by Norman Costa
As a child, arithmetic came easily to me. My father, though not well educated, had a facility with numbers. He had a fascination for tricks and short-cuts to computations. One day he brought home a book, "The Trachtenburg Speed System of Basic Mathematics," by Jakov Trachtenberg. He is a survivor of The Holocaust, who worked out his system in the camps, and saved his sanity in the process.
I tried some of the techniques. They worked very nicely, except that I lost interest, quickly. His system was great for very fast arithmetic calculations, but it did nothing to convey an understanding of numbers. That was the 8th grade. From another source I learned how to speed multiply, mentally, 2 numbers ending in 5. I still use it today because I developed an understanding of how the numbers worked.
"Anybody got a calculator? I need to know how much is 165 times 35." People are looking for a calculator or pencil and paper. About 15 to 20 seconds pass. I am quiet and unconcerned with finding anything. "Fifty-seven seventy five." I say. Silence. A few moments later the one in need of emergency calculation asks, "You sure." As I smile at him I tap my right temple with my right index finger. Someone finds a calculator, enters the information, and says, "Wow! You're good."
Another speed trick is multiplying any arbitrarily long number by 25. As long as I can look at the number I can start reciting the answer, beginning with the left most digit. "OK, smarty pants. How about 9 billion, 880 million, 981 thousand, 445." I right down the number so I can see it. I reply, "2 4 7 0 2 4 5 3 6 1 2 5."
As a junior in high school I fell in love with the book, "One Two Three ... Infinity," by George Gamow, and first published in the 1940s. It has been updated and improved and still selling. It did two things for me. It gave me a feel for numbers including an introduction to infinity. Also, it was the beginning of a life-long interest in Einstein's theories of Special and General Relativity. That life-long interest expanded into Cosmology and Quantum Mechanics.
In high school I read a number of the popular books on Einstein. Understanding the Lorentz transformation formulas was a near spiritual experience. College was statistics and calculus. Graduate School brought me into advanced statistics and a start on Bayesian Statistics. For lack of a better way to say this, statistics gave me a love for number play.
This came to the forefront one day when Bill O'Reilly ("The Factor" on Fox) responded to an AAAS scientist who said that scientists do not regard present knowledge as absolute. O'Reilly smirked at the scientist and thought he 'owned' him by saying that, of course, there are absolutes. There are only 24 hours in a day (actually more and getting longer, I was thinking,) there are only 4 seasons in a year (arbitrary demarcations, I thought,) and 1 + 1 is always equal to 2 (not in Boolean math, and not in summation of near light speed velocities, I said to myself.) It was a wonderful moment. The man is an idiot.
Today I read the non-mathematical explanations of both Cosmology and Quantum Mechanics. My calculus is too old and too rusty to take me any further. Yet, I am intrigued by how much more I would understand if I took the time relearn and surpass my earlier mastery of calculus. Hmmm.