There are 9 single-digit numbers (1 through 9) primarily due to the structure of the base-10 (decimal) numeral system. A single digit represents a value less than the base. Since base-10 uses ten as its base, single digits can only represent values from 0 to 9.
Understanding Base-10
The base-10 system is a positional numeral system, meaning the value of a digit depends on its position within a number. In base-10:
- We use ten distinct symbols (digits) to represent numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
- Each position in a number represents a power of 10. From right to left, these positions are the ones place (100), the tens place (101), the hundreds place (102), and so on.
Once we reach the value of the base (10), we need a second digit to represent it. Therefore, any value less than the base can be represented with a single digit. Since 0 is also considered a single digit, the single digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Analogy to Other Bases
The video reference alludes to the fact that other base systems have different number of single digits.
- Binary (Base-2): Binary uses only two digits: 0 and 1. Therefore, there is only 1 single-digit number greater than zero.
- Hexadecimal (Base-16): Hexadecimal uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here, A represents 10, B represents 11, and so on, up to F which represents 15. Therefore, there are 15 single-digit numbers greater than zero.
Summary
In conclusion, the number of single-digit numbers in a number system is determined by its base. In the decimal system, which uses base-10, there are nine single-digit numbers (1-9) greater than zero because, after 9, we need a second digit to represent the value 10.
