Thoughts on 1/7 and other rational numbers (Part 8)

In Part 6 of this series, I mentioned the following fact concerning the decimal representation of \displaystyle \frac{a}{b}: if neither 2 nor 5 is a factor of b, then the repeating block in the decimal representation of \displaystyle \frac{a}{b} has a length k that must be a factor of \phi(b). This function is the Euler toitent function or the number of integers less than b that are relatively prime with b.

In this post, I’d like to provide a justification for this theorem.

As discussed earlier, k is the least integer so that b is a factor of 10^k - 1. In the language of congruence, k is the least integer so that

10^k \equiv 1 (\mod b)

In other words, let G_b be the multiplicative group of numbers less than b that are relatively prime with b. By assumption 10 \in G_b. Then k is the order of 10 in G_b, and there’s a theorem that states that the order of an element of a group must be a factor of the order of the group, or the number of elements in the group. In our case, the order of G_b is the number of integers less than b that are relatively prime with b, or \phi(b).

In other words, using these ideas from group theory, we can prove that k \mid \phi(b).

green line

Naturally, we don’t expect middle school students seeing long division for the first time to appreciate this property of decimal representations. Still, my main purpose in writing this post was to give a concrete example of how ideas from higher-level mathematics — like group theory — actually can shed insight into ideas that are first seen in school — even middle school. In other words, there’s a reason why UNT (and other universities) requires that college students who want to earn mathematics teaching certification with their degrees must have a major in mathematics.

Leave a comment

1 Comment

  1. Thoughts on 1/7 and Other Rational Numbers: Index | Mean Green Math

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s

%d bloggers like this: