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Blast into Math! Mathematical perspectives: all aour mase are melong to us

We have found the largest power of b which is not larger than n +1, and we have found the digit

corresponding to this power of b : the power k and the digit x are both unique. Like in the examples,
once we’ve found the biggest power of the base and its digit, we subtract x ∗ b from n +1,
k
k
n +1 − x ∗ b .

We need to write n +1 − x ∗ b in base b , and show that there is only one way to do it. First, we
k
know that

k
n +1 − x ∗ b ≤ n,

k
because b ≥ 1, and x ≥ 1 since x is the largest element of T , and 1 ∈ T . We also know that

k
n +1 − x ∗ b ∈ Z,

because the integers are closed under addition, subtraction, and multiplication. Finally, by definition of
the set T , and since x ∈ T , we know that

k
k
x ∗ b ≤ n +1 whichmeans n +1 − x ∗ b ≥ 0.

It’s time to use the induction hypothesis. First, if

k
n +1 = x ∗ b ,

we’re done, because the remaining digits of n +1 must all be 0. Otherwise, if

n +1 >xb k

then

k
n +1 − x ∗ b ≥ 1,

and since

k
n +1 − x ∗ b ≤ n,

the induction hypothesis says the theorem is true for n + 1 – x*b . So there is a unique way to write
k
n +1 − x ∗ b in base b. We can just add this to x ∗ b and we end up with n +1 in base b . Since the
k
k
first part, x ∗ b is uniquely written in base b, and the remaining part n +1 − x ∗ b is also uniquely
k
k
written in base b, this means that

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