How does a base determine the symbols and place values used to represent numbers?

"As I mentioned, a base is simply a number that’s an anchor for our place value system. It represents how many distinct symbols we will use to represent numbers. This implicitly sets the value of the largest quantity we can hold in one digit, before we’d need to “roll over” to two digits. In base 10 (decimal), we use ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Consequently, the number nine is the highest value we can hold in a single digit. Once we add another element to a set of nine, we have no choice but to add another digit to express it. This makes a “ten’s place” because it will represent the number of sets-of-10 (which we couldn’t hold in the 1’s place) that the value contains. Now why is the next place over called the “hundred’s place” instead of, say, the “twenty’s place”? Simply because twenty — as well as every other number less than a hundred — comfortably fits in two digits. We can have up to 9 in the one’s place, and also up to 9 in the ten’s place, giving us a total of ninety-nine before we ever have to cave in to using three digits. The number one hundred is exactly the point at which we must roll over to three digits; therefore, the sequence of digits 1-0-0 represents one hundred. If the chosen base isn’t obvious from context (as it often won’t be in this chapter) then when we write out a sequence of digits we’ll append the base as a subscript to the end of the number. So the number “four hundred and thirty-seven” will be written as 437₁₀. The way we interpret a decimal number, then, is by counting the right-most digits as a number of individuals, the digit to its left as the number of groups of ten individuals, the digit to its left as the number of groups of hundred individuals, and so on. 5472₁₀ is just a way of writing 5 × 1000 + 4 × 100 + 7 × 10 + 2 × 1. If we use exponential notation (remember that anything to the 0th power is 1), this is equivalent to: 5472₁₀ = 5 × 10³ + 4 × 10² + 7 × 10¹ + 2 × 10⁰."