How Did We Come Up With Binary Numbers?

Actually, the Modern Binary system of numbers is by no means new. The credit for the creation of the modern binary system (base 2, using 0 and 1) goes to Gottfried Wilhelm von Leibniz, his publication in 1703 of Explication de l'Arithmétique Binaire (Explanation of Binary Arithmetic) outlined his system where he had worked out that by the use of just two numbers (0 and 1), any given number could be expressed, and how the system worked. In fact, he can be traced as the one of the earliest pioneers of modern computers, as there have been papers written by him that outlined some of his ideas and plans for a mechanical binary computer that would have used marbles, and the force of gravity to perform calculations. While micro-switches and transistors replaced his marbles, the basic principals are still there.
Frankly the man was truly a genius, if you do some research online about him it's very impressive the sheer number of mathematical processes he theorized or outlined.
Now, while von Leibniz did come up with the modern form of binary that we use today, the use of such numeric systems predates him by (almost or more than) 2000 years, to the ancient Indian writer Pingala, who developed advanced mathematical concepts for describing prosody (the study of poetic meter), and in the process gave us the first recorded description of a binary number system, but as the numbers used in this case were not actually 0s and 1s, but based on the Old Kingdom numeration system, it's not really considered to have been the origin of the modern form. I mention almost or more than 2000 years, because there is still some conjecture as to when he was alive and worked out his system (it seems to have been anywhere between 800 and 200 B.C.).
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Binary is a system allowing numerical quantities to be represented as a series of switches. A switch has two states: On and Off. If the switch is on, it represents a 1 and if a switch is off it represents a zero. In electronics this system is made possible by a simple device called a capacitor. Like a switch, a capacitor has two states charged and discharged. Charged = 1, discharged = 0. A single character in a modern electronic device is represented by a bank of 8 of these capacitors. There are 256 possible combinations of the 8. From 0000 0000 to 1111 1111. That's 2⁸ for you math guys. This is called base 2 numbering. All that is left is to assign some value to each of these 256 possible states. The big guys like IBM and DEC and others did that. They, however, did not always agree on the conversion. That is why we still have different systems like EBCDIC and ASCII.