The Complexity of Simplicity
By Dan Teague, posted October 12, 2015 —
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
We math teachers also insist that
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
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.
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
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.
firstname.lastname@example.org, 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.