R1- Exponentiation – Definition

This post covers the definition of exponentiation in R1. It also includes reciprocals and examples.

Introduction

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 where the base is an Opposite Value or Counter and the exponent or power is a Counter. So, the exponent can be applied to a Counter or to the complete Opposite Value.

For example:

• 10^2v = 100^v because the exponent only applies to the Counter
• 10^v2 = 100^{^} because the exponent applies to the Opposite Value and 1^v*1^v = 1^{^}

Reciprocals

In Wave Numbers the exponent is a Counter. A flipped exponent is used to represent the reciprocal or multiplicative inverse of an Opposite Value as follows in R1. 

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

• 10^{^-2} = 1^{^}/10^{^2} = 1^{^}/100^{^} = 1^{^}/100 
• 10^v-2 = 1^{^}/10^v2 = 1^{^}/100^{^} = 1^{^}/100
• 10^v-3 = 1^{^}/10^v3 = 1^{^}/1000^v = 1^v/1000  

As noted earlier, a flipped exponent can also be used with the Counter of an Opposite Value. In the expression a^–n? = 1‡a^‡a/|a|ⁿ, the flip applies to the Counter. Note that this is not a reciprocal, but just another implementation of exponentiation. For example:

• 10^-2v = 1^v/10²  = 1^v/100

So, 10^-2v   = 1^v/100, but the recciprocal 10^v-2 = 1^{^}/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:

• 10^3v * 10^{2^} = 1000^v  * 100^{^}  = 100,000^v
  • = 10^{(3 + 2)(v*^)} =  10^5v= 100,000^v

• 10^v2 * 10^{2^} = 100^{^}  * 100^{^}  = 10,000^{^}
  • = 10^{(2 + 2)(^*^)} =  10^{4^}= 10,000^{^}
    • Note that ^v multiplied by itself twice ends as a ^{^} giving ^{^*^}

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

• 10^v4 * 10^v-2 = 10,000^{^}  * 1^{^}/10^v2  = 10,000^{^}  * 1^{^}/100^{^} = 10,000^{^}  * 1^{^}/100 = 100^{^}
  • = 10^{(4 + -2)(^*^)} =  10^{2^}= 100^{^}
    • Note that ^v multiplied by itself 4 times ends as a ^{^} and 10^v-2 = 1^{^}/100

• 10^v3 * 10^-2v = 1,000^v  * 1^v/100  = 10^{^}
  • = 10^{(3 + -2)(v*v)} = 10^{^}
    • Note that ^v multiplied by itself 3 times ends as a ^v

Division

To calculate 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:

• 10^3v/10^{2^} = 1000^v/100^{^}  = 10^v
  • = 10^{(3 + -2)(v/^)} =  10^1v= 10^v

• 10^v2/10^{2^} = 100^{^}/100^{^}  = 1^{^}
  • = 10^{(2 + -2)(^/^)} =  1^{0^}= 1^{^}

• 10^{3^}/10^{-1^} = 1000^{^}/(1^{^}/10)  = 10,000^{^} 
  • = 10^{(3 + – -1)(^/^)} =  10^{4^}= 10,000^{^}

• 10^v4/10^v-2 = 10,000^{^}/(1^{^}/10^v2) = 10,000^{^}/(1^{^}/100^{^}) = 10,000^{^}/(1^{^}/100) = 1,000,000^{^}
  • = 10^{(4 + – -2)(^/^)} =  10^{6^}= 1,000,000^{^}
    • Note that ^v multiplied by itself 4 times ends as a ^{^} and 10^v-2 = 1^{^}/100

• 10^v3/10^-2v = 1,000^v/(1^v/100)  = 100,000^{^}
  • = 10^{(3 + – -2)(v/v)} = 100,000^{^}
    • Note that ^v multiplied by itself 3 times ends as a ^v

Further examples of R1 Exponentiation:

1. 2^{^2} = 2^{^}*2^{^} = 4^{^}                                                          
2. 2^v2 = 2^v*2^v = 4^{^}
3. 2^{^3} = 2^{^}*2^{^}*2^{^} = 8^{^}                                                      
4. 2^v3 = 2^v*2^v*2^v = 8^v
5. 1,000^{^}*10^v-2 =  1,000^{^}*1^{^}/10^v2 =  1,000^{^}/100^{^} = 10^{^} = 10^{(3 + -2)(^*^)} = 10^{1^}= 10^{^}
6. 1,000^{^}*5^v-2 =  1,000^{^}*1^{^}/5^v2 =   1,000^{^}/25^{^} =   40^{^} 
   • = 10^{(3 + -.1.4)(^*^)} = 10^{(1.6)^} =   40^{^}
7. 10^{^3}/10^v2 = 1,000^{^}/100^{^} = 10^{^}
   • = 10^{(3 + -2)}(^{^}/^{^}) = 10^{1^} = 10^{^}
8. 10^v3/10^{^3} = 1,000^v/1,000^{^} = 1^v
   • = 10^{(3 + -3)}(^v/^{^}) = 10^0v = 1^v
9. 10^v3/10^v3 = 1,000^v/1,000^v = 1^{^}
   • = 10^{(3 + -3)}(^v/^v) = 10^{0^} = 1^{^}
10. 10^3v/10^v2 = 1,000^v/100^{^} = 10^v
    • = 10^{(3 + -2)}(^v/^{^}) = 10^1v = 10^v

Conclusion

Finally, our online calculator does not yet support exponentiation.

Next: Fractions, Flip Sign and Rules

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