ADDITION AND SUBSTRACTIONN

"Numbers govern the universe" (Pythagoras).

"The One begets the Two. The Two begets the Three. The Three begets the ten thousand beings" (Lao-Tse).

"Arithmetic is the grammar of numbers" (Witgenstein).

"Number is to be found in another space, not in this space" (George Spencer−Brown).

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Addition
Semantics and Syntax

The sum of two expressions, specified by means of

x+y // reads "x plus y"

is as follows:

• If x and y are equal, the result is symbolized by 2*x, where 2 represents a multiplicative factor.
  
  
• If x and y are different, the result is the same expression, i.e., the expression is self-evaluating.

In general, the result of the expression (r1*x + r2*y), where r1 and r2 are real numbers, is as follows:

• If x and y are equal, the result is (r1+r2)*x, where r1+r2 indicates the traditional arithmetic sum.
  
  
• If the expressions are different, the result is the same expression (the expression is self-evaluating).

When in the general expression (r1*x + r2*y), x and y are equal to 1, then we have the traditional arithmetic sum: r1+r2.


Justification

The Sum primitive is the classical arithmetic operation, but generalized for any expressions.


Examples

1. abc+abc // ev. 2*abc
   
   
2. (2*x + y) // ev. 2*x+y
   
   
3. (2*x + 3*x) // ev. 5*x
   
   
4. (x/a + x/a) // ev. 2*(x/a)
   
   
5. (apple + 3*apple) // ev. 4*apple
   
   
6. (a b)+(a b) // ev. 2*(a b)
   
   
7. ({a b c} + 4*{a b c}) // ev. 5*{a b c}
   
   
8. ⟨(x y)⟩+⟨(x y)⟩ // ev. 2*⟨(x y)⟩
   
   
9. (x + y + x) // ev. (2*x + y)
   
   
10. (13 + 27) // ev. 40

Remarks

• The expression n*x represents x+...+x (n summands). For example,
  
  (3*x =: x+x+x)
  
  
• The form r*x allows you to formally specify quantities of objects and physical quantities. For example,
  
  23*apple 18*Kgr 42*meter

Axioms

1. ⟨( x+y ≡ y+x )⟩ // commutativity
   
   
2. ⟨( x+(y+z) ≡ (x+y)+z )⟩ // associativity
   
   By default, the associativity is to the left, as expressions are evaluated from left to right: ⟨ ( x+y+z ≡ (x+y)+z )⟩
   
   
3. ⟨( r*x ≡ x*r )⟩ The forms x*r and r*x, by definition, are equivalent: (2*x + x*3) // ev. 5*x
   
   
4. ⟨( x*r = r*x )⟩
   
   Any expression of the form x*r, where x is not a number, is replaced by r*x, with the numerical factor in front.
   
   
5. ⟨( (r1+r2)*x =: (r1*x + r2*x) )⟩ // by the definition of sum

⟨( (r1*x + r2*x) = (r1+r2)*x )⟩ // by the definition of sum
   
   
6. ⟨( r*(x+y) =: (r*x + r*y) )⟩ // distribution

⟨( (r*x + r*y) = r*(x+y) )⟩ // distribution
   
   
7. ⟨( x+θ = x )⟩ // null expression
   
   
8. ⟨(x ≡ 1*x)⟩ and ⟨(1*x = x)⟩
   By default, the multiplier attribute of an expression is 1.
   
   
9. ⟨( 0*x = 0 )⟩ // an expression with zero multiplier attribute, is zero
   
   Therefore, also: (0*0 = 0) (0*α = 0) (0*θ = 0) (0*Ω = 0)
   
   
10. ⟨( r1*(r2*x) ≡ (r1*r2)*x )⟩ // associativity in the multiplicative factor
    
    By default, the associativity is to the left, as expressions are evaluated from left to right: ⟨( r1*r2*x ≡ (r1*r2)*x )⟩

Theorems

1. ⟨((x+y)^2 = (x*x + 2*x*y + y*y))⟩
   
   By the axioms of distribution and commutativity (Newton's binomial).
   This theorem generalizes to any positive integer exponent.

Substraction
Syntax

(x +' y) or (x - y) // x minus y


Definition

⟨(((x−y = z) ↔ (x = z+y))⟩


Remarks

• Since addition is commutative, the left hand inverse operator is equal to the right hand inverse operator.
  
  
• It follows from the definition that subtracting two expressions, each with a corresponding multiplicative attribute, is:
  
  If the expressions differ only in the multiplier attribute, the result is the same expression with multiplier attribute equal to the traditional subtraction.
  
  If the expressions are different, the result is the same expression.

Examples

1. (5*x − x*3) // ev. 2*x
   
   
2. x−y // self-evaluates
   
   
3. (2*x − y) // self-evaluates
   
   
4. (2*x − 5*x) // ev. (−3)*x ev. −3*x

Axioms

1. ⟨( (x+y)−x = y )⟩
   
   
2. ⟨( (x+y)−y = x )⟩
   
   
3. ⟨( x−0 = x )⟩
   
   
4. ⟨( x−θ = x )⟩
   
   
5. ⟨( (r1*x - r2*x) = (< b>r1−r2)*x )⟩ // by the definition of subtraction

⟨( (r1−r2)*x =: (r1*x - r2*x) )⟩ // by the definition of subtraction
   
   
6. ⟨( r*(x−y) =: (r*x − r*y) )⟩

⟨( (r*x − r*y) = r*(x−y) )⟩

Theorems

1. ⟨( x−x = 0 )⟩
   
   Demonstration: x−x  eq.  (1*x − 1*x)  eq.   (1−1)*x  ev.  0*x  ev.  0

Sign of an Expression
By definition, the sign of an expression is the sign (+ or −) of the numerical value of its multiplying attribute, according to the following


Axioms

1. ⟨( (+r)*x ≡ +(r*x) ≡ r*x )⟩
   
   
2. ⟨( (−r)*x ≡ −(r*x) ≡ −r*x )⟩
   
   
3. ⟨( 0−x = −x )⟩
   
   
4. ⟨( θ−x = −x )⟩
   
   
5. ⟨( −(−r)*x) = r*x )⟩
   
   
6. ⟨( −(x+y) = −x−y )⟩
   
   
7. ⟨( x+(+y) = x+y )⟩
   
   
8. ⟨( x+(−y) = x−y )⟩
   
   
9. ⟨( x−(+y) = x−y )⟩
   
   
10. ⟨( x−(−y) = x+y )⟩

Examples

1. (−3)*x // ev. −3*x
   
   
2. (+3)*x // ev. 3*x
   
   
3. +x // ev. x