Our regular commentator Rafael Granero gives the following answers to last week’s questions:
The numbers whose decimal logarithms are between 0 and 2 are those that are between 1 and 100. This is because:
logd(1) = 0
logd(100) = 2
Therefore, any number x that satisfies 1 < x < 100 will have a decimal logarithm between 0 and 2.
Decimal logarithm of 0.01
The decimal logarithm of 0.01 is -2.
This can be understood as follows: logd(0.01) = logd(1/100) = logd(1) – logd(100) = -2
Decimal logarithms of 9, 30 and 1/3
Knowing that logd(3) = 0.477, we can calculate the others:
Decimal logarithm of 9: logd(9) = logd(3^2) = 2 * logd(3) = 2 * 0.477 = 0.954
Decimal logarithm of 30: logd(30) = logd(3 * 10) = logd(3) + logd(10) = 0.477 + 1 = 1.477
Decimal logarithm of 1/3: logd(1/3) = logd(1) – logd(3) = -0.477
For his part, Manuel Amorós finds in this ingenious way the value of x when x raised to the power x3 is equal to 3:
x^(x^3) = 3
(x^(x^3))^3 = 3^3
(x^3)^(x^3) = 3^3
x^3 = y
y^y = 3^3
y = 3, ergo x = cube root of 3
And Bretos Bursó proposes an interesting interpretation of the number e, the base of natural logarithms, which, although it is only suitable for people with certain mathematical knowledge, I have not resisted the temptation to include it:
Suppose we see a random line of people forming arbitrarily long, and we are always able to distinguish, given two people, which one is the tallest (no matter how small the difference). We count the number of people arriving until the last one to arrive is taller than the second-to-last one (this number will always be greater than or equal to 2). Then:
– the expected or average value of this variable quantity is the number e.
– the probability that the last person is also taller than all the previous ones is e-2.
(Each of the two statements above is equivalent to the sum of the series of 1/n! for n = 0, 1, 2, 3… being the number e).
The law of anomalous numbers
As we have seen, tables of logarithms, by allowing the conversion of multiplications into additions and divisions into subtractions, made calculations considerably easier in the days before computers; but they fell into disuse long ago, along with the wonderful slide rules that were in the top pocket of every self-respecting engineer.
In the 19th century, tables of logarithms were among the most consulted manuals in any technical or scientific library, and this continued use enabled the great astronomer and mathematician Simon Newcomb to notice that the first pages of all the tables he examined showed more signs of use than the next, and that the level of use decreased steadily as the pages were turned. This meant that there were more consulted numbers beginning with 1 than any other digit, followed in number by those beginning with 2, then those beginning with 3…
From his observations, Newcomb enunciated a law on the frequency of numbers in relation to the mantissas (decimal parts) of their logarithms, which allowed him to estimate that the probability of a number taken from the real world starting with 1 is approximately 30%, that of it starting with 2 is 18%, that of it starting with 3 is 12%… and so on, always decreasing, until reaching 9, whose probability of heading a number does not reach 5%.
Newcomb’s counterintuitive conclusions were forgotten until 1938, when the American engineer Frank Benford, after checking more than 20,000 numbers from 20 different samples (such as population numbers in a list of cities, stock quotes, physical constants, molecular weights, mortality rates, postal address numbers, etc.), enunciated what he called the “law of anomalous numbers,” today known as Benford’s law (although some of us prefer to call it the Benford-Newcomb law), according to which the first digit n in a sample of numbers taken from the real world appears with a probability given by the formula: logd(n+1) – logd(n). (Due to typographical limitations, the decimal logarithm is indicated as logd.)
Shouldn’t the first digits be distributed equally among the nine digits (obviously excluding zero)? Why is the number of inhabitants of a city more likely to begin with 1 than with 9? Can you think of any explanation for this seemingly arbitrary result?
#puzzling #anomalous #numbers