Page 123 - 'Blast_Into_Math

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

6 Mathematical perspectives: all

your base are belong to us

When you write a number like 43, you are writing the number in base ten. A computer would write
43 as 101011 because computers use base two. In base 10, each number is represented by a list of the
integers between 0 and 9. In base 2, each number is represented by a list of 0s and 1s. In this chapter

you’ll learn how to write any positive integer in any base b ≥ 2. You’ll also learn to write fractions and
decimal expansions in different bases. The way we write numbers influences how we think about and
understand them. For example, people often say that numbers ending in 0 are round numbers. Learning
to see numbers from different perspectives deepens our understanding of them.

6.1 Number bases: infinitely many mathematical perspectives

To write a number, we need to use a base.

Definition 1 A base is an integer b ∈ N such that b ≥ 2.

To write a number in a base, we list its digits.

Definition 2 Let b be a base. Then, for any integer x ∈ N its digits in base b are integers between 0
and b − 1 such that x is equal to the sum of these integers times powers of b . If the digits of x are
d 0 ,d 1 ,...,d k , then

k

0
1
k
j
x = d k b + d k−1 b k−1 + ... + d 1 b + d 0 b = d j b ,
j=0
and we write x in base b by listing the digits starting from d k and continuing until d 0 ,
x = d k d k−1 ... d 1 d 0 .

When we write a number in a base, that number is equal to the sum of its digits multiplied with the base
raised to corresponding powers. Since we’ll be working with the base raised to powers, now is a good
time to review the basic rules for exponents.

1. For any n ∈ N, for any non-zero number x , the definition of x is x multiplied with itself
n
n times.
2. For any non-zero number x , x is defined to be 1.
0

123