# Expression -- the class of all expressions

## Description

An expression is a symbolic representation of a mathematical expression. It retains some of the semantics of the mathematical expression, as well as enough information to print the expression nicely. In Macaulay2 expressions have two main functions: they are an intermediate phase in the conversion of a mathematical object to a net that can be printed; and they are a way of holding and displaying a mathematical expression in an unevaluated form that can be both printed and evaluated.

Internally, each expression is a basic list whose elements may also be expressions. The elements that are not expressions are interpreted as themselves, and may be strings, symbols, numbers, etc. There are several types of expression that correspond to various sorts of mathematical entities, such as sums of class Sum, products, of class Product, fractions of class Divide, etc.

Expressions are produced with the function expression. The various methods installed for it try to bring as much of the semantic structure of the mathematical object to light. The following examples illustrate that, using peek and peek' to display the internal structure.

 ```i1 : expression 4 o1 = 4 o1 : Expression of class Holder``` ```i2 : peek oo o2 = Holder{4}``` ```i3 : d = expression (-4) o3 = -4 o3 : Expression of class Minus``` ```i4 : peek oo o4 = Minus{4}``` `i5 : QQ[x];` ```i6 : f = (x+1)^5 5 4 3 2 o6 = x + 5x + 10x + 10x + 5x + 1 o6 : QQ[x]``` ```i7 : peek f o7 = QQ[x]{x5+5x4+10x3+10x2+5x+1}``` ```i8 : e = expression f 5 4 3 2 o8 = x + 5x + 10x + 10x + 5x + 1 o8 : Expression of class Sum``` ```i9 : peek e 5 4 3 2 o9 = Sum{x , 5x , 10x , 10x , 5x, 1}``` ```i10 : peek'_2 e 5 4 3 2 o10 = Sum{Product{x }, Product{5, x }, Product{10, x }, Product{10, x }, ----------------------------------------------------------------------- Product{5, x}, 1}``` ```i11 : peek'_11 e o11 = Sum{Product{Power{x, 5}}, Product{5, Power{x, 4}}, Product{10, Power{x, ----------------------------------------------------------------------- 3}}, Product{10, Power{x, 2}}, Product{5, x}, 1}```
The function factor returns an expression.
 ```i12 : c = factor f 5 o12 = (x + 1) o12 : Expression of class Product``` ```i13 : peek'_2 c o13 = Product{Power{x + 1, 5}}``` ```i14 : factor 240012 2 2 o14 = 2 3 59*113 o14 : Expression of class Product```
Expressions can be evaluated using value.
 ```i15 : value e 5 4 3 2 o15 = x + 5x + 10x + 10x + 5x + 1 o15 : QQ[x]``` ```i16 : value e == f o16 = true``` ```i17 : value c 5 4 3 2 o17 = x + 5x + 10x + 10x + 5x + 1 o17 : QQ[x]```
The following operators can be applied to expressions: SPACE, *, **, +, -, /, ==, ^, and _. They are contagious, in the sense that when applied to an expression and a non-expression, the non-expression will be converted to an expression and the operator will be applied. Only the most trivial algebraic simplications are applied.
 ```i18 : d + e 5 4 3 2 o18 = - 4 + x + 5x + 10x + 10x + 5x + 1 o18 : Expression of class Sum``` ```i19 : d + 4 o19 = - 4 + 4 o19 : Expression of class Sum``` ```i20 : d / 4 -4 o20 = -- 4 o20 : Expression of class Divide``` ```i21 : d / 1 o21 = -4 o21 : Expression of class Minus``` ```i22 : d == e 5 4 3 2 o22 = -4 == x + 5x + 10x + 10x + 5x + 1 o22 : Expression of class Equation```

## Functions and methods returning an expression :

• expression -- convert to an expression
• hold -- hold something in a holder expression

## Methods that use an expression :

• Equation == Expression, see == -- equality
• Expression == Equation, see == -- equality
• - Expression
• Expression * Expression
• Expression ** Expression
• Expression + Expression
• Expression - Expression
• Expression / Expression
• Expression == Expression
• Expression ^ Expression
• Expression _ Expression
• Expression Expression
• value(Expression)
• Expression ..< Expression (missing documentation)
• Expression ..< Holder (missing documentation)
• Expression ..< Thing (missing documentation)
• Expression and Expression (missing documentation)
• Expression or Expression (missing documentation)
• Holder ..< Expression (missing documentation)
• Thing ..< Expression (missing documentation)

## For the programmer

The object Expression is a type, with ancestor classes BasicList < Thing.