Numeral Systems: Overview & Positional Notation
Numeral Systems: Overview & Positional Notation
This page introduces numeral systems, explains why different bases exist, and outlines the principle of positional notation.
Decimal system
Humans use the decimal system with base 10.
- Digits are 0–9.
- A number is a sequence of digits, where each digit has a positional value.
- Example: 123 = 1 × 100 + 2 × 10 + 3 × 1.
Other bases are also used in daily life:
- Base 12 for hours on a clock.
- Base 60 for minutes and seconds.
* Example: 50 minutes and 33 seconds = 50 × 60 + 33 = 3033 seconds.
Binary system and bits
Computers work electronically, using two states: voltage present or absent. This gives the binary system with digits 0 and 1.
The smallest unit of information is the bit. Each additional bit doubles the number of possible states:
- 1 bit = 2 states
- 4 bits = 16 states (nibble)
- 8 bits = 256 states (byte)
- 16 bits = 65 536 states (word)
- 32 bits = over 4 billion states (double word)
- 64 bits = over 18 quintillion states (quad word)
The issue with binary notation
Even moderate numbers require many digits in binary. A 64-bit value is too long to be readable, even when grouped into bytes. For this reason, shorter notations are used by grouping bits:
- 3 bits → octal (base 8)
- 4 bits → hexadecimal (base 16)
Essence of a number base
Roman numerals were not column-based, but modern systems use positional notation. In positional systems, the value of a digit depends on its position:
- In any base, the number written as 10 represents the base itself.
- Column 0 (rightmost) is multiplied by base^0 = 1.
- Column 1 is multiplied by base^1.
- Column 2 is multiplied by base^2, and so on.
Example in decimal:
- 532 = 5 × 10^2 + 3 × 10^1 + 2 × 10^0.
This principle applies to all bases: binary, octal, hexadecimal, and beyond.