Math.Arithmetic.Multiplication

Math.Arithmetic.Multiplication is the tag for the expression representing the mathematical multiplication.

Definition

Math.Arithmetic.Multiplication is the tag for the expression representing the mathematical multiplication.

Behavior

This expression must have at least two children.

Children names

The subexpressions should be named "Factor 1", "Factor 2", etc. First factor must be numbered as one, contrary to several programming languages, in which first element is numbered as zero.

Localization packages could be created in order to provide the names in languages other than English.

Visualization

There could be different visualizations, being the most popular the infix one, that us, using the multiplication symbol between each consecutive pair of factors, followed by the prefix form and rarely the postfix form.

The following symbols are commonly used for infix visualization:

Symbol Example Use  Arithmetic  Algebra
(a single space) High mathematics

Edition

There could be several editions. The most obvious is: Given a selected expression, it is replaced by a new multiplication expression with two children, the first one is the expressions that was selected, and the second one is a new Null expression. The latter becomes the new selected expression.

Reduction

Ther could be several reductions, see below.

Known implementations

The Standard Arithmetic Package

Visualization

There are three visualizations on this package. All of them are infix.

These visualizations align the horizontal baselines of the factors, such like the following example:

Infix cross (×) symbol among factors

This visualization shows the multiplication in infix form, using a cross (×) as a symbol, such the following example: Infix dot (•) symbol among factors

This visualization shows the multiplication in infix form, using a dot (•) as a symbol, such the following example: Infix space among factors

This visualization shows the multiplication in infix form, using a single space as a symbol, such the following example: Parentheses awareness

The visualizations in this package awares whether some of its children belong to the parentheses category Parentheses as operator or it is also a Math.Arithmetic.Multiplication. In such that case, these children are drawn between parentheses.

In the following example, the second factor is an addition. Because this expression belongs to the parentheses category Parentheses as operator the parent multiplication shows this subexpression between parentheses.

In the next example, the second factor is also a multiplication, so the parent multiplication shows this subexpression between parentheses.

Parentheses categories

The visualizations in this package belongs to the category Parentheses as super/subscripted. It means that if this visualization is a child of a parent visualization that awares of this, that such parent expression will show this child between parentheses.

In the following example, the Math.Arithmetic.Exponentiation visualization awares of either its base or exponent to belong of that category, to show either the base or the exponente between parentheses (in this case is the base):

Edition

There are several editions in this package, which are the combinations of a default edition with two variantions:

Variation Meaning
default (no variation) Creates a new Math.Arithmetic.Multiplication Expression, the current selected expression becomes the first factor, a new Null expression is created and becomes the second factor.
Before The new expression (usually a Null expression) will be created as the first factor, instead of the second one
Forced Usually, if the current selected expression is already a Math.Arithmetic.Multiplication expression, or a factor of a Math.Arithmetic.Multiplication expression, a new factor is not created. With this option a multiplication is always created.

In the following examples, the selected expression is neither a Math.Arithmetic.Multiplication or a factor, so the "Forced" variant behaves as the default edition.

Example Edition performed Result  Multiplication  Multiplication (forced) Multiplication (before)  Multiplication (forced, before)

In the following examples, the selected expression is a Math.Arithmetic.Multiplication or a factor, so the "Forced" variation applies:

Reduction

Multiplication of numeric factors

This reducer searches on the Math.Arithmetic.Multiplication children, for numeric expressions. If there are more than one, performs a numeric multiplication. According to Arithmetic canon, a numeric expression is either:

The following are examples. In these examples, expressions , and are supposed not to be numeric expressions.

Expression before reduction Expression after reduction Explanation The same It does not reduce, there are no numeric expressions as addends. The same It does not reduce, there are not two or more numeric expressions as addends.    According to Arithmetic canon, if any operand is a decimal number, the result must be either.    According to Arithmetic canon, if any operand is a decimal number, the result must be either.  If there is a reduction, and the numeric result of this reduction is one, it will be removed of the Math.Arithmetic.Multiplication expression, unless it remains as the only expression.

The following are examples. Again, and are supposed not to be numeric expressions.

Expression before reduction Expression after reduction Explanation  One is removed.  One is not removed, it remains as the only child.

According to Arithmetic canon, if any operand is a decimal number, the result must be either.

Flatten nested factors

This reducer searches into the current Math.Arithmetic.Multiplication for children being also Math.Arithmetic.Multiplication expressions. For each one of them, it integrates their factors as factors of the current Math.Arithmetic.Multiplication expression.

For example, the following -three factor- multiplication: reduces to the following -five factor- multiplication: In the above example, and expressions are supposed not to be Math.Arithmetic.Multiplication expressions.

Elimination of negative factors, maybe producing a negative multiplication

This reducer searches into the current Math.Arithmetic.Multiplication for children being a Math.Arithmetic.Negative expression and removing it (the Math.Arithmetic.Negative, not the entire expression). If there were an odd number of such these removals, the entire current Math.Arithmetic.Multiplication will be put under a Math.Arithmetic.Negative.

The following are examples. , , and are supposed not to be negative expressions.

Expression before reduction Expression after reduction Explanation The same No negative factors  An even number of negative factors removed  An odd number of negative factors removed

Numeric factors first

This reducer searches for factors being numeric expressions to move to the beginning, so the following expression ( , and are suppose not to be numeric expressions): Will be reduced to Left distributivity of numeric factors

This reducer searches for expression of the following form ( and are suppose not to be numeric expressions): To be converted to The Standard list package

This package contains neither visualizers nor editors for the Math.Arithmetic.Multiplication expression.

Reduction

Matrix multiplication

In mathematics, A matrix is an array of rows and columns. It is also said that this is a matrix. According to the List canon in the Fōrmulæ framework, a matrix is represented as a Structure.List expression containing Structure.List expressions, each one with elements. In other words, it is a list of equal-size sublists.

Matrix multiplication between an matrix and a matrix results in a matrix.

This reducer searches for contiguous pairs of expression representing matrices as previously defined --a list of equal-size sublists-- that match the size requeriments, and performs a matrix multiplication, substituting them by the new resulting multiplaction matrix.

The following are examples. In these examples, expressions , and are supposed not to be matrices, according to the definition already given.

Expression before reduction Expression after reduction Explanation The same It does not reduce, there are no matrix expressions as factors. The same It does not reduce, there are not two or more matrix expressions as factors. The same It does not reduce, number of columns in the first matrix is not equal to the number of rows of the second one.  The expressions shown in orange color are the result given by this reducer. If these expression are reduced to another expressions will be the result of application of other reducers, but not by this.