Page 135 - 'Blast_Into_Math

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

If x ∈ Q is a fraction, then there exist natural numbers p and q such that

p
x = and p< q.
q

For example,

3
4

is a fraction. We can also write

3
=0.75.
4

This means

3 7 5 7 5
= + = + .
4 10 100 10 1 10 2

When we write a fraction in base 10, we write the fraction as the sum of fractions whose denominators
are powers of the base. Then, we list the digits; these are the numerators.

How would look in base 4? Since
3
4
3 3
=
4 4 1

3
we would write in base 4 by listing the numerator, so we would write as 0.3 in base 4.
3
4 4
We can follow the same steps for writing integers in different bases to write fractions in different bases.
3
To illustrate this, let’s write in your computer’s favorite base, 2.
4
3
1
1. First, we check if is larger than or not. In this case, it is. So, we begin by writing
4 2
3 1 3 1
= + − .
4 2 4 2
3
The first digit of in base 2 is 1, because it is the numerator of the first fraction .
1
4 2
2. Next, we look at the remainder
3 1 1
− = .
4 2 4

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