Conic Completion Day

Today I worked through poor concentration to finish the exercises from the review of conics. There wasn’t much left to do, in fact, because the five final exercises all called for tools from calculus that I have not reviewed yet.

Next up will be trigonometry. I didn’t start reading the appendix on that topic today, but I looked over it. I also did a little review of the Unit Circle (which, oddly, isn’t pictured in the review of trigonometry). I remember Mike, my high school math teacher for two out of three years, was very strong on the Unit Circle. He used to say that if he ever met us again as adults, his first question would be, “What’s $\sin\left(\frac{\pi}{6}\right)$?” I did remember the answer to that before my review today, so perhaps threatening years-late pop quizzes is an effective teaching technique.

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I reckon it’s about time I present my hand-waving argument that repeating decimals must represent rational numbers. I proved the converse in my post Rational-a-Rama with what I think most of my audience found a boring amount of rigor. Here I’m going to err far on the other side.

Consider the repeating decimal $x=0.\overline{142857}$.

Since $x$ repeats every six digits, multiplying $x$ by $10^6$ will yield another number with the same decimal part: $10^6x=142857.\overline{142857}$

Observe that $10^6x-x=142857.\overline{142857}-0.\overline{142857}=142857$.

Thus $(10^6-1)x=142857$ and $x=\frac{142857}{10^6-1}=\frac{142857}{999999}\text{.}$

Both $142857$ and $999999$ are integers, so it follows that $x=0.\overline{142857}$ is rational.

(In fact, $x=0.\overline{142857}=\frac{142857}{999999}=\frac{1}{7}$.)

A similar argument can be made about any repeating decimal, so any repeating decimal represents a rational number.

(Note that some repeating decimals do not begin repeating right after the decimal point. An example is $y=0.58\overline{3}$ (which is $\frac{7}{12}$). Multiplying $y$ by ten and subtracting $y$ from the result yields $10y-y=5.83\overline{3}-0.58\overline{3}=5.25$. This is not an integer, but it can be made one by multiplying by 100. This leaves the equation $100(10y-y)=525$, which can be used to show that $y$ is rational in the same manner as above.)