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
In a place value system with a base, b, the digits represent different values depending on their position within a number. The value of each digit is determined by multiplying it with the corresponding power of the base.
For example, let's consider a base 10 system, which we commonly use in our everyday lives. In this system, we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The value of each digit is determined by its position in the number. The rightmost digit represents ones, the next digit to the left represents tens, the next one represents hundreds, and so on.
Now, let's extend this explanation to a generalized base, b. In this case, the digits available to us are from 0 to b-1. For instance, in a base 5 system, the digits are 0, 1, 2, 3, and 4.
Why are these digits limited to the range of 0 to b-1?
The reason for this limitation lies in the principles of positional notation. In a given base, b, each position represents a different power of the base. The rightmost position corresponds to b^0 (which is 1), the second position from the right corresponds to b^1, the third digit corresponds to b^2, and so on.
To avoid ambiguity or confusion, we need to ensure that each digit represents a unique value based on its position within a number. Therefore, we limit the available digits to be less than the base itself (b-1). This ensures that no digits greater than b-1 are needed because any digit greater than or equal to b would result in a conflict with the next higher position's value.
For example, if we were to include the digit "5" in a base 5 system, it would create ambiguity because we already use the digit "1" to represent the value of 1, equivalent to b^1. Introducing the digit "5" would make it unclear whether it represents 5 units or 5 times the base (b^1), potentially leading to confusion.
Therefore, by restricting the available digits to the range of 0 to b-1, we ensure a clear and unambiguous representation of numbers in a place value system.