Explain why the digits in a place value system with base, b, are 0,1,... b-1 and why no digits for numbers greater than b-1 are needed

Base b only needs b symbols. After all b are used up, the next number is 10, and the bottom digit cycles through the other b-1 digits again, before going on to 20 (assuming b >= 2!)

In a place value system with base, b, the digits represent the values at each position in a number. The digits range from 0 to b-1 because each digit's value corresponds to the power of b raised to that position.

For example, in base 10, the digits range from 0 to 9. This is because each digit's place represents a power of 10. The rightmost digit is the ones place, the next digit to the left is the tens place, then the hundreds place, and so on. Each place represents a different power of 10: 10^0 for the ones place, 10^1 for the tens place, 10^2 for the hundreds place, and so on.

Similarly, in a base b system, each digit's place represents a power of b. Therefore, the range of digits can be from 0 to b-1. For example, in base 5, the digits range from 0 to 4 because each place represents a power of 5: 5^0 for the ones place, 5^1 for the fives place, 5^2 for the twenty-fives place, and so on.

Since the digits represent values up to b-1, it is not necessary to have digits for numbers greater than b-1. The next number in the sequence can be represented by adding a digit to the left in the same way as in base 10. For instance, in base 10, after 9 comes 10, which is represented by adding a digit to the left.

In summary, the digits in a place value system with base b range from 0 to b-1 because each digit's place represents a power of b. No digits are needed for numbers greater than b-1 because they can be represented by adding a digit to the left.

In a place value system with base, b, the digits range from 0 to b-1 because we use the decimal numbers 0 to 9 in our everyday base 10 system.

To understand why we don't need additional digits for numbers greater than b-1, let's take a look at how the place value system works. In any base, each digit's position represents a certain power of the base. For example, in base 10, the rightmost digit represents the ones place (10^0), the next digit represents the tens place (10^1), then hundreds place (10^2), and so on.

When we reach the largest digit for a given base (b-1), the next number is formed by increasing the value in the next position to the left. For example, in base 10, when we reach 9 in the ones place, we increase the tens place to 1, and the ones place becomes 0. This way, we can represent higher numbers by "carrying over" to the next position.

Since we only have b-1 digits, we will always need to carry over to the next position when we reach that limit. For instance, in base 10, when the ones place reaches 9, we carry over to the tens place, making the ones place reset to 0. This process allows us to represent any number without needing additional digits.

So, the digits from 0 to b-1 are sufficient to represent all the numbers in a place value system, as we can always carry over to the next position when we reach the maximum digit value. No additional digits are needed because they can be expressed using the existing digits.