Section 22–3 Abstraction and generalization

(Negative integers / Fractional numbers / Irrational numbers)

In this section, Feynman discusses the need for three classes of numbers: negative integers, fractional numbers, and irrational numbers.

1. Negative integers:

“In fact, we can show that one can make all subtractions, provided we define a whole set of new numbers: 0−1, 0−2, 0−3, 0−4, and so on, called the negative integers (Feynman et al., 1963, section 22–3 Abstraction and generalization).”

According to Feynman, we can use the rules as the definition of the symbols, which then represent a more general kind of number. He also shows that it is possible to make use of subtractions to define negative integers: 0−1, 0−2, 0−3, 0−4, and so on. Instead of using the subtractions, one may suggest the need for negative numbers in solving the problem x + n = 0 in which n is a positive integer. Importantly, the main idea is not about proving the existence of negative numbers but defining mathematical objects using consistent rules. In other words, it is a matter of mathematical definitions and consistent mathematical structure, but physicists need to make sense of the definitions.

Feynman elaborates that one cannot say “−2 times 5” really means to add 5 together successively −2 times. In addition, “it means nothing.” One may find it strange that Feynman did not try to explain the meaning of “−2 times 5” or a negative quantity. On the contrary, Feynman (1949) suggests “the ‘negative energy states’ appear in a form which may be pictured in space-time as waves traveling away from the external potential backward in time (p. 749).” It means that a positron having negative energy is equivalent to an electron that is moving backward in time. Historically, mathematicians also had difficulties in making sense of negative numbers. One may clarify that calculations can be done with or without negative numbers, but using negative numbers is a matter of convenience and abstraction.

2. Fractional numbers:

“But if we suppose that all fractional numbers also satisfy the rules, then we can talk about multiplying and adding fractions, and everything works as well as it did before (Feynman et al., 1963, section 22–3 Abstraction and generalization).”

Feynman explains that a⁽³⁻⁵⁾ = a³/a⁵ using the definition of division. He adds that it can be reduced to 1/a², but it is a meaningless symbol. The square of a can be greater than 1, and this results 1/a² in a number that is not an integer. Feynman’s explanation is potentially confusing because fractional numbers are not defined using 1/a². The main point should be how the division operation can generate fractional numbers (a/b) instead of integers. For example, we can use 1 ¸ 2 to explain that it is equal to 0.5 or ½ that is a fractional number. Alternatively, one may illustrate the need for a new class of numbers using a simple equation x ´ b = a in which a and b are non-zero integers.

Note that Feynman uses the term fractional numbers instead of rational numbers. In general, all fractional numbers are rational numbers, but rational numbers are not definitely fractional numbers. Specifically, a rational number is a ratio of an integer to a non-zero integer, and a fractional number is a fraction a/b in which a and b are natural numbers or positive integers. For example, 1/-2 is a rational number, but it is not a fractional number because -2 is not a natural number. The number Ö2/2 is also written as a fraction, but it is not a fractional number since Ö2 is not a natural number. One may prefer rational numbers because the types of numbers that are commonly used are integers, rational numbers, irrational numbers, real numbers, and complex numbers. (There may not be agreement whether zero is a natural number.)

3. Irrational numbers:

“That is good enough for what we wish to discuss, and it permits us to involve ourselves in irrational numbers… (Feynman et al., 1963, section 22–3 Abstraction and generalization).”

Feynman explains that it is impossible to solve the equation b = √2 because it is impossible to find a rational number whose square is equal to 2. However, he did not prove that √2 is an irrational number. To prove it, one may assume √2 is a rational number and express it as p/q that is reduced to lowest terms, i.e., there are no common divisor other than 1 for p and q that are non-zero integers. After squaring it, we have p² = 2q². If p² is even as suggested by the equation, one may deduce that p is also even and represent p by 2c. By substituting p = 2c into the equation, we have 4c² = 2q² or simply 2c² = q². This suggests that q² and q are also even and thus, it contradicts the assumption that √2 is a rational number that is reduced to lowest terms. (According to a legend, Hippassus discovered irrational numbers on a boat, but his colleagues threw him overboard.)

Two common definitions of irrational numbers are: (1) a number that cannot be expressed as a quotient of two integers; (2) a number whose decimal part is not periodic and has an infinite number of digits. A rigorous definition of an irrational number is based on the concept of Dedekind cut such that a number a corresponds to this cut, or that it produces the cut. Feynman says that a precise definition of an irrational number requires the concept of continuity and ordering, and it is the most difficult step in the processes of generalization. One may add that the number line (e.g., between 0 and 1) is continuous and infinitely dense with rational numbers. The existence of irrational numbers implies that there are “holes” in the number line that cannot be described as a ratio of two integers (thus, Dedekind cuts are possible).

Questions for discussion:

1. How would you explain the meaning of negative numbers?

2. Would you use the term fractional numbers or rational numbers? 

3. How would you explain the existence of irrational numbers?

The moral of the lesson: the existence of negative numbers, fractional numbers, and irrational numbers are a result of abstraction and generalization.

References:

1. Feynman, R. P. (1949). The theory of positrons. Physical Review, 76(6), 749-759.

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.