The Complexity of Simplicity

  • The Complexity of Simplicity

    By Dan Teague, posted October 12, 2015 —

    I distinctly remember the first time I thought about what I was teaching. I had often thought about how I was teaching, but I had never really thought about the content. The content was always whatever was in the text I was assigned. In the summer of 1984, in the middle of a talk by Henry Pollak, then head of the mathematics division of Bell Labs, that changed.

    Henry said something like, “To be a high school teacher of mathematics, you must learn to say, with a straight face, that 1/a + 2/b + 3/c, an expression involving five operations, could be simplified to (bc + 2ac + 3ab)/(abc), an expression requiring ten operations.” I had never considered that before. How can something get more complex when you simplify? What does it even mean to simplify?

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    We math teachers also insist that 10/√(99), a number easily seen to be just larger than 1, could be simplified to 10√(11)/33, a number less obviously just larger than 1. In what world was either of these expressions truly simplified?

    Henry suggested that we consider a world in which we must pay for each mathematical operation we use. Imagine a world where addition and subtraction are essentially free; multiplication costs only a little; but division is very expensive, and division by bad numbers—you can’t afford that. In that world, (bc + 2ac + 3ab)/(abc) is much less expensive than 1/a + 2/b + 3/c, for we have traded several expensive divisions for cheaper multiplications. By similar reasoning, 10√(11)/33 is far less costly than 10/√(99) because, although we haven’t reduced the number of divisions, we have reduced the complexity of the divisor. 

    But what is the world being described? In what world do we simplify an expression by reducing the number and complexity of division? 

    It is the world of paper-and-pencil arithmetic.

    When we compute with paper and pencil, division is hard, so reducing the number of divisions indeed simplifies the calculation. In the world of paper-and-pencil arithmetic, (bc + 2ac + 3ab)/(abc) is less expensive than 1/a + 2/b + 3/c.

    Likewise, dividing by 33 is far easier than dividing by the infinite decimal √(99) ≈ 9.94987. . . . Much of what we teach in algebra under the term simplify is an attempt to reduce the complexity and number of divisions, so that, when the calculation is eventually done by hand, it will be easier to do without error.

    Ray’s Algebra (published in 1866) says on page 186, following a half page of rationalizing denominators, “The object of the above is to diminish the amount of calculation in obtaining the numerical value of a fractional radical. Suppose it is required to obtain the numerical value of the fraction √2/√3 true to six decimal places. We may extract the square root of 2 and of 3 to seven places of decimals and then divide the first result by the second. This operation is very tedious. If we render the denominator rational, the calculation merely consists in finding the square root of 6 and then dividing by 3.” The textbooks since 1866 have left out that little explanation of the purpose of the process (to save space, I suppose) but have retained all the problems for the practice of the process.  

    This simple example started a line of thought that continues today. It has led to major changes in the content of mathematics at my school. After that summer, my department got together, decided which century we were preparing our students for, and made some important decisions about the content of our courses. We decided to focus on mathematical modeling and data analysis in all our courses. This content decision required us to use technology whenever and however it was appropriate, which led quickly to other pedagogical changes that made our classes more active and our students more engaged. Teaching now is more challenging, more open-ended, and more fun. It just shows how, when you think about simplifying, things can get more complex.

    2015-09 Teague  

    DAN TEAGUE,, teaches at the North Carolina School of Science and Mathematics in Durham. He is interested in mathematical modeling and finding problems that connect concepts from different areas of mathematics.


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