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

So, 2 n+1 is greater than 2n which is greater than or equal to n +1. This means that

2 n+1 >n +1.

n
n
Since any base b ≥ 2, for any n ∈ N , b ≥ 2 . Therefore, for any base b ,
b n+1 ≥ 2 n+1 >n +1.

This proves the inequality also holds for n +1, so by induction, it is true for every n ∈ N .

♥

To get familiar with the mathematical definitions of base and digits, let’s play out some examples.

In our usual base b =10, when we write a number like 43, we would say that the tens digit is 4, and
the ones digit is 3. This means

1
0
43 =4 ∗ 10 +3 ∗ 10 .
The number 1357 means the thousands digit is 1, the hundreds digit is 3, the tens digit is 5, and the

ones digit is 7, and

2
0
3
1357 =1 ∗ 10 +3 ∗ 10 +5 ∗ 10 +7 ∗ 10 .
What does it mean when some of the digits are 0? Sometimes the digit multiplying a certain power of
the base is 0, like 200 which we can write as

2
1
0
200 =2 ∗ 10 + 0 ∗ 10 + 0 ∗ 10 .
The main theorem of this chapter, the All Your Base Theorem, tells us that we can uniquely write any
natural number in any base. This theorem is a cornerstone of modern computing, because when we
apply the theorem to base b =2, it says that a computer can store any number uniquely as a list of 0s
and 1s. The uniqueness is important because otherwise computers could confuse different numbers and
wreak havoc! The proof of the theorem is based on an algorithm which we can use to write any positive
integer in any base b . Let’s use this algorithm to write 100 in base 2.

1. The first step is to take the base and raise it to powers until you reach the highest power

of the base that is not bigger than 100. For base 2, the powers of 2 are 2 =1, 2 =2,
0
1
7
2
3
2 =4, 2 =8, 2 =16, 2 =32, 2 =64, 2 = 128. So, the largest power of 2 which
4
5
6
is not bigger than 100 is
6
2 =64 ≤ 100.
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