Formulas for Converting Tk to Exentation Number n
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The original function defining the exponent k for an exentation n is:
\(e(n) = begin{cases} 0 & text{if } n=1 (-1)^n cdot lfloor n/2 rfloor & text{if } n>1 end{cases}\)To find the exentation number n from an exponent k, we derive the following three conditions based on the definition of e(n):
- If the exponent k is equal to 0: This only occurs when n=1.
If the exponent k is positive (k>0): Positive exponents are generated when n is an even number. In this case, (−1)n=1, and the formula simplifies to k=⌊n/2⌋. Since n is even, ⌊n/2⌋=n/2. Therefore:
\(k = n/2 rightarrow n = 2k quad text{if } k > 0\)If the exponent k is negative (k<0): Negative exponents are generated when n is an odd number. In this case, (−1)n=−1, and the formula is k=−⌊n/2⌋. For an odd n, ⌊n/2⌋=(n−1)/2. Substituting this into the equation:
\(k = -(n-1)/2\)Multiplying by −2:
\(-2k = n-1\)Solving for n:
\(n = -2k + 1 quad text{if } k < 0\)Summary of Inverse Formulas
Combining these three conditions, the formula for finding the exentation number n given an exponent k is:
\(n = begin{cases} 1 & text{if } k = 0 2k & text{if } k > 0 -2k + 1 & text{if } k < 0 end{cases}\)