Integer(10))
    -0.5440211109
    >>> numerical_approx(pi, prec=Integer(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

    sage: a = 5   # a is an integer
    sage: type(a)
    <class 'sage.rings.integer.Integer'>
    sage: a = 5/3  # now a is a rational number
    sage: type(a)
    <class 'sage.rings.rational.Rational'>
    sage: a = 'hello'  # now a is a string
    sage: type(a)
    <... 'str'>

Python

    >>> from sage.all import *
    >>> a = Integer(5)   # a is an integer
    >>> type(a)
    <class 'sage.rings.integer.Integer'>
    >>> a = Integer(5)/Integer(3)  # now a is a rational number
    >>> type(a)
    <class 'sage.rings.rational.Rational'>
    >>> a = 'hello'  # now a is a string
    >>> 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.