Understanding binary numbers

Instructive video that simply shows how binary numbers work compared to decimal ones.


Decimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111

A binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically “0” (zero) and “1” (one).

The base-2 numeral system is a positional notation with a radix of 2.

Each digit is referred to as a bit.

Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices.

The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz.

However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India.

Leibniz was specifically inspired by the Chinese I Ching.

Leibniz studied binary numbering in 1679; his work appears in his article “Explication de l’Arithmétique Binaire” (published in 1703).

The full title of Leibniz’s article is translated into English as the “Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi”. (1703).

Leibniz’s system uses 0 and 1, like the modern binary numeral system.

An example of Leibniz’s binary numeral system is as follows:

0 0 0 1 numerical value 20
0 0 1 0 numerical value 21
0 1 0 0 numerical value 22
1 0 0 0 numerical value 23

Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus.

As a Sinophile, Leibniz was aware of the I Ching, noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111 and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.

Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary.

Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own religious beliefs as a Christian.

Binary numerals were central to Leibniz’s theology.

He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.

(A concept that) is not easy to impart to the pagans, is the creation ex nihilo through God’s almighty power. Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing.
— Leibniz’s letter to the Duke of Brunswick attached with the I Ching hexagrams