Let's start with the system you already know. We usually work in base 10. In base 10, the place values are ones, tens, hundreds, thousands and so on. So when we see a number like 437, it really means 'four hundreds, 3 tens and 7 ones.' We understand that to be worth 'four hundred thirty seven'. The place values are determined by raising the base to powers. In base 10, ones is 10^0, tens is 10^1, hundreds is 10^2, thousands is 10^3 and so on. When we start to count in base 10, we can write 1, 2, 3, 4, 5, 6, 7, 8, 9. Each of those stands for how many ones we have. The number 8 means 8 ones, or 8 * 1. But when we go past 9 to the number 10, we don't have a single digit that stands for '10 ones.' So instead, we use a two-digit number, 10, which stands for '1 ten and 0 ones.' Once we get to 99, we have reached '9 tens and 9 ones.' Going past that, we move to a three-digit number, 100, which means '1 hundred, 0 tens and 0 ones.' It's kind of hard to think about this, because your brain just does it without thinking about it, but that's what's really going on. So what happens in base 4? The place values are again given by raising 4 to powers. 4^0 = 1 4^1 = 4 4^2 = 16 4^3 = 64 So, the number 23 in base 4 is NOT worth twenty-three. It's only twenty three in base 10, where it means '2 tens and 3 ones.' In base 4, 23 (which is read as 'two-three') means '2 fours and 3 ones.' So it has a value of 2*4 + 3*1 or 8 + 3 or 11. Now think about how we count in base 4. We start with 1, 2, 3. But there is no digit '4' to use--the number 4 is written '1 four and 0 ones,' so it's 10.
GLaDOS was right.
