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

2. Next, we need to find the digit for this power of the base. This is the largest integer between
0 and 11 so that when multiplied with 12, the result is not bigger than 100. This integer is
8 because 8 ∗ 12 =96 but 9 ∗ 12 = 108 > 100. So, we write

1
1
100 =8 ∗ 12 +(100 − 8 ∗ 12) =8 ∗ 12 +4.
3. Now, since 4 < 12, it will be the ones digit because

0
4= 4 ∗ 12 .

In general, for any x ∈ N which is smaller than the base, we write x in base b just as x .
This is because

0
x = x ∗ b andsince x< b, x is a digit in base b.

4. Finally, we have

0
1
100 =8 ∗ 12 +4 ∗ 12 .
This means we would write 100 in base 12 as 84.

With these examples in mind, we can follow the same steps to prove the All Your Base Theorem. The
theorem is named after an internet meme from the late 1990s.

Exercise: Look up “all your base” on the internet and find how the meme began. (Hint: look for the
introduction of a certain video game.)

Theorem 6.1.4 (All your base). Let b be a natural number such that b ≥ 2. Then, there is a unique way

to write each n ∈ N in base b.

Proof: This theorem is a statement for all n ∈ N . So, it makes sense to prove the theorem by induction.
Let’s start with n =1. We can only write 1 as 1 in any base because

0
b =1 forall b ∈ N.

So, the theorem is true for n =1, because there is one unique way to write 1 in any base, and that’s 1.

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