July 22, 2024
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R2 – Exponentiation – Definition

Introduction

This post covers the definition of R2 Exponentiation in Wave Numbers.  It also includes reciprocals and examples.

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 Number Definition

In Wave Numbers the exponent uses the same definition. 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. For example:

  1. 102iv = 100iv as the exponent only applies to the Counter
  2. 10iv2 = 100v because the exponent applies to the Opposite Value and iv*iv = 1v
  1. 102v = 100v as the exponent only applies to the Counter
  2. 10v2 = 100^ as the exponent applies to the Opposite Value and 1v*1v = 1^
  3. 1,000^*1,000 = 1,000,000v =103^*103v = 106v because 1^*1v = 1v
  4. 1,000i^*1,000iv  = 1,000,000^ =103i^*103iv = 106^ because i^*iv = 1^
  5. 10^2*10v2 = 100^*100^ = 10,000^ = 104^ because 1^2 = 1^ and 1v2 = 1^
  6. 10^2*10iv2 = 100^*100v = 10,000v = 104v because 1^2 = 1^ and iv2 = 1v and 1^*1v = 1v
  7. 103v*10v3 = 1,000v*1,000v = 1,000,000^= 106^
  8. 103iv*10iv3 = 1,000iv*1,000i^ = 1,000,000^= 106^

The examples above show that the exponents are added when multiplying. In order to get the correct sign for the result, the signs for each element need to be calculated and the results multiplied.

Reciprocals

In R2 exponentiation the exponent is a Counter. So, a flipped exponent is used to represent the reciprocal or multiplicative inverse as follows. 

a??-n = 1^/a‡aa – The flip applies to the Opposite Value. For example:

  • 10v-2 = 1^/10v2 = 1^/100^ = 1^/100 
  • 10iv-2 = 1^/10iv2 = 1^/100v = 1v/100 

 In the expression a-n?? = 1??/an – The flip applies to the Counter. For example:

  • 10-2v = 1v/102  = 1v/100
  • 10-2iv = iv/102  = iv/100

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

  • 103^ * 10-1^ = 1000^  * 1^/10  = 100^ 
  • = 10(3 + -1)(^/^) =  102^= 100^

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:

  • 103iv * 102^ = 1000iv  * 100^  = 100,000iv
    • = 10(3 + 2)(iv*^) =  105iv= 100,000iv

  • 10iv2 * 102^ = 100v  * 100^  = 10,000v
    • = 10(2 + 2)(v*^) =  104v= 10,000v
      • Note that iv2v 

  • 103i^ * 10-1^ = 1000i^  * 1^/10  = 100i^
    • = 10(3 + -1)(i^*^) =  102i^ = 100i^

  • 10iv4 * 10iv-2 = 10,000^  * 1^/10iv2  = 10,000^  * 1^/100v = 10,000^  * 1v/100 = 100v
    • = 10(4 + -2)(^*v) =  102v= 100v
      • Note that iv4^  and 10iv-2 = 1v/100

  • 10iv3 * 10-2iv = 1,000i^  * iv/100  = 10^
    • = 10(3 + -2)(i^*iv) = 10^
      • Note that iv3 = i^ 

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:

  • 103iv/102^ = 1000iv/100^  = 10iv
    • = 10(3 + -2)(iv/^) =  101iv = 10iv

  • 10v2/102i^ = 100^/100i^  = 1iv
    • = 10(2 + -2)(^/i^) =  10iv = 1iv

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

  • 10v4/10iv-2 = 10,000^/(1^/10iv2) = 10,000^/(1^/100v) = 10,000^/(1v/100) = 1,000,000v
    • = 10(4 + – -2)(^/v) =  106v= 1,000,000v
      • Note that v4^ and 10iv-2 = 1v/100 giving ^/v

  • 10iv3/10-2i^ = 1,000i^/(1^/100i^) = 1,000i^/(iv/100)  = 100,000v
    • = 10(3 + – -2)(i^/iv) = 100,000v
      • Note that iv3 = i^  

Further examples of R2 Exponentiation:

  1. 2^2 = 2^*2^ = 4^                                                          
  2. 2v2 = 2v*2v = 4^
  3. 2^3 = 2^*2^*2^ = 8^                                                      
  4. 2v3 = 2v*2v*2v = 8v
  5. 1,000^*10v-2 =  1,000^*1^/10v2 =  1,000^/100^ = 10^ 
  6. 1,000^*5v-2 =  1,000^*1^/5v2 =   1,000^/25^ =   40^ 
    • = 10(3 + -.1.4)(^*^) = 10(1.6)^ = 40^
  7. 10^2*10iv2 = 100^*100v = 10,000v = 104v
    • = 10(2 + 2)(^*v) = 104v
  8. 10i^3*10v2 = 1,000iv*100^ = 100,000iv = 105iv
    • = 10(3 + 2)(iv*^) = 105iv
  9. 10iv3*10i^3 = 1,000i^*1,000iv = 1,000,000^
    • = 10(3 + 3)(i^*iv) = 106^
  10. 10i^3/10v2 = 1,000iv/100^ = 10iv
    • = 10(3 + -2)(iv/^) = 101iv = 10iv
  11. 10iv3/10i^3 = 1,000i^/1,000iv = 1v
    • = 10(3 + -3)(i^/iv) = 100v = 1v
  12. 10iv3/10iv3 = 1,000i^/1,000i^ = 1^
    • = 10(3 + -3)(i^/i^) = 100^ = 1^
  13. 103iv/10iv3 = 1,000iv/1,000i^ = 1v
    • = 10(3 + -3)(iv/i^) = 100v = 1v
  14. 104^/10v3 = 10,000^/1,000v = 10v
    • = 10(4 + -3)(^/v) = 10v

Next: Fractions, Flip Sign and Rules

Previous: Roots

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