0.9999999999999999....

Today is the birthday of Nikolai Chebotaryov born 1894 in Kamenets-Podolsk, Ukraine. Chebotaryov was a mathematician contributed to number theory. He created a density theorem which generalized Peter Gustav Lejeune Dirichlet's prime number theorem.

Today's quote is from Jack Nitzsche and Sony Bono.

"I saw her today, I saw her face

It was the face I loved, and I knew

I had to run away

And get down on my knees and pray, that they go away

And still it begins

Needles and Pins"

My first love was books. I was at an early age, a library groupie. Our city librarian was Mrs. Herman and in today's vernacular, was a rock star. She was the wife of Vern Herman, a legendary educator, a coach of a multitude of sports, a physical education teacher, and a driver's education instructor during the twilight of his career. Mrs. Herman became my literacy advisor many years before Vern guided me through the travails of parallel parking. I visited the library at least twice a day during the summer months of my preteen years. Each time I was greeted with a warm smile and was shepherd through a list of intriguing possibilities. The back of each possible book was secured with a code.  That code was a dewey decimal which gave the appearance of the "Good Housekeeping Seal of Approval", assuring that this budding reader would be gratified.

Reading is one of my sanctuaries. When I read, I am transported to exotic locales, struggle with vile scoundrels, peek into the minds of historic figures, and am stunned by plot twists as in Gone Girl by Gillian Flynn, Fight Club by Chuck Palahniuk or The Red Wedding in A Game of Swords by George R. R. Martin. I cherish the unexpected. The surprise twist or ending in a book is like the shock that vibrates through your brain as an unsuspecting snowball strikes the side of your head as you turn the corner of garage and contains that shiver. The shiver that trickles down your spine like the slow melting snowball whose cold stream meanders the rise and fall of the quivering vertebra. This shiver, this prickly sensation, feels like needles and pins.

This blog is the second in my series of Point A to B. My calculus students would be remiss if I didn't identify this particular moment in my progression as C1, somewhere in the interval, [A, B], but closer to A than to B.

In 1972, I was in 8th grade and a student at Immanuel Lutheran School in Plainview, Minnesota. My teacher was Evan Schiller. His enthusiasm as he taught math captured my attention. I was an average mathematics student. My scores in previous grades in the subject oscillated between B's and C's. The concentration of my math experience to this point was with arithmetic and I had been applying those skills for 4 years in my dad's welding shop. My teacher was introducing algebra and generalizing arithmetic concepts with letters. He was moving numbers to a different and more abstract level.

The lesson of the day was fractions and decimals. All rational numbers (fractions) can be written as decimals and vice versa. 0.5 is an example of a terminating decimal because it ends with a repeating 0. All decimals that end this way are called terminating decimals. The fraction equivalent for 0.5 is 1/2. 0.666... is a repeating decimal because it ends with a repeating digit other that 0. Its fractional representation is 2/3. I knew that, but Schiller continued, "What about 0.999...?" He extended the algebra that he just taught us to demonstrate.

x = 0.999...

10x = 9.999...

10x = 9.999...

- x =  -.999...

9x = 9

x = 1

1 = 0.999...

I was flabbergasted. Really, how can this be true? 1 is the same as 0.999... I experienced for the first time the exact same feeling I treasured in reading, the staggering "needles and pins". This meant that 2.999... was equivalent to 3 but more importantly, mathematics contained a mystery, a twist and turn, and I could find the joy of surprise. I could feel "needles and pins".

In  1978, I found out that 89.999996 was not 89.999... and definitely not 90. I was in college and taking a computer programming class entitled FORTRAN. Prior to the final, my percentage was 91. 91% was important. To earn an A, I had to have a 90% or greater. 89% grade was the same as 80%, a B. I did not do well on the final and when I received my grade report, the course notated a grade of a B. I did a little arithmetic and determined that I had a percentage of 89.999996. I mustered some courage and approached my instructor, Dr. Jim. Dr. Jim was amenable when I nervously suggested that I would like to examine the computation of my grade. Dr. Jim pulled out his red grade book and his Hewlett Packard 35 pocket calculator. "Mmm," he pondered. His eyes squinted and an eye brow raised to form a question mark. "Mmm... 89.999996%," he summarized, "What is your question?" I looked into his clear, blue, stoic eyes. "Nothing," I murmured. I knew that 89.999996% is not 90%.

This nightmarish decimal would resurface again in college when a calculus discussion about divergent and convergent series emerged. The instructor was demonstrating how a repeating decimal could be written in its fractional form using a series and the example being used was 0.999....  He wrote it as a sum of a sequence of 9's, each multiplied by 0.1. 0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.999... This series is called a geometric series and using limits its sum could be found by the following computation: 0.9/(1 - 0.1) and yes, that sum is 1!

Finally, in 1998, I was giving same lesson to my class of seventh graders as Schiller gave me 26 years prior when one of my students, Ben Stommes, raised his hand and offered his proof. "Mr. Kruger, it appears quite obvious," Ben confidently pointed out.

1/3 = 0.333...

+ 2/3 = 0.666...   

1  =  0.999...

There have been other mysteries that have opened their cocoon unsuspectedly. The derivative of e^x is e^x and e^(iπ) + 1 = 0. These moments I have cherished and have attempted to replicate in my teaching. This year in my BC Calculus class when I used Taylor Series to show e^(iπ) + 1 = 0, my students hushed and then in unison murmured "wow". This was my third greatest moment this year. I had two presentations for finals. My greatest were yet to come. One was entitled Calculus in Nature and the other, The Gaussian Integral. Both of these gave me "needles and pins". Nature connected for me Ecology and mathematics using the Lotka-Volterra Equations. These differential equations model prey versus predator. The Gaussian Integral bridged the assumed crevice that separated Calculus and Statistics. The ending of this presentation redirected my isolationist view of statistics.

The unsuspected realities that arise in literature and mathematics should not be confused with the "aha" moments that can also come to fruition in both subjects. Those epiphanies are topics to be examined at a later date. The mysteries that result in "needles and pins" abound in literature and mathematics. Those unexpected nuggets invite me to explore the continuous depth and complexities in those disciplines and have moved me along my path. The path that I have followed to this point in time.

Point A Began In the Shop

Today is the birthday of Juan Caramuel, born 1606 in Madrid, Spain. Caramel was a mathematician and a member of the Cistercians religious order. He developed and explained the general principle of numbers to base n and the benefits of using bases other than 10.

Today's quote is a joke that I told my father while working with him in his welding shop. I was a 22 year old college graduate and believed that I knew the solutions to all the world's problems.

"The parts of the human body were having an argument about who should be the boss of the body. The legs proposed that they should be the boss because they carried the body from home to work each and every day. The hands retorted, 'We should be the boss because we build homes, hold children's hands, and feed the body.' The brain countered, 'I should be the boss because I coordinate the systems of the body and solve a variety of problems.' Finally, the rectum spoke up and stated, 'I should be the boss because of all the crap I have to deal with every day.' The other body parts just laughed so the rectum stopped working. After a few days, the body parts were in panic and relented and the rectum became the boss. The moral of the story, you don't have to be a genius to be a boss, just an asshole."

My dad chuckled and replied, "Yep, that's about right."

Recently my editor, Maria Burnham, suggested that I reflect about my journey in becoming a mathematics teacher. I was hesitant at first but I was then asked in class if I always wanted to be a math teacher. I teach juniors and seniors and many are exploring their occupational options. This question floated in my mind and I found the answer more complicated than I anticipated. It remained in my thoughts like a sliver working its way through my epidermis, constantly pestering me until I addressed it and removed it. This blog begins my series of reflections that I entitle from Points A to B. Between two points, there are an infinite number of points which form a smooth curve that is continuous and differentiable. Similarly, the teacher I am today is a result of many mathematical experiences, each forming a link to the beginning. That beginning, that point A, began in my dad's welding shop.

I started working in my dad's welding shop at the age of 9. I earned a quarter an hour and received a $10 bill at the end of each week. The shop was dirty. The walls and floor were covered with a dingy, grayish, brown film that was a mixture of oil, grease, and toxins created by a firework's display of acetylene torches and arc welding. One of my duties was to clean the inside of windows to provide additional light into the shop. However, the walls were never cleaned. These blacked walls reduced the chance of receiving flash burns. Amidst the cacophony of rings provided by a hammer striking an anvil, the sizzle of water cooling iron, and the pop liquid metal exploding on the cement floor, I was introduced to the power of mathematics.

Each day was a lesson in geometry. Terms like square, rectangle, area, volume, height, perpendicular, and plum were added to my vocabulary with an example of what each looked like and how they were to be applied in the task at hand. The 3, 4, 5 rule was introduced early and the properties of rectangles such as equal diagonals were considered truths long before I was to prove them in my 10th grade geometry class. I was introduce to orientation when I made fifty, 4 foot, cylindrical rods each with a 6 inch, right handed, course thread on one end. My jobs weren't complicated but rather repetitive. I was only told once and the repetition was helpful.

The tape measure was my first introduction into fractions. My dad told to halve any fraction, just double the denominator. From that nugget of advice, I was quick to realize that a progression occurs if I kept halving, 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ... This sequence, I would later determine was a geometric sequence and when written as a series, 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... , ad infinitum, had a sum of 2.

I became quick with fractions because it was a job requirement. I remember I was to cut a 37-inch piece of 4-inch flat iron, center punch two holes (1/2 inch in diameter) 3 inches from each end, and a hole (3/4 in diameter) centered between the two end holes. I was doing my computation on a piece of paper and my dad asked what I was doing. After I responded, he gruffly remarked that he wasn't paying me to think and proceeded to explain to me how I was to do that in my head. He continued, "One center punch is at 3 inches and another at 34. Their difference is 31. Half of 31 is 15 1/2, added to 3 is 18 1/2. Done. Center punch your holes 3 inches, 18 1/2 inches, and 34 inches. Remember, I'm not paying you to think."

A few weeks later, I had completed a different job and my dad surveyed my work. He shook his head in dismay and asked, "What were you thinking?"

After I had gained some experience. My dad would leave the shop and travel to onsite jobs. The busywork of the shop was in my care, a curious, pre-teen boy. I recall two stories of when I was left unattended.

One day a man entered the shop and wanted a 6-ft. piece of 4-inch flat iron. I retrieved the piece of metal, measured it, and cut it on our band saw. This piece was to have two holes at each end, so I again measured, and centered punch the location for the holes. The piece was heavy for me, and I lugged it to the standing drill press. The bit was large, perhaps, a 3/4-inch bit. I knew that I was to oil the bit, but I was unaware that I also needed to change the ratio of the pulleys that rotated the bit, slowing its rotation. This knowledge would not have been of any use because I was too short. I barely could reach the handles used to lower the spinning bit on to the piece of metal. As I applied pressure, the bit bound, wrenching the 6 ft piece of metal from my hands, and almost decapitating the customer. I dropped to the ground and pushed the foot pedal to off. I stood up, stepped away shaking with tears welling in my eyes. The kind and patient man offered to finish the job and did. My dad anticipated that I might need to assistance in completing a job because I was told to give any customer a 10% discount for helping finish the job.

The other story involves my curiosity. I wondered how everything worked and I investigated everything; the torch, the welder, the grinder, and the forge. Now that I look back, the explosion that never occurred on the north end of Plainview, Minnesota, is in itself, quite remarkable. Leaning in the corner of the shop, my father had an old 22-caliber rifle. I admired it from afar for a few weeks and then had an opportunity to examine it when my dad was again asked to make a "house call". I found a screw driver and took the rifle apart. After scrutinizing each piece, I started to put the gun back together. I was 99% successful. The trigger had fallen to the floor and easily assimilated with the various shards that surrounded me. I scoured the floor, relentlessly looking at each piece of shining metal, praying for a miracle, but none intervened. I shrugged my shoulders and put the gun in the corner. I reassured myself that a missing trigger on an old gold would avoid detection. That evening, a man came to our house and my dad greeted him eagerly and quickly walked to the shop. They were back just as quick and my dad asked if I had "played" with the gun. My face became grim and I nodded. He inquired, "Did the trigger fall to the floor?" I again nodded. "Where did it fall?" he gravely countered. "Among the chains by the door," I whispered, again praying for this last chance of redemption. They never found the trigger and my dad never discussed the topic further.

My dad would always tell me he wasn't a teacher but he held the position in high regard. He was a man of few words. His directions were only given once. I can recall four gems of wisdom. 1) Be smarter than the tool you're using. He told me this right after I hit my finger with a hammer. I have used the quip many times as students try to negotiate the result on their calculator with the mathematics they are trying to learn. Unfortunately, the digital estimator usually wins. 2) The boss may not always be right but he is something you're not, the boss. He told me this when I was disagreeing with some procedure done at the shop. I have carried this into my career. The school district can sometimes create a policy that I disagree with, such as a no-hat policy. My obligation as an employee is to enforce that policy. If all teachers adhere to the policies given by their "bosses" the school system runs much more smoothly. 3) Keep your pecker in your pants. My brothers and I were working together the summer after I graduated from college. A farmer strolled into the shop and announced to dad that he had seen his El Camino parked in the long drive way leading to the farm house. Dad chuckled and told the farmer it must have been one of his boys. After the farmer left, Dad raised his eyebrows and asked which of us were on a date last night. After an awkward amount of silence, he chuckled again, and gave his only sex education talk with that one line. 4) Find a job you love to do and it for the rest of your life. I heard this bit of advise the most often. My dad loved what he did. When I asked him why, he stated that every day there was a new problem to solve and he loved solving problems. I love what I do. Every day is different and every day I have a new problem to solve.

A few days ago, I was interrupted by a colleague as I was tutoring a student. The concept I was reteaching was complex and required a great deal of effort on my part to make the idea attainable. I snapped at the intruder like an old dog focused on a bone and then followed with a confused response to his question. He was taken aback. I have reacted like this before. He accepted my apology but I was left upset with my behavior. At lunch, my confessor (a member of my department) approached me. He had heard the reverberations of the incident. "Your passion," he explained,"is teaching mathematics. You're oblivious to your environment when you enter your passion and you don't transition well. Those people that know you, understand. You are really asking us to enter your passion." My dad had the same reputation. When he was working on a project, you could wait for what seemed like hours before he would acknowledge your presence. He would growl, bark, and snapped as he transitioned from his problem solving to interpersonal communication. The growl, bark, and snap were really invitations to his passion. My final lesson of the shop was a lesson of passion.

Happy Birthday Numero Unus

Today is the birthday of Theodore Olivier born 1793 in France. Olivier was a mathematician who specialized in geometry. He is known for his string models of ruled surfaces.

Today's quote is by Al Jolson, Billy Rose, and Dave Dreyer.

"Like wall paper that sticks to the wall

Like the seashore clings to the sea

Like you'll never get rid of your shadow

You'll never get rid of me

Let all others fight and fuss

Whatever happens, we've got us

We're closer than pages that stick in a book

We're closer than ripples that flow in a brook"

There are many problems that deal with shadows. Usually these problems use the relationships found in trigonometry or similar triangles. I have listed below three of my favorite shadow problems,

1) If I am 6 feet tall (in my dreams) and I cast a 5 foot shadow, how tall is a flagpole that casts a 10 foot shadow? Answer: 12 feet

2) A street light is mounted at the top of a 15 foot pole. I am walking away from the pole at a rate of 5 ft/second. How fast is the tip of my shadow moving when I am 40 feet from the pole? Remember: I am 6 feet tall. :-) Answer: 25/3 ft/second

3) My favorite problem that applies the use of shadows is the calculation of the Earth's circumference by Eratosthenes over 2000 years ago. Eratosthenes estimate was 25,000 miles. The actual circumference is 24,902 miles.

Today, I complete my Sibling Bond series. My previous blogs were tributes to my sister and younger brothers. The day to day conflicts and agreements with these three siblings acted as a rock tumbler, shaping and polishing me in my progression to adulthood.

Today is my brother, Russ' birthday. Today, he is a decade older than I am. Unlike my other siblings, Russ was a comet, following an unpredictable, elliptical path that when his intersected mine, his effect on my growth was more transitory but not any less impactful.

Russ was sixteen when I was six. The memories of my family became more distinct for me at that age. My younger brothers were energetic, four year old playmates and my sister was the ever present baby sitter. My brother, however, had his driver license, was involved in sports in high school, and was working with my father in his shop. He flew in my view like a bird at my window feeder, who departs to locations unknown as soon as I am aware of its presence.

I treasured any time I had with him. I adored this mysterious, independent, good natured brother. He occasionally included me in his excursions. I was oblivious to the task at hand and eagerly followed behind at a pace that could quite never keep up to his confident gate. I recall many times that I was introduced to his friends, as "his little shadow", a description that caused me to grin and swell with pride.

When I was nine, Russ was a senior in high school and his orbit grew. I saw him less often but desired his attention even more. During this time, I was watching shows such as Dennis The Menace and the Little Rascals. Each of these shows had booby traps involved in their plots. I was captured with the idea of setting traps and Russ became my victim. For a period of three months, a series of pranks greeted him when he came home from his late night excursions. As he entered his room, a barrage of novelties fell from a precarious bucket perched at the top of the door. Books, shredded paper, and water were all items that landed on my brother's unsuspecting head. Throughout this period of time, Russ never raised his voice. He took it in stride. He was patient with his annoying brother.

I became a teenager in the early 1970's and Russ was in his early 20's. He invited me to stay with him in Minneapolis for a weekend. I recall walking through Dinky Town, encountering the sights, sounds, and smells that defined the 70's. Those sensory inputs opened the eyes of this naive, small town kid. That evening, we decided to go to a movie. Our choices were: Magnum Force, Paper Moon, The Paper Chase, and Papillon. We chose Papillon. The night ended, falling asleep while watching Rat Patrol, and I, dreaming of my urban life.

As I have grown older, Russ is no longer the super hero that I thought he was when I was younger. He, like us all, is flawed and complex, but he is wise. He is patient and calm. He has the gift of diminishing my failings and applauding my achievements. He is someone I aspire to be.

As I write this blog about my brother, I am surprised about the details I can recall about our times together. I feel his interactions with me were intentional. I believe he wanted to impact me differently than what I experiencing at home.

I have learned from him that family allows its members to find their own way and encourages them to stitch their own block on the family quilt. He has taught me that family welcomes all its members, and, if necessary, rejoices on the return of its members to the fold, and that as members of the family, we should refrain from casting judgement.

Happy Birthday Numero Unus!