Leibniz’ different base

This is a selection from a new book, Science is Like That, which is yet to find a publisher.

I’ll teach you differences.
—William Shakespeare, King Lear, I, iv, 90.

They say Gottfried Leibniz taught himself Latin when he was eight, and by fourteen, he could read Greek as well. We may as well believe this as not: these legends of precocity in scientific greats are as tenacious as any urban myth. It matters little if Leibniz taught himself Basque while hanging upside down like a bat at the age of two: what really counts is what he did later on.

The son of a professor of moral philosophy, Leibniz was interested in the mathematical side of philosophy. In his lifetime, he introduced the use of the dot to indicate multiplication, popularised the decimal point, the equals sign, the colon for division and ratio, and the use of numerical superscripts for exponents (like x² and x³) in algebra. We also owe the elongated sigma for summation in calculus, and the way we use the letter d in differential calculus (as in dy/dx) is his idea also.

He and Newton argued over who invented calculus, but whatever else Newton did, he never designed a calculating machine as Leibniz did. Only Blaise Pascal had done so before Leibniz, and Charles Babbage did so later. Leibniz’ design was used in building the first totalisator (also called a tote or pari-mutuel, a machine used to manage betting on horse races), because it could multiply and divide.

Leibniz wanted to create a united Europe, even before Germany was a single nation, and long before anybody dreamed of the European Union. In the end, he was librarian to the court of Hanover, but when the Elector of Hanover went off from there to England to take up his new throne as King George I in 1714, Leibniz was left in Germany, presumably because of his disputes with Newton.

Whatever the reason, he was definitely left behind, and he died a couple of years later, leaving a ‘sleeper’ in the form of a letter written to the French Academy of Sciences in 1701, in which he outlined the binary number system which is used by all modern computers.

I enclose an attempt to devise a numerical system that may prove to be entirely new. Briefly, here is what it is…By using a binary system based on the number 2 instead of the decimal system based on the number 10, I am able to write all of the numbers in terms of 0 and 1. I have done this not for mere practical reasons, but rather to allow new discoveries to be made…This system can lead to new information that would be difficult to obtain in any other way…

Talking of bases, there is a conundrum that depends on readers understanding the significance of the apparently erroneous sum: 6x9=42. (This will only make serious sense to people who have read the works of Douglas Adams, especially The Hitch Hiker’s Guide to the Galaxy and its successors.)

Something over two decades ago, I observed on an Internet list that the relationship 6x9=42 is true if the calculations are performed in base-13 notation. A list member, known only as Merlyn, noted that there is a pattern to be observed. If we take “six times x = forty-two,” and vary the value of the base, we find a number of values of x which satisfy the statement, and these form a pattern when we examine both x and the base used. Six times x equals forty two is true when x is:

5 and the base for the calculation is 7;

7 and the base for the calculation is 10;

9 and the base for the calculation is 13;

11 and the base for the calculation is 16;

13 and the base for the calculation is 19;

15 and the base for the calculation is 22;

17 and the base for the calculation is 25;

19 and the base for the calculation is 28;

21 and the base for the calculation is 31;

23 and the base for the calculation is 34;

25 and the base for the calculation is 37;

27 and the base for the calculation is 40;

29 and the base for the calculation is 43…

The pattern continues beyond that, and it is an elegant pattern. Explaining it requires finding a formula for each of x and the nominated base in terms of its order n, in the pattern. These days, most computing is based on binary (base 2) or hexadecimal (base 16) numbers, but we didn’t start out that way, because we have 10 digits on our hands (old sawmillers excepted, sometimes).