How does binary work and how does it relate to decimal?

Answer 1

Unlike decimal, which uses base 10, binary uses base 2. What this means is that a number like eleven is represented in decimal by 11. This can be broken up into two parts. The second 1 represents 10^0. You take the 1 and multiply it by 10^0, which is 1. The first 1 represents the 10^1 spot. You take it and multiply it by 10^1, which equals 10. You then add up the 1 from the 10^0 spot and the 1 from the 10^1 spot, which is 1 + 10 = 11. In binary, each spot represents 2^i instead of 10^i. So eleven would be represented by 1011. The ones represent 1 * 2^i and the zeros represent 0 * 2^i. So by breaking it up into its parts, we get the following:
(1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8+0+2+1 = 11.

To illustrate how this works further, take the number 23. In decimal, the 2 represents 2 times 10^1, which is 20. The 3
represents 3 times 10^0, which is 3. You add these two together: (2 * 10^1) + (3 * 10^0) = 20 + 3 = 23.
Binary works the same way, only with each digit representing
base 2. The following would be a binary representation of 23:
10111
The following is an expansion of 10111:
2^4 + 0^3 + 2^2 + 2^1 + 2^0 = 16 + 4 + 2 + 1 = 23.

Answer 2

To expand on above, you set bits with binary, think changing the on/off state like a light switch. If you are counting in binary you are setting bits to an on or off state.
For instance if you are doing a binary counter you are turning on the bits that would equal the number you are trying to generate so,

Values: 1 2 4 8 16
to generate the number 5 you would turn on the following bits: 10100 (1 + 4 = 5) another example, 11 = binary 11010 (1 + 2 + 8 = 11)

Why would you do this? everything you do on a computer ultimately ends up (is converted to) binary.

Answer 3

Decimal is 10 base while binary is 1 base. You can only use 1 or zero when counting in binary
In decimal, 1, 2, 3, 4, 5
in binary 1, 10, 11, 100, 101 etc