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I have a favorite quote from Karen Smith which I would like to share. She states, "One cool thing I remember was learning how to "cast out nines"* to check my multi-digit multiplication in elementary school. ... This trick intrigued me already in elementary school, but later in middle school, when we learned about arithmetic in bases other than the standard base 10, I tried to apply this trick to check my work in base 8, and realized it no longer worked. I was very excited to realize that in base 8, one needs to cast out sevens instead! Finally, in high school, when I learned a little modular arithmetic ..., I was thrilled when I finally figured out the complete generalization to any base, including a formal proof. I finally fully understood something that has intrigued me for years! A great feeling."
I have had cool moments like Ms. Smith's in my education that intrigued me to study mathematics and become a mathematics teacher. A simple example is when my eighth grade teacher, Mr. Schiller did an algebra proof of how 1 and 0.999... are equivalent. The proof is as follows:
Let x = 0.999...
then 10x = 9.999...
thereby 10x - x = 9.999... - .999...
resulting in 9x = 9
so x = 1 and x = 0.999...
I later learned of infinite series and limits was able to derive that 1 and 0.999... are the same. The best, and possibly the simplest proof that was given to me was by a seventh grader named Ben. Ben, looking very studious, stated, "Mr. Kruger is seems quite obvious 2/3 is 0.666... and 1/3 is 0.333... so their sum, 1, is equal to 0.999..." A professor was impressed when I told him of this story.
*Cast Out Nines
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