DECIMAL POINT

"Number is plurality or combination of units" (Aristotle).

"Number is the exponent of an operation" (Wittgenstein, Tractatus 6.021).

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Semantics

Converting a composite number to a simple number (simple sequence of digits). A composite number is a sequence of numbers, which in turn, can contain numbers.


Syntax

The "decimal point" (.) is used in three possible ways:

1. Decimal point to the right (postfix): x.
2. Left decimal point (prefix): .y
3. Middle decimal point (infix): x.y

Definition

1. ⟨( x. =: (x ← (x# = 1) →' (x\(x#))). +
10*([x\[1…((x#)−1)]).) )⟩ // (recursive definition)
   
   For example,
   (13 5 71 4). = 4 + (13 5 71).*10
(13 5 71). = 71 + (13 5).*10
(13 5). = 5 + 13*10
   
   
2. ⟨( .y =: (y. ÷ 10^(y#)) )⟩
   
   For example,
   .1234 = 1234. ÷ 10^4
   
   
3. ⟨( x.y =: (x. + .y) )⟩
   
   For example,
   1234.5678 = 1234. + .5678

Examples

1. (7 12 4). // ev. (7*(10^2) + 12*10 + 4) ev. 824
   
   
2. (42 57). // ev. (42*10 + 57) ev. 477
   
   
3. (7 (12 13) 4). // ev. (7*100 + (12 13).*10 + 4) ev. (7*100 + (12*10 + 13)*10 + 4) ev. 2034.
   
   
4. (1 −23 4). // ev. (100 − 230 + 4) ev. −126
   
   
5. .(4 (5 6) 3) // ev. .(400 + 560 + 3) ev. .963
   
   
6. (3 .4 5). // ev. (300 + 4 + 5) ev. 309
   
   
7. .(3 .4 5) // ev. .(300 + 4 + 5) ev. .309
   
   
8. .(3 .41 5) // ev. .(300 + 4.1 + 5) ev. .(309.1) ev. .3091
   
   
9. (12 13).(1 4 15) // ev. 12*10 + 13 + 1*10−1 * 4*10−2 + 15*10−3 ev. 133.155
   
   
10. 123. // self-evaluates
    
    
11. .123 // self-evaluates
    
    
12. 123.456 // self-evaluates

Remarks

• The definition of "decimal point" uses potential substitution (representation), but it could be defined as immediate substitution.
  
  
• A simple sequence of digits is interpreted as a sequence or as a number, depending on the context, that is, on the type of operation that affects it. This is the only case of polymorphism admitted in MENTAL, justified by the greater ease of use [see MENTAL Language - Principles - Polymorphism]. For example,
  
  (123 ∪ 456) // ev. 123456 (interpreted as sequences)

(123 + 456) // ev. 579 (interpreted as numbers)

(123 * 456) // ev. 56088 (are interpreted as numbers)

((1 2 3) + (4 5 6)) // ev. 579 (interpreted as numbers)

((1 25 3). + (4 5 65).) // ev. 353 + 515 ev. 868
  
  
• A particular case of composite number is that of a sequence formed by digits (each with positive or negative sign). Examples:
  
  (3 −4 −5). // ev. (300 −40 − 5) ev. 255
.(3 −4 −5) // ev. .(300 −40 - 5) ev. .255
  
  In Vedic Mathematics the negative sign is represented above each digit. For example, 345 and .345

Flexible representation of numbers

Since a number can be composed of other numbers, a flexible representation system is available, allowing a number to be represented in multiple ways.

For example, the number 17 can be represented as:

17 (2 −3). (3 −13). (4 −23). (1.7 0). (1.6 1). (1.5 2). (1.44 2.6). (1.92 −2.2). etc.

Analogously, the number .17 can be represented as:

.17 .(2 −3) .(3 −13) .(4 −23) .(1.7 0) .(1.6 1) .(1.5 2) .(1.44 2.6) .(1.92 −2.2) etc.


Normalization of a number

A number is said to be in normalized format when:

1. The sequence is composed of digits only.
2. There is only one sign for the entire sequence.

To switch to normalized format, the operator "." (decimal point) is applied. Examples:

1. (x = (1.44 2.8))
x. // ev. 1.44*10 + 2.8 ev. 17.2
   
   
2. ((x = 13) (y = 122))
(x y). // ev. 13*10 + 122. ev. 252
   
   
3. (x = 12)
(x★3). // ev. (12 12 12 12). ev. 1332

Variable number

It is a sequence (hierarchical or not) variables may appear. When those variables have concrete values, the sequence is evaluated as a concrete normalized number. Examples:

1. (2 x 1). // rep. (2*100 + x + 1) ev. 201+x
   
   
2. (x = ab1)
(a=3 b=2)
x // ev. 321
   
   
3. (x = (a (b c)).
x/(a=1 b=2 c=3) // rep. 10 + (2*10 + 3) ev. 33

Properties

1. ⟨( (m. = m) )⟩
   
   The "decimal point" operation has no effect on an integer. Examples:
   
   123. // ev. 123
3. // ev. 3
−12. // ev. −12
   
   
2. ⟨( .(x.) = .x )⟩
   
   
3. ⟨( x.. = x. )⟩ // additional dots on the right have no effect
   
   
4. ⟨( ..x ≡ (.x ÷ 10) )⟩
   
   The additional dots on the left play the same role as the zeros. Examples:
   
   (..123 ≡ .0123)
(...123 ≡ .00123)
   
   
5. ⟨( x.θ = x. )⟩
   
   
6. ⟨( θ.x = .x )⟩
   
   
7. ⟨( x.0 = x )⟩
   
   
8. ⟨( 0.x = .x )⟩
   
   
9. ⟨( (−x).y = (−x + .y) )⟩
   
   
10. ⟨( x.(−y) = (x − .y) )⟩
    
    
11. ⟨( (−x).(−y) = (−x + . −y) = −(x.y) = −x.y) )⟩
    
    
12. ⟨( ((x y). + (z u)).) ≡ ((x+z). (y+u).). )⟩
    
    Examples:
    ((5 4). + (8 9).) // ev. 54 + 89 ev. 143
((5+8). (4+9).). // ev. (13 13). ev. 143.
    
    
13. ⟨((x y). * (z).)) ≡ ((x. * z.) (y. * z.)).⟩ // product distribution
    
    Examples:
    ((5 4). * 3) // ev. 54*3 ev. 162
((5*3) (4*3)). // ev. (15 12). ev. 15*10+12 ev. 162
    
    
14. ⟨( −(x y). ≡(−x −y). )⟩ // sign distribution
    
    This property is a consequence of the previous one. Example:
    −(7 6). // eq. (−7 −6). ev. −76
    
    
15. ⟨( (x.y − .y) = x. )⟩
    
    
16. ⟨( (x.y − x.) = .y )⟩