LL first meta-model – ISA\LL\Elementary\Numeral

Parent

Elementary

Data
Numeral
Aliases None.
Information This assembly is about the elementary numbers.
Assembly data
Edition
Version 1
Country None.
Language  None.
Tirage 1
Dependencies None.
Inhibit all invariants False.
Scope Ideas.
Diagram

Classes
Decimal
Information A real number (ℝ set) represented as a decimal value. The Decimal is composed of a 2 fixed parts: an integer on the left of the separator and a mantissa on the right. This representation support a lower range of values than floating point one: Approximate range from ±1,84467× 10-19 to ± 9,22337× 10+18

However it provides exact arithmetic computing except for division (fractions) and irrational numbers. This type is useful when  accuracy is more important that performance. Its performance is nevertheless better than string based representation.
Fields
Integer The integer part of the decimal as an Integer sbyte.
Mantissa The mantissa part of the decimal as an Integer sbyte.
Properties
Shared
+∞ ,
-∞		 Positive and negative infinity.

The “last” number, at ordinal 263-1, is noted +∞. The “first” number, at ordinal -263, is noted -∞. These ordinal values are a facility to supports the potential of infinity.
+0,
-0		 Positive and negative zero.
π Ratio of a circle’s circumference to its diameter: 3.14159…
e Euler’s number: 2.71828…
φ The golden number: 1.61803…
Unshared
String The string follows this regular expression: \-∞|\+∞|(0|\-[0-9]+|(\+)?[0-9]+)\.[0-9]+
Attribute
$ (Alias Value) The decimal value
Constructors
Decimal (◈)
 Base constructor.
Decimal (Number: Number)
 Conversion constructor from another Number if the origin type and Value are compatible.
Commands
+ Addition of the operands to the current object if they are ⊥.
– Subtraction of the operands to the current object if they are ⊥.
×
 Multiplication of the operands to the current object if they are ⊥.
÷
 Division of the operands to the current object if they are ⊥.
Round to
 Round with a given number of decimal following a rounding mode.
Round to multiple
 Round to the desired multiple.

Example: Rounding 10.1 to the nearest multiple of 3.0 gives 9.0.
 Queries
=,⊥ Overrides the Type queries.
<, >, ≤, ≥ Overrides the Ordered queries.
Invariants
Name Value greater that (263)-2 (~ 9,22337× 10+18) is not allowed
Information  Upper value limit.
Name
Value least that (-263)+1 (~ -9,22337× 10+18) is not allowed
Information  Lower value limit.
Name
The minimum absolute value and the resolution are 2-64 (~1,84467× 10-19)
Information  Mantissa domain extend.
Decimal
Floating point
Information A real number (ℝ set) represented as a floating point following the IEEE 754 standard. A Floating point representation can support a much wider range of values over fixed point representation: Approximate range from ±5.0 × 10−324 to ±1.7 × 10308
However it do not provides exact computing. This type is useful when performance is more important that accuracy. Its performance mainly come for of its support by a wide range of hardware components.
Properties
Shared
+∞, -∞ Positive and negative infinity.
+0,-0 Positive and negative zero.
Not a number The not a number value (NaN).
π Ratio of a circle’s circumference to its diameter: 3.14159…
e Euler’s number: 2.71828…
φ The golden number: 1.61803…
Unshared
String The string follows this regular expression: \-∞|\+∞|NaN|(0|\-[0-9]+|(\+)?[0-9]+)\.[0-9]+
Attribute
$ (Alias Value) The floating point value
Constructors
Floating point (◈)
 Base constructor.
Floating point (Number: Number)
 Conversion constructor from another Number if the origin type and Value are compatible.
Commands
+ Addition of the operands to the current object if they are ⊥.
– Subtraction of the operands to the current object if they are ⊥.
×
 Multiplication of the operands to the current object if they are ⊥.
÷
 Division of the operands to the current object if they are ⊥.
 Queries
=,⊥ Overrides the Type queries.
<, >, ≤, ≥ Overrides the Ordered queries.
Invariants
Name
Value follows the IEEE 754 standard
Information Value domain extend.
Floating point
Integer element
Information The Integer element abstract class provides the common features to integers concrete classes.
Properties
Shared
+∞ The “last” positive number, at ordinal 263-1, is noted +∞.  This ordinal value is a facility to support the potential of infinity.
-∞ The “last” negative number, at ordinal 263, is noted -∞.  This ordinal value is a facility to support the potential of infinity.
Attribute
$ (Alias Value) The integer value
 Query
Mod The modulo operation.

Given two integers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n.
Commands
+ Addition of the operands to the current object if they are ⊥.
– Subtraction of the operands to the current object if they are ⊥.
×
 Multiplication of the operands to the current object if they are ⊥.
÷
 Division of the operands to the current object if they are ⊥.
Invariants
Name Value greater that (2^63)-2 is not allowed
Information  Upper value limit.
Note
Note kind Explanation
Text As most Elements, the integers are mutable in particular for efficiency reasons.

The basic operations like addition or multiplication are Commands. This implies that the value of the current object is altered by these operations. So there are no “flying” operators like in most languages where they are some kind of keywords. There is also not something almost similar implemented as Queries, like: +(Operand: Integer) : Integer.
; because this kind of operation is definitely not a query.
Integer element
Integer
Information An integer number of the mathematics ℤ set.
Property
String The string follows this regular expression: 0|\-∞|\+∞|\-[0-9]+|(\+)?[0-9]+
Constructors
Integer (◈)
 Base constructor.
Integer (Number: Number)
 Conversion constructor from another Number if the origin type and Value are compatible.
 Query
⊥ Overrides the Type queries.
Invariants
Name Value least that (-2^63)-1 is not allowed
Information Lower value limit.
Integer
Natural
Information A natural integer superior or equals to zero (ℕ set).
Property
String The string follows this regular expression: 0|\+∞|(\+)?[0-9]+
Constructors
Natural (◈)
 Base constructor.
Natural (Number: Number)
 Conversion constructor from another Number if the origin type and Value are compatible.
 Query
⊥ Overrides the Type queries.
Invariants
Name
Value least that 0 is not allowed
Information Lower value limit.
Natural
Natural*
Information A natural integer strictly superior to zero (ℕ* set).
Property
String The string follows this regular expression: \+∞|(\+)?[1-9][0-9]*
Constructors
Natural* (◈)
 Base constructor.
Natural* (Number: Number)
 Conversion constructor from another Number if the origin type and Value are compatible.
 Query
⊥ Overrides the Type queries.
Invariants
Name
Value least that 1 is not allowed
Information Lower value limit.
Natural*
Real element
Information The Real element abstract class is the common root of reals concrete classes.
Real element
Other modules
Rounding mode
Information The mode a real number, typically a Decimal is rounded.
Field
Names Information
Nearest Rounding to number that is the closest following the tie-breaking rules with round half up.
Up Rounds a number up, away from 0 (zero).
Down Rounds a number down, toward zero.
Truncate Round towards zero by cutting the ending digits.
Rounding mode
Assemblies

None.

 

Version and status

Version Date Description Status
V1R1 20-Jan-18 Initial. Candidate

Related posts:

  1. Elements Assembly
  2. Textuals Assembly
  3. Temporal Assembly
  4. Metamodel Assembly
  5. Nominals (Elements) Assembly

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