Simple expressions

A cornerstone of a programming language is the  ability  to compute expressions.  We will start here  with  arithmetic expressions, just to take the  concept.   An expression consists  of a sequence of 'tokens' that show the application of operators to data component. Each expression has a number,  which  may be computed recursively by evaluating primitive expressions, and then using  standard principles to combine their values  to get new variables. Numerals, such -3.7 as or 6  are expressions, whose  numbers  are  numeric constants. Their numeric values  can be integers, within some  range  dictated by the  programming language, or ?oating point  variables. Floating  point  variables are used  to represent non-integer values,  but they  are not related, in many  important types,  from  the  real  numbers.  There  are in?nitely many real numbers within a ?nite  difference, but  only  finite various  ?oating-point numbers exist at all.  In fact, the normal laws of real arithmetic (transitivity, associativity, etc.)  are created in ?oating-point arithmetic, because  the results of any provided sub-computation may not be representable in the given number of bits.

We will explain the calculation of expressions in Python by showing short  transcripts of interactive sessions  with the Python shell : the shell is a computer program that

• Prompts the user for a relation, by typing »>,
• Reads what the user types in, and changes it into a set of tokens,
• Parses the tokens into a data structure showing the syntax of the expression,
• Computes the parsed expression using an interpreter, and
• Prints out the resulting value

So, for example, we might  have this interaction with Python:

>>> 2 + 3

5

>>> (3 * 8) - 2

22

>>> ((3 * 8) - 2) / 11

2

>>> 2.0

2.0

>>> 0.1

0.10000000000000001

>>> 1.0 / 3.0

0.33333333333333331

>>> 1 / 3

0

There are a couple  of things  to look here.  First, we can look how  ?oating point  values only approximately represent real numbers: when  we write in 0.1, the closest Python can take it in ?oating point  is 0.100000000000001.  The end interaction is particularly troubling: it seems like the value of the expression 1 / 3 should be something like 0.3333.  However, in Python, if both registers to the / operator are integer number, then it will operate an integer division.

These expressions can be arbitrarily deeply nested combinations of primitives. The laws used  for calculation are essentially the same as the ones you learned in school; the interpreter proceeds by applying the operations in precedence order13,  calculating sub-expressions to get new variables,  and then calculating the expressions those values  takes in, until in one value  results.