Bases

The most commonly used number systems are:

• Decimal (Base-10): Uses digits 0-9. It is the standard system for human counting and arithmetic.
• Binary (Base-2): Uses only two digits, 0 and 1. It is the fundamental language of computers.
• Octal (Base-8): Uses digits 0-7. It is sometimes used in computing as a more compact way to represent binary numbers.
• Hexadecimal (Base-16): Uses digits 0-9 and letters A-F. It is widely used in programming and digital electronics for representing large binary values concisely.

Computers use the binary system because their internal hardware, like transistors, operates in two distinct states: on or off. These two states can be perfectly represented by the two digits of the binary system, 1 (on) and 0 (off). This simple, two-state logic is far more reliable and easier to build into electronic circuits than a system requiring ten different states for the decimal system.

The concept of place value is fundamental to all number bases, but its calculation changes. In any base, the value of a digit is determined by its position multiplied by the base raised to the power of that position. For example, in the decimal (base-10) number 123, the place values are (1 × 10²) + (2 × 10¹) + (3 × 10⁰). In the binary (base-2) number 111, the place values are (1 × 2²) + (1 × 2¹) + (1 × 2⁰), which in decimal is 4 + 2 + 1 = 7.

The hexadecimal number system is a base-16 system. It uses 16 unique symbols: the digits 0-9 and the letters A-F to represent the values 10-15. It is crucial in computing because it provides a human-friendly way to represent long binary strings. One hexadecimal digit can represent a four-digit binary number, making it much easier for programmers to read and write values for memory addresses, colour codes, and other data.

To convert a number from any base to the decimal system, you multiply each digit by its corresponding place value (the base raised to the power of the position) and then sum the results. For example, to convert the binary number 1011₂ to decimal:

• (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
• = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)
• = 8 + 0 + 2 + 1
• = 11

So, 1011 in base-2 is equal to 11 in base-10.

The octal (base-8) and hexadecimal (base-16) systems have a direct and simple relationship with the binary (base-2) system because 8 and 16 are powers of 2 (8 = 2³ and 16 = 2⁴). This allows for easy conversion:

• One octal digit corresponds to exactly three binary digits.
• One hexadecimal digit corresponds to exactly four binary digits.

This relationship is why programmers often use octal or hexadecimal as a shorthand for writing long binary numbers.

Yes, the simplest is the unary (base-1) system. However, it is not a positional system like binary or decimal. In unary, a number 'n' is represented by repeating a single symbol 'n' times (e.g., 5 is represented as '11111'). It is highly inefficient for any practical calculation and is mainly a theoretical concept. For a positional number system to work effectively and represent zero, a base of at least 2 is required.