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.

Section 22–5 Complex numbers

(Mathematical definition / Mathematical properties / Complex powers)

In this section, Feynman discusses a mathematical definition of complex numbers, mathematical properties of complex numbers, and how to compute complex powers of complex numbers.

1. Mathematical definition:

“It is just as good a solution, and since the only definition that i has is that i² =−1, it must be true that any equation we can write is equally true if the sign of i is changed everywhere (Feynman et al., 1963, section 22–5 Complex numbers).”

Using rectangular co-ordinates, Feynman defines a complex number as the most general number, a, that is of the form p+iq in which i² =−1. Using polar co-ordinates, he elaborates that a complex number a = x + iy = re^iθ, where r² = x² + y² = (x + iy)(x − iy) in the next chapter. One should clarify that the imaginary number i has nothing to do with the concept of a number as a means of counting. In general, there are three different definitions of complex numbers: (1). The sum of a real number and an imaginary number a + ib. (2). A vector that has a directed magnitude and a real number with a direction in two dimensions. (3). An ordered pair (a, b) of real numbers (Eves, 1997).

Feynman says that the greatest miracle of all is: we do not have to invent new numbers and the complex number is the last invention. He adds that the rules still work with complex numbers, and says that we have finished inventing new things. This is incorrect because Hamilton invented quaternions (or hypercomplex numbers in four-dimensional spaces) as an extension of complex numbers. The usual form of a quaternion is x = x₀ + x₁i + x₂j + x₃k in which i.i = -1, j.j = -1, k.k = -1, i.j = 0, j.k = 0, and k.i = 0. However, Feynman was aware of the concept of quaternions and used it to solve some problems related to quantum electrodynamics (Mehra, 1994, p. 174). When he was young, he studied quaternions on his own and read books that are about Hamilton.

2. Mathematical properties:

“We have already discussed multiplication, and addition is also easy; if we add two complex numbers, (p+iq)+(r+is), the answer is (p+r)+i(q+s) (Feynman et al., 1963, section 22–5 Complex numbers).”

Feynman discusses two properties of complex numbers in terms of addition and multiplication. In short, an addition of two complex numbers can be expressed as (a + bi) + (c + di) = (a + c) + (b + d)i, whereas an multiplication of two complex numbers can be expressed as (a + bi)(c + di) = (ac − bd) + (bc + ad)i in which a, b, c, and d are real numbers. As a suggestion, one should state that complex numbers also satisfy the commutative, associative, and distributive laws for addition and multiplication. Furthermore, we should not assume that the rules are strictly valid for all operations such as square root and logarithm. For instance, if we assume that Ö(mn) = Ö(m) ´ Ö(n) is applicable to all numbers, we will have a paradox: 1 = √1 = (√ −1)(√ −1) = (i)(i) = −1.

According to Feynman, when the power of 10^is is small, we can suppose that the “law” 10^ϵ = 1+2.3025ϵ is right, as ϵ gets very small and ϵ may be a complex number. In other words, this law is true in general, that is, 10^is = 1+2.3025is, if s→0. However, one should not assume all rules are still applicable to complex numbers. Firstly, the rule Ö(mn) = Ö(m) ´ Ö(n) is only valid for positive real numbers instead of all complex numbers. Similarly, the rule log (mn) = log m + log n is only valid for positive real numbers. On the other hand, the complex exponential function is not injective (one to one), because e^i(θ+2π) = e^iθ for any θ, since adding i2p has the effect of rotating e^iθ counterclockwise by 2p radians.

3. Complex powers:

“But the real problem, of course, is to compute complex powers of complex numbers. (Feynman et al., 1963, section 22–5 Complex numbers).”

Feynman mentions that the real problem is to compute complex powers of complex numbers and concentrate on the problem of calculating 10^(r+is). One may be surprised that his main purpose is to determine log₁₀ i using a logarithm table again, but his method is rather laborious using trial and error repeatedly. In essence, we need to keep calculating (a + bi)(c + di) = (ac − bd) + (bc + ad)i such that the product of two numbers is eventually almost an imaginary number. In other words, the real part (ac − bd) has to become smaller and get closer to zero. Alternatively, one may prefer a direct and fast method to determine log₁₀ i that is exactly equal to (p/2)i/(ln 10).

It was easy for Feynman to get log₁₀ i = 0.68184i possibly because he memorized the logarithm table and noticed some patterns in numbers (Feynman, 1997). It is likely difficult for students to use trial and error to get i(512+128+64−4−2+0.20)/1024. However, the logarithm is a multiple-valued function and the answer 0.68184i should be known as the principal value. For simplicity’s sake, we can use natural logarithm to show why it has an infinity of values: e.g., in polar co-ordinates, ln z = ln (r e^iθ) = ln [r e^i(θ+2kπ)] = ln |r| + i(θ+2kπ) where k is an integer. Applying this relationship, the general value of log₁₀ i can be easily calculated as log₁₀ i = (ln i)/(ln 10) = [ln 1 + i(p/2+2kπ)]/(ln 10) = 0 + (3.141592/2)i/(2.302585) + 2kπi/(ln 10) = 0.682188 i + 2kπi/(ln 10), but the principal value is 0.682188 i.

Questions for discussion:

1. How would you define a complex number and explain the concept of an imaginary number?

2. What are the mathematical properties of complex numbers?

3. Would you use Feynman’s method to determine log₁₀ i?

The moral of the lesson: a complex number has nothing to do with the concept of a number as a means of counting, but it does have useful properties for calculation purposes.

References:

1. Eves, H. (1997). Foundations and Fundamental Concepts of Mathematics (3rd. ed.). New York: Dover.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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. Mehra, J. (1994). The Beat of a Different Drum: The life and science of Richard Feynman. Oxford: Oxford University Press.

Section 22–4 Approximating irrational numbers

(Successive square roots of 10 / Logarithm of 2 in base 10 / Euler’s number e)

In this section, Feynman discusses how to approximate successive square roots of 10, the logarithm of 2 in base 10, and Euler’s number e. Before reading this section, it is advisable to read the chapter “Lucky numbers” in Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. It will help to understand or appreciate this section, but computers have made the logarithm tables obsolete.

1. Successive square roots of 10:

“So we evaluate ten successive square roots of 10, and that is the main work which is involved in the calculations (Feynman et al., 1963, section 22–4 Approximating irrational numbers).”

Feynman explains that we can easily work out one-half power of 10 because it is the square root of 10 and there is a simple process for taking square roots of any number. In an appendix, Feynman provides a formula that can improve the value of a square root: a′ = ½ [a + (N/a)]. The process of improving the value of square root can also be achieved using the iteration formula: x_n+1 = ½ (x_n + S/x_n). For the square root of 10, we have S = 10 and we may first “guess” x₀ » 3. By repeatedly using the formula, we have x₁ = ½ (3 + 10/3) = 3.16667, x₂ = ½ (3.16667 + 10/3.16667) = 3.16228, and x₃ = ½ (3.16228 + 10/3.16228) = 3.162277 » 3.16228. Note that the second iteration gives a reasonably accurate value.

Feynman mentions that there is a definite arithmetic procedure to find the square root of any number N, but one easy way is to choose some a fairly close, and repeat using a′= ½ [a+(N/a)] for the next choice for a. Using this numerical method, the convergence to an approximate value is very rapid as shown above. The iterative formula x_n+1 = ½(x_n + S/x_n) is also known as the Heron’s method, after the Greek mathematician Hero of Alexandria. One may explain that: if x_n is an overestimate to the square root of S, then S/x_n will be an underestimate, or vice versa. Thus, the average of these two numbers ½(x_n + S/x_n) provides a better approximation. This formula can be proved using Newton-Raphson’s method: x_n+1 = x_n - f(x_n)/f¢(x_n) in which f(x) = x² – N.

2. Logarithm of 2 in base 10:

“Suppose we want the logarithm of 2. That is, we want to know to what power we must raise 10 to get 2 (Feynman et al., 1963, section 22–4 Approximating irrational numbers).”

According to Feynman, the problem of solving 10^x = 2 is merely a computational problem. He elaborates that the answer (x = log₁₀ 2) is simply an irrational number, an unending decimal, that is not a new kind of a number. This is not correct because x = log₁₀ 2 is a new kind of number that is known as a transcendental number. In mathematics, every transcendental number is also an irrational number and it is not a root (or solution) of a nonzero polynomial equation with rational coefficients. On the contrary, the square root of 2 is an irrational number, but it is not a transcendental number because it is a root of the polynomial equation x² − 2 = 0.

In making an approximation of log₁₀ 2, Feynman identifies one factor 1.000573, which is beyond the range of the logarithm table. To find the logarithm of this factor, he uses the result of 10^Δ/1024 ≈ 1+2.3025(Δ/1024) to get Δ = 0.254. This method may pose difficulties in understanding because some students are likely unfamiliar with numerical analysis. Although Feynman writes Δ→0, one should be cognizant that Δ lies between 0 and 1/1024, i.e., 0 < Δ < 1/1024. One may clarify that this approximation method assumes a linear proportionality within a small range of values. Furthermore, the use of (10^s−1)/s is reasonable because the limiting value of 10^s is 1 as s gets closer to 0 (i.e., 10^s → 1).

3. Euler’s number:

“This then corresponds to using some other base, and this is called the natural base, or base e. Note that log_e(1+n) ≈ n, or eⁿ ≈ 1+n as n→0 (Feynman et al., 1963, section 22–4 Approximating irrational numbers).”

Feynman says that we can calculate 10^ϵ easily for very small powers ϵ because 10^ϵ = 1+2.3025ϵ and this also means that 10^n/2.3025 = 1 + n if n is very small. He also mentions that logarithms to any other base are merely multiples of logarithms to the base 10. To have a better understanding, one may specify the series e^x = 1 + x + x²/2! + x³/3! + … Perhaps Feynman did not include this series in his explanation because he knew this series very well even when he was a kid. We may rewrite 10^n/2.3025 ≈ 1 + n as eⁿ ≈ 1 + n because e = 10^1/2.3025 by using a property of power x^a/b = (x^1/b)^a. It could also be clearer if Feynman writes down the rule for logarithm log₁₀ x = (log_e x)/(log_e 10) = (ln x)/(ln 10) and simplify it into (ln x)/2.3025.

Feynman concludes that e is a product of numbers and it is equal to 2.7184. He adds that it should be 2.7183, but it is good enough. However, it is an irrational number that can be expressed as 2.71828182845904523536028…… It should be worth mentioning that e is also a transcendental number that is commonly known as Euler’s number. This number is not only expressible as a product of numbers, but it is also a sum of numbers: 1 + 1/1 + 1/1´2 + 1/1´2´3 + … … Although the work on logarithm tables had come very close to recognize the number e, it was “discovered” through a study of compound interest. In 1683, Bernoulli was thinking about how an amount of money would converge toward a number by essentially determining the limit of (1 + ¹/_n)ⁿ as n approaches infinity.

Questions for discussion:

1. How would you prove the validity of the method a′= ½ [a+(N/a)] in approximating the square root of a number?

2. How would you explain x = log₁₀ 2 is a new kind of number? 

3. How would you approximate the Euler’s number e?

The moral of the lesson: the logarithm of 2 in base 10 can be approximated by using successive square roots of 10, whereas Euler’s number can be approximated by using the logarithm table.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. 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.