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.