ARITHMETIC OPERATIONS

Product, Power, Hyperpower, Division, Root, Logarithm
"Arithmetic does not talk about numbers, but works with numbers" (Witgenstein. Philosophical Investigations).

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Repetitive Sum (Product)
Semantics and Syntax

The expression x+x+...+x (n summands), where n is a natural number, is a repeating sum and evaluates to n*x. In turn, n*x represents x+x+...+x (n summands).

When x is a real number, the operation is the product of x times n.


Definitions

Reduction:
⟨( +⊣( x★n ) = n*x ) )⟩

Representation:
⟨( n*x =: (x ←' n>1 → (x + (n−1)*x) )⟩


Examples

1. a+a+a // ev. 3*a
2. 3*a // rep. a+a+a

Properties

1. ⟨( 1*x = x )⟩
   
   
2. ⟨( (n*x + n*y) = n*(x+y) )⟩
⟨( (n*x − n*y) = n*(x−y) )⟩
   
   
3. ⟨( (n1*x + n2*x) = (n1+n2)*x )⟩
⟨( (n1*x - n2*x) = (n1−n2)*x )⟩
   
   
4. ⟨( 0*x ≡ (n−n)*x ≡ (n*x − n*x) = 0 )⟩

Remarks

• By definition, the expression x*n is evaluated as n*x, i.e., the numeric factor is prefixed: ⟨( x*n = n*x )⟩. For example, a*3 evaluates to 3*a.
  
  
• The repetitive addition operation can be generalized to any real number r (expressible in finite form): r*x. For example,
  
  3.7*a // rep. (a + a + a + a + 0.7*a)
  
  If the real number is negative, we have property
  
  ⟨( −r*x ≡ (−r)*x) ≡ −(r*x) )⟩

Repetitive Product (Power)
Syntax and semantics

The expression x*x*...*x (n factors), where n is a positive integer, is a repeated product and evaluates to x^n. In turn, x^n represents x*x*...*x (n factors).

When x is a number, the operation is called "power of base x and exponent n".


Definitions

Reduction:
⟨( *⊣( x★n ) = x^&n ) )⟩

Representation:
⟨( x^n =: (x ←' n>>1 → (x * x^(n−1))) )⟩


Examples

1. a^3 // rep. a*a*a
2. a*a*a // ev. a^3

Properties

1. ⟨( x^1 = x )⟩
⟨( 1^n = 1 )⟩
   
   
2. ⟨( (x^n1)*(x^n2) = x^(n1+n2) )⟩
   
   
3. ⟨( (x^n1)÷(x^n2) = x^(n1−n2) )⟩
   
   
4. ⟨( x^0 ≡ x*(n−n )≡ (x^n)÷(x^n) = 1) )⟩
   
   
5. ⟨( (0^n = 0) )⟩
   
   
6. ⟨( x^n)*x = x^(n+1) )⟩
   
   
7. ⟨( x^n)÷x = x^(n−1) )⟩

Remarks

• As in the case of repetitive addition, the repetitive product operation can be generalized to any real number r: x^r. For example,
  
  a^3.7 // rep. (a * a * a * a * a^0.7)
  
  If the real number is negative, we have property
  
  ⟨( −r*x ≡ (−r)*x) ≡ −(r*x) )⟩
  
  
• There is an analogy between the definitions of the product and power operations.

Repetitive Power (Hyperpower)

A repetitive power is a higher order power, which we can call hyperpower.

Suppose we want to represent the expression

((((x^x)^x)^x^...) (n times x)

Let's specify this expression for the cases n=1, 2, 3, etc. until we get to the general case.

n=2) x^x eq. ^⊣( x★2 )
n=3) (x^x)^x eq. ^⊣( x★3 )
n=4) ((x^x)^x)^x eq. ^⊣( x★4 )

In general, you have the expression ^⊣( x★n ), where the evaluation is performed from the left.

On the other hand, the expression

(x^(x^(x^(x^...)))) (n times x)

would be specified by the expression (^⊣( x★n ))∼, with the evaluation starting from the right. For example, for n = 4, you have x^(x^(x^x)).

Using the notation ⟨( x(^^)n =:  ^&⊣( x★n ) )⟩, the two types of hyperpowers are:

x(^^)n (left evaluation)
(x(^^)n)∼ (right evaluation)


Higher-order hyperpowers

Using the general notation: ^^ for hyperpower of order 1, ^^^ for hyperpower of order 2, etc., we have the general definition:

Left-hand evaluation:

⟨( (x ( ^★k )n) = (^★(k−1) )⊣( x★n )⟩

Evaluation by the right:

⟨( (x ( ^★k ) n)∼ = ((^★ (k−1) )⊣( x★n ))∼ )⟩

Examples:

1. (x ^^ 4) = x^x^x^x^x = ((x^x)^x)^x&x
   
   
2. (x ^^^^ 4) = (x ^^^ x ^^ x^^ x) = (((x ^^ x) ^^ x) ^^ x) ^^ x)
   
   
3. (2 ^^^ 3) = (2 ^^^ 2 ^^ 2) = ((((2 ^^ 2) ^^ 2) = (2^2)^(2^2) = 4^4 = 256

Contrary Arithmetic Operations: Division, Root, Logarithm

Contrary Multiplication

According to hyper-semantics,

⟨(x(*')y = z) ★(x = z*y)⟩ ⟨(x('*)y = z) ★ (x = y*z)⟩

The *' operator is also symbolized as ÷, and is the division operation.

Examples:

1. 21.(*')3 // ev. 21.÷3 ev. 7
21.('*)3 // ev. 7
   
   
2. x = 3*a
x(*')a // ev. x÷a ev. 3
x('*)3 // ev. x÷3 ev. a

Properties:

1. Hyper-semantics.
   
   ⟨( (x*y)(*')y = x )⟩
⟨( (x*y)('*)x = y )⟩
   
   
2. In case x and y are numbers, the opposite multiplication on the right and on the left are equal:
   
   ⟨( x(*')y ≡ x('*)y )⟩
   
   
3. ⟨( x÷x ≡ (1**x)(*')x = 1 )⟩
   
   
4. ⟨( x÷1 ≡ (1*x)(*')1 = x )⟩
   
   
5. ⟨( (x+y)÷z ≡ (x÷z + y÷z) )⟩
   
   
6. ⟨( x*(y÷z) ≡ (x*y)÷z )⟩
   
   
7. ⟨( 0÷x ≡ (y−y)÷x = (y÷x − y÷x) = 0 )⟩

Contrary power (root and logarithm)

According to hyper−semantics, we have seen in the generic operation "Contrary":

x(^')y es y√x
x('^)y es logyx
⟨( x((^')')y ≡ x^y )⟩ // x^y
⟨( x('(^'))y ≡ y('^)x )⟩ // logxy
⟨( x((('^)')y ≡ y^x )⟩ // y^x
⟨( x('('^))y ≡ y(^')x )⟩ // x√y

Properties:

1. ⟨( x(^')1 = x )⟩
   
   
2. ⟨( x('^)x = 1 )⟩
   then logxx = 1
   
   
3. ⟨( 1('^)x = 0 )⟩
   then logx1 = 0
   
   
4. ⟨( (x^y)('^)x = y )⟩
   then logx(x^y) = y
   
   
5. ⟨( (x^y)(^')y = x )⟩
   
   
6. ⟨( (x^y)('^)y ≡ y*(x('^)y) )⟩
   then logy(x^y) = y*logyx
   
   
7. ⟨( (y^z)('^)x ≡ z*(y('^)x) )⟩
   then logx(y^z) = z*logxy

Arithmetic operations depending on whether the numbers are integers or real numbers

The rule is:

If the operands are integers, the operation is performed in integer.

If any operand is real, the operation is performed in real.

By definition, a real number is one that contains a decimal point.

For example, in integer division n1÷n2, the result is an integer.

The remainder of the division is:
(n1 - (n1÷n2)*n2)

Real division: (r1÷r2). The result is a real number.

Conversion to integer: (n = r).

Conversion to real: (r = n.).
The integer variable n is converted to real by adding a decimal point to the right.

Examples:

1. (n = 13)
n÷2 // ev. 6
n.÷2 // ev. 6.5 (the operation is performed in real)
(13 - (13÷2)*2) // ev. 1 (remainder of division)
   
   
2. (n = 13.8) // conversion to integer
n // ev. 13
   
   
3. 2(^')2 = 1 // square root of 2 (integer operation)
2.(^')2 = 1.4142 // square root of 2 (operation in real)
2(^')2 = 1 // base 2 logarithm of 2 (integer operation)
2(^')2. = 1. // logarithm in base 2 of 2 (operation in real)

Contrary hyperpowers

We will use the following notations:

Contrary-right hyperpower of order k:

⟨( ( ((x ( ^★k )' y) = z) ← (x ( ^★k ) z) = y) )⟩

Contrary-left hyperpower of order k:

⟨( ((x '( ^★k ) y) = z) ← (x ( ^★k ) z) = x) )⟩

A simpler notation is achieved by using the following definitions:

(v =: '^') (v' =: '^)

⟨( (v★k ) =: (^★k )' )⟩ // contrary-right hyperpotential

⟨( (v★k )' =: '( ^★k ) )⟩ // countrary-left hyperpotency

Examples:

1. (xv2) // square root of 2
2. (x v' 2) // logarithm of 2
3. (x vv 2) // inverse hyper power to the right
4. (x vv' 2) // inverse hyperpower to the left