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.