User:Thepigdog/Value

In mathematics, value is a Canonical form of a mathematical expression. This means a value is an expression written in an agreed standard form such that two expressions may be compared for equality by converting each expression into a canonical form (by valid mathematical steps) and then literal comparing the values obtained from the two expressions. The two expressions are equal if the values are literally the same.

Converting an expression into the canonical form is called evaluating the expression.

Historically a value has referred to quantity or worth, or numerical value. However common usage uses the term value to apply to other mathematical objects. Boolean values such as true and false are not related to quantity.

In the more limited sense, a numerical value of a natural number is the canonical representation of a number, as a series of digits. A value may occur as:
 * A value of a variable or a constant is any number or other mathematical object assigned to it.
 * A value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are replaced by some values.
 * A value of a function is the result associated to a value of its argument (also called variable of the function).

For example, if the function $$f$$ is defined by $$f(x) = 2x^2-3x+1$$, then, given the value 3 to the variable x yields the function value 10 (since indeed 2 · 32 – 3 · 3 + 1 = 10). This is denoted $$f(3)=10.$$

Value of numbers
Starting with numbers we may want to know if,


 * $$ 55 + 67 = 85 + 37 $$

This may be achieved by evaluating each side of the equation,


 * $$ 122 = 122 $$

Then as 122 on the left side is identical to 122 on the right side the expressions are equal.

122 may be regarded as a shorthand for,
 * $$ 1 * 10^2 + 2 * 10^1 + 2 * 10^0 $$

This is the agreed canonical form that we put numbers into. Each number is uniquely identified by its canonical form.

Value of sets
The value of sets may be considered as a canonical form of a set. Using these forms sets may be tested for equality.

, a set represented as,


 * $$\{1, 3, 2, 2\}$$

is not a value. It does not uniquely identify the set. So if we were to compare,


 * $$\{1, 3, 2, 2\} = \{1, 2, 3\}$$

The two expressions are not the same, but the values are equal. So a canonical form or "value" for a set, has each element represented once, and in a sorted order. For example,


 * $$\{1, 3, 2\} \cup \{3, 8, 9\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \cap \neg \{4, 5, 6, 7\} $$
 * $$\{1, 2, 3, 8, 9\} = \{1, 2, 3, 8, 9\} $$

As the left and right hand sides are in sorted order and each element only appears once the two expressions are equal because the canonical forms are the same.

Value of functions
There is no standard form which you can can convert functions into that allow you to compare them. Function equality is defined by two functions being equal if they give the same value for each argument value in the domain.

Two expressions defining the same function may look completely different, but calculate the same result due to potentially deep reasons. In fact the equality of functions is undecidable.

This makes functions potentially difficult to deal with, as values of variables.

Alternative definition
A value may also be considered as a normal form. However a normal form may be defined as the result left when no more rules are left that will match the expression. This definition is unsound mathematically, as it does not guarantee the form is unique. So this definition may make constructs that are inconsistent.