Section 22–6 Imaginary exponents

(Remarkable formula / Relation to geometry / Abstraction and generalization)

In this section, Feynman discusses the most remarkable formula in mathematics, its relation to geometry, and summarizes the accomplishments due to abstraction and generalization. He delivered a similar lecture titled “Some Interesting Properties of Numbers” during the Manhattan Project in Los Alamos. One should not feel bad if she or he has difficulties in understanding the lecture because it was targeted to “mighty minds.” In a letter to his mother: “all the mighty minds were mighty impressed with my little feats of arithmetic… they should have known it all the time – of course (Feynman, 1944, p. 32).”

1. Remarkable formula:

“We summarize with this, the most remarkable formula in mathematics: e^iθ = cos θ + isin θ.This is our jewel (Feynman et al., 1963, section 22–6 Imaginary exponents).”

Feynman concludes the chapter by stating the most remarkable formula in mathematics: e^iθ = cos θ + isin θ and calls this formula “our jewel.” However, mathematicians prefer Euler’s identity e^pi + 1 = 0 that is also described as the most beautiful equation. This equation is related to five fundamental mathematical constants: (1) 1 is the multiplicative identity: a ´ 1 = 1 ´ a = a. (2) 0 is the additive identity: a + 0 = 0 + a = a. (3) π is the ratio of the circumference of a circle to its diameter (= 3.141...). (4) e is Euler’s number (= 2.718...). (5) i is the imaginary number that satisfies i² = -1. It is remarkable that the equation connects whole numbers, complex numbers, and transcendental numbers. In Gleick’s summary of Feynman’s lecture on Numbers, Gleick (1992) ends by saying, “he had written elatedly in his notebook at the age of fourteen, that the oddly polyglot statement e^pi + 1 = 0 was the most remarkable formula in mathematics (p. 183).”

Feynman uses a table of “Successive Powers of 10^i/8” to show how the numbers x and y oscillate, that is, 10^is repeats itself as a periodic thing. He explains that all of the various properties of these remarkable functions, e.g., e^iθ, which have complex powers, are the same as the sine and cosine of trigonometry. Thus, a complex number (re^iθ) can be defined as a vector that has a magnitude r and phase angle θ. Similarly, in a public lecture on QED, Feynman (1985) describes wave functions of photons moving through space using arrows and “imaginary stopwatch hands” instead of complex numbers. He clarifies that it may seem impressive to use the phrase complex number, but it is possible to use an arrow or imaginary stopwatch hand as a different language for explanations.

2. Connection to geometry:

“We wake up at the end to discover the very functions that are natural to geometry. So there is a connection, ultimately, between algebra and geometry (Feynman et al., 1963, section 22–6 Imaginary exponents).”

According to Feynman, we may relate the geometry to algebra by representing complex numbers in a plane. That is, we can represent every complex number, x+iy such that the horizontal position of a point is x, whereas the vertical position of a point is y. This way of representation is commonly known as an Argand diagram, Gauss plane, or complex plane. Perhaps Feynman could have used it to illustrate a multiplication of two complex numbers. In the public lecture on QED, Feynman (1985) explains multiplying complex numbers by saying: “multiplying arrows can also be expressed as successive transformations (for our purposes, successive shrinks and turns) of the unit arrow … (p. 62).” In other words, we can use a complex plane to show how 10^is repeats itself as a periodic thing because of successive rotations.

Feynman mentions that the discovery of complex exponential functions is natural to geometry. He also adds that there is a connection between algebra and geometry. Mathematicians may not agree with Feynman because it took a long time for them to accept complex numbers. In 1831, Gauss suggests that “[i]f, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question (Dubisch, 1952, p. 91).” One may elaborate on how the geometric representation of a complex number z as a point in an Argand diagram helps to visualize complex numbers. This is due to the rotational property of complex numbers, e.g., multiplying z by e^iπ (= -1) is equivalent to a rotation by π radians, whereas multiplying by e^iπ/2 (= i) is equivalent to a rotation by π/2 radians.

3. Abstraction and generalization:

“… we had little idea of the power of the processes of abstraction and generalization (Feynman et al., 1963, section 22–6 Imaginary exponents).”

Feynman began this chapter with only basic notions of integers and counting and it allowed him to show the power of the processes of abstraction and generalization. Using the set of algebraic “laws,” properties of numbers, and the definitions of inverse operations, he has demonstrated how to manufacture numbers. Mathematicians may emphasize that it leads to the fundamental theorem of algebra: every algebraic equation of any degree n with real or complex coefficients, f(x) = x_n + a_n-1x^n-1 + … + a₁x + a₀ = 0, has solutions in the field of complex numbers. In addition, Hamilton’s generalization of the complex numbers to quaternions has contributed to the development of abstract algebra. Using abstraction and generalization, it also results in definitions of all kinds of numbers that are stranger than the complex numbers.

Feynman ends the chapter by saying we have been able to manufacture not only numbers but useful things like tables of logarithms, powers, and trigonometric functions. These accomplishments were achieved with the help of extracting ten successive square roots of ten. As a suggestion, one should clarify that it requires a lot of hard work to accurately calculate successive square roots of ten and even Briggs made many mistakes in determining the logarithm table. However, it requires even much longer time and more pain for mathematicians to extend the definition of “number” to include negative numbers. The same painful process was repeated to discover and appreciate the usefulness of complex numbers (Gardner, 1991).

Questions for discussion:

1. What is the most remarkable formula in mathematics?

2. What is the connection between algebra and geometry? 

3. What have mathematicians achieved using the set of algebraic “laws,” properties of numbers, and the definitions of inverse operations?

The moral of the lesson: we have been able to manufacture not only numbers but useful things like tables of logarithms, powers, and trigonometric functions by extracting ten successive square roots of ten.

References:

1. Dubisch, R. (1952). Nature of Number: An Approach to Basic Ideas of Modern Mathematics. New York: Ronald Press Co.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Gardner, M. (1991). Fractal Music, Hypercards and more: Mathematical Recreations from Scientific American. New York: W.H. Freeman & Co Ltd.

5. Gleick, J. (1992). Genius: The Life and Science of Richard Feynman. London: Little, Brown and Company.