Page 137 - 'Blast_Into_Math

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

So the first step is to write

3 4 3 4
= + − .
4 6 4 6

2. Next, we look at the remainder
3 4 1
− = .
4 6 12

2
The next power of 6 is 6 , so the next digit will be the numerator corresponding to denominator
6 =36. To find this digit, we find the largest x ∈ Z such that 1 ≥ x . This is x =3, because
2
12 36
1 3 3
= = .
12 36 6 2

So, we write

3 4 3
= + .
4 6 1 6 2

3. In base 6 we would write as 0.43.
3
4
We can write any fraction in any base. However, sometimes when we write a fraction in a base,
it does not end. For example, in base 10,

1
=0.3333333333 ...
3

To understand this, we need to learn about limits. We’ll do this in the next chapter.

6.3 Exercises

1. Write 7, 13, and 21 in base 2 and in base 7.
2. Write an algorithm that takes a natural number in base 10 and outputs the same number in
base 2.
3. How do computers add and multiply numbers using only the digits 0 and 1? Write an
algorithm to add and multiply numbers using only base 2 (without switching back to base
10).

137