It is thought that we use the decimal system because we have 10 fingers, but the truth is we could have just as easily used the binary system naturally. The Mayans used a base-20 system, perhaps because they used their toes also. The reason we want to be able to think in binary is for programming and circuit design. The most basic TTL component is the flip-flop (a switch). It is High or Low, On of Off, 1 or 0. If we wanted a computer to think in base-10 we would need 9 different states not including a 0. With Radix-2 we only need 2 states, 1 or 0. This is the most efficient numeration system a digital computer can work with at the lowest level. As a matter of fact this will remain true for ALL computers until analog, or multi-state computers are invented.
The question is; if it's so great, why don't we use it all the time? Well it can get very tedious to write, for instance 36771 in decimal notation only takes 5 characters to describe. In binary notation it is: 1000000000000011, the same number takes 16 characters to describe, over 3 times as many characters! The reason is simple, decimal notation has 10 characters to choose from to describe a number 0-9, binary only has 2 characters, 1 and 0. There are systems that can shorten decimal numbers also, such as base-16, known as the hexadecimal system that we will learn about later. The same number in hex is 8003, this is because in hex notation there are 16 characters to choose from, 0-F! For instance 3A in hex is equal to 58 in decimal. This is because A is in the 1's column (160), and 3 in the 16's column (161) so (3*16+10 = 58, 10 = A in hex, carry the 1 in decimal), more on that later.
To fully understand binary, we will need to study another type of numeration system as well, this system is referred to as the octal system. Yes you guessed it, the octal system is Radix-8, using the characters 0 though 7. The reason we need to learn about counting in base-8 will become clear later. Working in octal numbers is an easy way to convert binary numerations to decimal notation, octal numbers are sort of a mid point. Every three bits of a binary number represent an octal digit. This makes it easy to look at a binary number and see what it is as an octal. Again more on this later, the writing here is just to tell you what is coming so you are not thrown into the world of unknown numeral systems.