Show that every nonzero integer can be uniquely represented

Show that every nonzero integer can be uniquely represented in the form e_k 3^k + e_k - 1 3^k - 1 + middot middot middot + e_1 3 + e_0 where e_j = -1, 0, or 1 for j = 0, 1, 2, ..., k and e_k notequalto 0. This expansion is called a balanced ternary expansion.

Solution

Using Mathematical Induction:

We\'ll prove that every non-zero Integer has a unique ternary expansion in the range [3^j, 3^ (j+1)] where j0

Basis: Clearly, the statement holds for j=0

1=3^0, 2=3^13^0, 3=3^1

Induction: If the statement is true until some integer k1, then it true for k.

Let\'s consider the range [3^k, 3^ (k+1)]. Since, it is true until k1, hence every Integer in the range [1, 3^k] has a unique ternary representation. For each representation in this range, we add 3^k. We get a unique representation for the integers in the range [3^k+1, 2.3^k].

Consider adding 3^ (k+1) - 3^k=2.3^k to each integer representation in the range [1, 3^k]. We have a unique representation for integers in the range [2.3^k+1, 3^ (k+1)]

Combining both, we have a unique representation for integers in the range [3^k, 3^ (k+1)] which implies the statement is true for the integer k.

Hence Proved.