That’s the expansion of the fraction in base , as opposed to base .

In the previous post, I verified that the above infinite series actually converges to :

Still, a curious student may wonder how one earth one could directly convert into binary without knowing the above series ahead of time.

This can be addressed by using the principles that we’ve gleaned in this study of decimal representations, except translating this work into the language of base . In the following, I will use the subscripts and so that I’m clear about when I’m using decimal and binary, respectively.

To begin, we note that . (In other words, ten is equal to two times five.) So, following Case 3 of the previous post, we will attempt to write the denominator in the form

, or

If , then , but is not an integer.

If , then , but is not an integer.

If , then , but is not an integer.

If , then . This time, . Written in binary,

We now return to the binary representation of .

Therefore, the binary representation has a delay of one digit and a repeating block of four digits:

Naturally, this matches the binary representation given earlier.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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One thought on “Thoughts on 1/7 and other rational numbers (Part 7)”

## One thought on “Thoughts on 1/7 and other rational numbers (Part 7)”