Rational and Irrational Numbers
Name: _____________________________________
On a crisp Saturday morning in late November, an unusually dense marine layer blanketed the hillsides of Burbank on its
way towards conquering the skies of Los Angeles. As the invasion of aquatic fog continued its march inland, I stood
beneath its imposing shadow of dullish gray. A swift and frigid draft swept across Baldwin Avenue. The sea of spectators
packed on the sidewalk all braced their bodies in response to the sudden chill. Not a single muscle on my body moved as
I looked towards the sidewalks to see the people in the crowd tighten their jackets and windbreakers. My thick uniform
jacket and tight-fitting helmet sheltered me from the autumn gusts. The draft felt like a gentle and comforting breeze as
it splashed my face, washing away some of the tension and anxiety that had been building up for the last five minutes.
The fatigue of supporting the weight of my 30-pound tenor saxophone began to wear on me, but I disciplined my mind
to ignore it as I continued standing at attention. In just four minutes from now, I will have completed the final
performance of a parade that I have trained for every day after school since late August. I recall sweeping my peripheral
vision with my eyes and seeing my fellow band mates lined up in perfect rows and diagonals. The two hundred of us had
woken up at 5:00 in the morning and boarded buses that transported us 115 miles from our school in San Diego to the
quiet and tranquil L.A. suburb of Arcadia, where we annually compete in the most renowned marching band festival in
Southern California. Our school had won the championship every year for the past four years. Adorned in our school
colors as we fielded white jackets with neatly polished buttons, military hats with bleached feathery plumes, and royal
blue plants with parallel white stripes streaking vertically down to our white marching shoes, we were poised to bring
home the six-foot tall trophy for a fifth consecutive year.
"Tweeeeeeett. Tweet, Tweet, Tweet!" After captivating the audience with his baton twirl, the drum major signaled to
band that the stage was ours. The orchestrated cadence of drums and cymbals counted down the final seconds before
our performance, and then, on a syncopated off-beat, we played our first note of the march and a split second later, our
band of 200 stepped off in unison to the low brass opening of "Washington Greys." Two measures later, the woodwinds
joined in and the fanfare of trumpets cut through the air. Down the street we marched to the fast-tempo tune in the key
of F-minor. I marveled at how impeccably straight and precise the rows, columns, and diagonals of our parade block
remained as we glided down the street, perfectly harmonious in both sound and motion. Just two months earlier, my
eyes frequently witnessed the exact opposite.
When the marching season began, our band seemed years away from the orderly and rationally cohesive performance
ensemble we had shaped into by the end of November. Our first few practices were chaotic and confusing. Musicians
who had not yet memorized the music kept playing off-beat, often influencing their entire instrument section to play in
disagreement with the rest of the band. Our marching was flawed, and I remember the ends of some rows would curve
off by an entire yard from the center of the row. On extreme cases, we even had musicians bump into each other and
trip then fall down when we were practicing our halftime field shows for football games. At times, the various horn
sections of the band were playing two to three seconds apart from where the band was supposed to be during a song...
Our band was a disorderly and chaotic mess on some days, unable to perform together rationally as part of a unified
ensemble. Our band director held us to a high level of excellence, and had no tolerance for the disorganized display of
confused and unorganized musicianship. So, he cracked down hard on us for the irrational mistakes we made, as he
increased practice times and frequencies of practices. His hard and uncompromising approach towards subpar and
irrational mistakes reminded me the Greek mathematicians Pythagoras and Hippasus.
Thousands of years ago, Pythagoras taught the brightest minds in Ancient Greece the most advanced topics in math of
that era. One of the mathematical ideas that Pythagoras advanced was the idea of rational numbers. Pythagoras
believed that all numbers are rational, meaning that they can be written as a ratio of two numbers. For example, the
number 7.5 is a rational number, because it can expressed as a ratio of the numbers 15, and 2. If we write 15 and 2 as a
fraction, we get 7.5. Likewise, the number 2.33333333..., even though it is a repeating decimal, is also a rational number
because it can be written as the ratio of two numbers. In the case of 2.333333…, the two numbers are 7 and 3. If we
place 7 on the numerator, and 3 on the denominator, then we come up with the fraction. Dividing 7 by 3, we end up
with the answer: 2.3333333... proving that 2.33333… is a rational number.
However, one of his students named Hippasus started creating disorder, and began disturbing the idea that all numbers
can be expressed a ratio of two integers. Hippasus said that some numbers, like the square root of 2, cannot be
expressed as a ratio of two integers. The value of 2 is 1.41421356237309504880168872420969807856967187537694
807317667973799... and so it continues without any pattern forever. When Hippasus tried to write the value of as a
fraction of two integers, he found there is no solution. He could come close to the answer by solving for 99/70, but the
answer was merely a close approximation, not an exact match. Other number values, such as the commonly used Pi, are
also irrational. Pi is equal to 3.14159265... and the numbers continue without any pattern. Thus, Pi cannot be written as
a ratio of one integer over another.
Pythagoras was deeply angered and disappointed in his student Hippasus. Just like my band director, Pythagoras would
not tolerate a single student who operated out of alignment and in disagreement with the rest of the group. When
Hippasus created disharmony with Pythagoras' teaching by pursuing the idea of irrational numbers, Pythagoras realized
that he could not tolerate Hippasus' disorderly conduct. Since Pythagoras viewed the idea of an irrational number to be
illogical and harmful towards his orderly organization of number theory, he knew that Hippasus could no longer be a
member of his group. Pythagoras invited Hippasus out a on a sailing trip one afternoon, and threw Hippasus into the
sea, drowning his student.
Although our band director was often mean and yelled at us for even the most minor errors in our musical
performances, we knew that we were fortunate students because he would work with us on improving our techniques
until we could perform as a rationally cohesive group rather than just drown us like Pythagoras did to Hippasus when he
discovered the existence of irrational numbers. After all, the ocean was just a few minutes away from our school...
Video link: http://www.youtube.com/watch?v=nuX_6KLyj6Y
Assignment (Due Tuesday, October 16th)
This assignment is worth 20 points in Quizzes/Tests/Projects category.
Write your own story, relating it to our lesson on rational and irrational numbers. In your story, you must:
1) Define what rational and irrational numbers are
2) State at least one example of a rational number and one of an irrational number
3) State the criteria for rational numbers (what does it take to be a rational number)?
Hint: Use your packet from class. Also, your story can be fictional. Your stories do not have to be tied to a life experience
or a historical event like parts of mine are. In fact, I look forward to reading stories that are brilliant works of fiction.
Remember: You have complete creative licensing over this assignment. Your story can be set in outer space, the Triassic
period, or a fictional universe. You can personify numbers into characters of your story if you'd like...maybe even create
historical satire where one society is depicted as rational numbers, and another society as irrational numbers and
explore the differences between the societies... or a league of rational numbered super heroes that is recruiting new
members only if they meet the criteria for being a rational number... The possibilities are endless! Be creative, and let
your imagination take you to new places!