How do you do binary numbers? - Answers

In order to understand how to do binary numbers, you must first understand clearly what binary is. Binary is just another way to express numbers, using the same system we normally use. There is only one difference, and that is the number of digits available. In decimal, we have ten different digits. In binary, we have only two. Other than that, the two systems work exactly the same way. Consider the way we express a number. Let's take my birth year, nineteen-seventy-six. we normally express that as: 1976 Now consider the individual digits in that number, 1, 9, 7 and 6. As we're all taught in our earliest years, those digits each fall into a column, often referred to as "ones", "tens", "hundreds", etc. We call them that because the digit in that column does not actually represent it's own value, but a multiple of it's own value. Instead of "1 + 9 + 7 + 6", which is obviously incorrect, we actually see that as: 1 × 1000 + 9 × 100 + 7 × 10 + 6 × 1 which can also be expressed as: 1 × 103 + 9 × 102 + 7 × 101 + 6 × 100 In other words, each column represents a power of ten that is multiplied by the number in column. Those powers increment with each column from right to left. Binary works exactly the same way. The only difference between them is the exponent used. We normally use powers of ten because we have ten unique symbols that represent the first ten values, 0 through 9. There is nothing preventing us from using other numeric bases though. In the case of binary, or "base two" numbers. These work exactly the same way as decimal, but instead of having ten unique symbols, it only has two, "0" and "1". As a result, the columns used in expressing the numbers are not powers of ten, but powers of two. Instead of ones, tens, hundreds, etc. the columns in a binary number are ones, twos, fours, eights and so on. So the same number we used above, rather than being expressed as "1976", would instead be: 11110111000 If you break it down, you'll see that this is: 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 or converting back to decimal, 1024 + 512 + 256 + 128 + 0 + 32 + 16 + 8 + 0 + 0 + 0 = 1976 This same technique of expressing numbers works in any numeric base. For example, in "hexadecimal", or base 16, we would use 16 unique symbols, rather than 10. The standard there is to use letters as additional symbols, so the digits in use would be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. That same number would then be expressed as "7B8", or: 7 × 162 + B × 161 + 8 × 160.