Page 130 - 'Blast_Into_Math

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

We have just proven the base case. To use induction, we assume the theorem is true for all positive
integers less than or equal to n , for some n ∈ N . Then we show there is only one way to write n +1
in base b . What was the first step in our examples? We found the largest power of the base which is not
bigger than the number. So, in this case, we are looking for the largest power of b which is not bigger
than n +1. To find this power, we can use a set. Let’s define the set

k
S = {k ∈ Z such that k ≥ 0and b ≤ n +1}.

We’re looking for the largest element of S. First, we need to show that S = ∅. . Since we proved the base
case which was n =1, we know that n +1 ≥ 2, which means that

0
0 ∈ S because b =1 ≤ n +1.

Therefore, S = ∅. Next, we need to show that S is bounded above. To do this, we can use the Base
Lemma, which says that for any n ∈ N,

n
n
b ≥ 2 >n.

By the Base Lemma, if k ∈ N with k> n +1, then

k
b ≥ k> n +1.

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