Page 131 - 'Blast_Into_Math

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

Therefore, each element of S is less than or equal to n +1. This means that S is a non-empty set of

integers which is bounded above, so by the LUB Property, S has a unique largest element. Let’s call it
k . Since k is the largest element of S, this means that

k +1 /∈ S.

Since k +1 ∈ Z , and k +1 >k ≥ 0 (because k ∈ S which means k ≥ 0), the only reason for
k +1 /∈ S is that b > n + 1. So, we know that
k+1

k
b ≤ n +1 <b k+1 .

We also know that k is unique because it is the largest element of S .

k
Following the steps of our examples, we now need to find the digit which goes with b . We can do this
with a set. Let

k
T = {x ∈ N such that x ∗ b ≤ n +1}.

We are looking for the largest x ∈ T. . As usual, we need to check that T is not empty. Well, we already
know that b ≤ n +1. So,
k

k
1 ∗ b ≤ n +1,

which means that

1 ∈ T.

Next, we need to check that T is bounded above. We also know that

b k+1 >n +1.

This means that every element of T must be less than b, because for any x ≥ b ,

k
k
x ∗ b ≥ b ∗ b = b k+1 >n +1.
So, since T is a non-empty set of integers which is bounded above, by the LUB Property it contains a
unique largest element. Let’s call it x . Since x ∈ Z and 1 ≤ x ≤ b − 1, x is a digit in base b.

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