R3 – Exponentiation – Definition

Introduction

This post covers the definition of R3 Exponentiation in Wave Numbers.

In Wikipedia, exponentiation is defined as follows:

‘Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base  b and the exponent or power n, and pronounced as “b raised to the power of n”. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:

bⁿ = b * b … * b * b 

———————

<—- n times ——>’

Wave Numbers Definition

In Wave Numbers the exponent uses the same definition where the base is an Opposite Value or Counter and the exponent or power is a Counter. The exponent can be applied to a Counter or to the complete Opposite Value.

Note: There is no distinction between exponents applied to Counters and Opposite Values in R3. This is due to the multiplication and division tables where multiplication or division of an Opposite Value by an Opposite Value with the same Opposite Type and Sign results in an Opposite Value of the same Opposite Type and Sign. As a result the two formats are interchangeable. For example:

1. 10²j^v = 100j^v  = 10j^v2
2. 10i^v2 = 100i^v = 10²i^v 
3. 1,000i^{^}*1,000i^v  = 1,000,000i^v =10³i^{^}*10³i^v 
   • = 10⁶i^v = 10i^v6 because i^{^}*i^v = i^v
4. 1,000j^{^}*1,000j^v  = 1,000,000j^v =10³j^{^}*10³j^v 
   • = 10⁶j^v = 10j^v6 because j^{^}*j^v = j^v
5. 10^{2^}*10^2v = 100^{^}*100^v = 10,000^v 
   • = 10^4v = 10^v4 because 1^{^}*1^v = 1^v
6. 10^{2^}*10i^2v = 100^{^}*100i^v = 10,000j^v 
   • = 10⁴j^v = 10j^v4 because 1^{^}*i^v = j^v
7. 10^3v*10^v3 = 1,000^v*1,000^v = 1,000,000^v= 10^6v = 10^v6
8. 10³i^v*10i^v3 = 1,000i^v*1,000i^v = 1,000,000i^v= 10⁶i^v = 10i^v6

The examples above show that the exponents are added when multiplying. To get the correct sign for the result, the product of the Opposite Types and Signs of each element needs to taken from the multiplication table.

Reciprocals

In R3 exponentiation the exponent is a Counter. According to the theorem of Multiplicative Inverses, R3 does not have multiplicative inverses. However there are reciprocals in R3 and a flipped Counter is used to represent them as follows. 

a?^?-n = 1‡a^‡a/aⁿ – The unitary numerator and the divisor have the same Opposite Type and Sign. For example:

• 10j^v-2 = j^v/10² = j^v/100 
• 10 i^v-2 = i^v/10² = i^v/100  

Flipped exponents can be added in order to allow for the calculation of equations without multiplication or division, such as:

• 10j^{^3} * 10j^{^-1}= 1000j^{^}  * j^{^}/10  = 100j^{^} 
• = 10^{(j^/j^)(3 + –1)} =  10²j^{^}= 100j^{^}

Adding Exponents

Exponents can be added in order to allow for the calculation of equations without doing a full multiplication or division.

Multiplication

To calculate a multiplication, add the exponents. The Opposite Sign of the result needs to be calculated based on the multiplication taking place. For example:

• 10³i^v * 10²j^{^} = 1000i^v  * 100j^{^}  = 100,000^v
  • = 10^{(3 + 2)(iv*j^)} =  10^{5^}= 100,000^v

• 10j^v2 * 10^{2^} = 100j^v  * 100^{^}  = 10,000i^v
  • = 10^{(2 + 2)(jv*^)} =  10⁴i^v = 10,000i^v

• 10³i^{^} * 10^-1j^{^} = 1000i^{^}  * 1j^{^}/10  = 100^{^}
  • = 10^{(3 + -1)(i^*j^)} =  10^{2^} = 100^{^}

• 10i^v4 * 10i^v-2 = 10,000i^v  * 1i^v/100  = 100i^v
  • = 10^{(4 + -2)(iv*iv)} =  10²i^v = 100i^v

Division

To calculate a division, add the flip of the divisor’s exponent to the numerator’s. The Opposite Sign of the result needs to be calculated based on the division taking place. For example:

• 10³i^v/10²j^{^} = 1000i^v/100j^{^}  = 10^v
  • = 10^{(3 + -2)(iv/j^)} =  10^1v = 10^v

• 10^v2/10²i^{^} = 100^v/100i^{^}  = 1j^v
  • = 10^{(2 + -2)(v/i^)} =  1⁰j^v = 1j^v

• 10³i^{^}/10^-1i^{^}  = 1000i^{^}/(i^{^}/10)= 10,000i^{^}
  • = 10^{(3 + – -1)(i^/i^)} =  10⁴i^{^} = 10,000i^{^}

• 10j^v4/10i^v-2 = 10,000j^v/(i^v/100) = 1,000,000^v
  • = 10^{(4 + – -2)(jv/iv)} =  10^6v= 1,000,000^v

• 10i^v3/10^-2i^{^} = 1,000i^v/(i^{^}/100) = 100,000i^v
  • = 10^{(3 + – -2)(iv/i^)} = 100,000i^v

Further examples of R3 Exponentiation

1. 2^{2^} = 2^{^}*2^{^} = 4^{^}                                        
2. 2^2v = 2^v*2^v = 4^v
3. 2^{3^} = 2^{^}*2^{^}*2^{^} = 8^{^}                                                      
4. 2^3v = 2^v*2^v*2^v = 8^v
5. 1,000i^{^}*10^–2 i^v =  1,000i^{^}*(i^v/10²) 
   • =  1,000i^v/100 = 10i^v = 10^(3+–2)i^{(i^*iv)} = 10i^v
6. 10^–2 i^v*1,000i^{^} =  (i^v/10²)*1,000i^{^} 
   • =  1,000i^{^}/100 = 10i^{^} = 10^{(–2 +3)}i^{(iv*i^)} = 10i^{^}
7. 1,000j^{^}*5^{–2 v} =  1,000j^{^}*(1^v/5²) 
   • = 1,000j^{^}*1^v/25 =  1,000i^v/25 = 40i^v 
   • = 10^{(3+–1.4)(j^*v)} = 10^(1.6)i^v = 40i^v 
8. 10^{2^}*10²i^v = 100^{^}*100i^v = 10,000j^v 
   • = 10⁴j^v = 10^{(2+2)(^*iv)} = 10⁴j^v
9. 10³i^{^}*10²j^v = 1,000i^{^}*100j^v = 100,000^v 
   • = 10^5v = 10^{(3+2)(i^*jv)} = 10^5v
10. 10³i^{^}/10²j^v = 1,000i^{^}/100j^v 
    • = 10^v = 10^(3+–2)(i^v/j^{^}) = 10^1v = 10^v
11. 10³i^v/10³i^{^} = 1,000i^v/1,000i^{^} 
    • = i^v = 10^(3+–3)(i^v/i^{^}) = 10⁰i^v = i^v

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