Feynman and Elementary Algebra

When you compare Feynman’s treatment of the subject with that in regular textbooks, you will realize why so many students fear algebra, are bored by it and why some of them drop out when they come in contact with algebra for the first time in their lives. The Feynmans of the world are rare but that shouldn’t keep us lesser mortals – particularly math teachers - from taking cues from Feynman about how to make the subject come alive through connections, associations and serendipity.  May 11, 2018, marks the centennial birth anniversary of the iconic Nobel-laureate physicist Richard Feynman. Born on May 11, 1918, in Far Rockaway, NY, and regarded as a child prodigy, Feynman taught ‘freshman physics’ at California Institute of Technology from 1961 to 1963. The course was published as the three-volume The Feynman Lectures on Physicsin 1964. It brought him world-wide fame not only as a physicist’s physicist but also as a peerless teacher. (The Lectures are available for free at feynmanlectures.caltech.edu)

One of the lectures Feynman gave was on elementary algebra. Unlike the subject’s usual presentation in textbooks, Feynman’s insight and intuition, and his uncanny ability to focus on what is important, shines through the lecture as much as it does in his lecture on, say, Principle of Least Action or Superconductivity.

Feynman introduces algebra (http://www.feynmanlectures.caltech.edu/I_22.html) by describing the basic arithmetic operations of addition, subtraction, multiplication and division and then seamlessly transitioning to powers and exponentials. He relates each operation to its inverse: addition and subtraction are the inverses of one another, as are multiplication and division. But it quickly becomes more challenging when he shows that the inverse of a power is the root, and the inverse of an exponential is the logarithm.

It is then a fairly simple matter to relate rational exponents to roots. Along the way he distinguishes between the distinct approaches of mathematicians and physicists: “The subject of algebra will not be developed from the point of view of a mathematician, exactly, because the mathematicians are mainly interested in how various mathematical facts are demonstrated, and how many assumptions are absolutely required, and what is not required. They are not so interested in the result of what they prove. For example, we may find the Pythagorean Theorem quite interesting, that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse; that is an interesting fact, a curiously simple thing, which may be appreciated without discussing the question of how to prove it, or what axioms are required. So, in the same spirit, we shall describe qualitatively, if we may put it that way, the system of elementary algebra. We say elementary algebra because there is a branch of mathematics called modern algebra in which some of the rules such as ab=ba are abandoned, and it is still called algebra, but we shall not discuss that.”

Before we know it, Feynman is approximating irrational numbers and logarithms, all integral part of an intricate mathematical mosaic that only he can weave. He gives tips on how to come up with these approximations in your head simply by following some general principles. He gives us insight into how he approximates and then gradually increasing the precision of his approximations. He moves on to complex numbers, periodic functions and waves, ending on Euler’s formula as the crown jewel of the discipline.

Here is his simple but powerful conclusion:

“When we began this chapter, armed only with the basic notions of integers and counting, we had little idea of the power of the processes of abstraction and generalization. Using the set of algebraic “laws,” or properties of numbers, and the definitions of inverse operations, we have been able here, ourselves, to manufacture not only numbers but useful things like tables of logarithms, powers, and trigonometric functions (for these are what the imaginary powers of real numbers are), all merely by extracting ten successive square roots of ten!”

(Compare Feynman’s ideas about algebra in this delightful video https://www.youtube.com/watch?v=VW6LYuli7VU. You will laugh when Feynman says that “I learned algebra fortunately by not going to school!”)
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When you compare Feynman’s treatment of the subject with that in regular textbooks, you will realize why so many students fear algebra, are bored by it and why some of them drop out when they come in contact with algebra for the first time in their lives. The Feynmans of the world are rare but that shouldn’t keep us lesser mortals – particularly math teachers - from taking cues from Feynman about how to make the subject come alive through connections, associations and serendipity.