Number Systems

Understand binary, decimal, octal and hexadecimal bases used in digital electronics.

Why several bases?

A digital circuit manipulates electrical levels interpreted as bits: 0 and 1. For humans, long binary strings quickly become hard to read. Hexadecimal and octal are compact notations that still map directly to bits.

The rule is always the same: each position has a weight. In base 2, the weights are 1, 2, 4, 8, 16, and so on.

Reading a binary number

The number 101101₂ is read by adding the weights of the bits set to 1:

101101₂ = 32 + 8 + 4 + 1 = 45₁₀

In digital electronics, this is how you interpret a bus value, a memory address or a counter.

Decimal to binary

To convert a decimal integer to binary, divide by 2 and read the remainders from bottom to top.

Example with 45₁₀:

45₁₀ = 101101₂

The rightmost bit is the LSB (least significant bit). The leftmost bit is the MSB (most significant bit).

Hexadecimal: bus notation

Hexadecimal is common because one hexadecimal digit represents exactly 4 bits.

Example:

0x2D = 0010 1101₂ = 45₁₀

On an FPGA, a 32-bit bus is often written as 8 hexadecimal digits:

constant C_MASK : std_logic_vector(31 downto 0) := x"0000_FF00";

Octal: groups of 3 bits

Octal groups bits by sets of 3. It is less common than hexadecimal in modern FPGA projects, but it is useful to recognize.

101101₂ = 101 101₂ = 55₈

Each group ranges from 000₂ to 111₂, therefore from 0₈ to 7₈.

Binary fractions

Bits to the right of the binary point have fractional weights:

101.01₂ = 4 + 1 + 0.25 = 5.25₁₀

This idea appears in digital filters, PWM, fixed-point arithmetic and DSP blocks.

VHDL points to watch

The binary value is not enough: you also need its type and width.

signal count_u : unsigned(7 downto 0);
signal data_s  : signed(7 downto 0);
signal flags   : std_logic_vector(7 downto 0);

• unsigned represents a positive integer.
• signed represents a two's-complement signed integer.
• std_logic_vector is only a packet of bits: its meaning depends on context.

Two buses may contain the same bits while representing different values. This is a common source of mistakes in comparisons, additions and width extensions.

Key points

• Hardware manipulates bits, but humans use compact notations.
• Hexadecimal is the natural format for wide buses.
• A numeric value is understood through its base, width and interpretation.
• In VHDL, choosing unsigned, signed or std_logic_vector clarifies the circuit intent.

📝 Test your knowledge - Chapter quiz