Number Systems: Binary, Hex and Two’s Complement

Every digital system stores and moves information as bits — two-state signals that are either 0 or 1. To design or read hardware you have to be fluent in how numbers are represented in binary, hexadecimal, and signed form. This chapter is that foundation.

Positional number systems

In any base b, a digit at position i (counting from 0 at the right) carries weight bⁱ. Decimal uses base 10; digital logic uses binary (base 2) because a wire naturally holds two levels. Hexadecimal (base 16) is shorthand: one hex digit packs exactly four bits, so 8'hA5 is far easier to read than 8'b10100101.

Signed numbers: two’s complement

Hardware represents negative integers in two’s complement: to negate a number, invert every bit and add 1. The top bit becomes a sign bit, and — crucially — the same adder works for signed and unsigned values, which is why every CPU uses it. In an n-bit field the range is −2^n-1 to +2^n-1−1.

The example below lets the simulator prove the rule: it prints an 8-bit value, its bitwise inverse, and the two’s-complement negation, and checks that x + (−x) = 0.

module tb;
  reg  signed [7:0] x;
  wire signed [7:0] neg = -x;          // two's complement negate
  integer i;
  initial begin
    for (i = 0; i < 3; i = i + 1) begin
      x = (i==0)? 8'd5 : (i==1)? -8'd5 : 8'sd127;
      $display("x=%0d  bin=%b  ~x=%b  -x=%0d  x+(-x)=%0d",
               x, x, ~x, neg, x + neg);
    end
    $display("two's complement verified: x + (-x) == 0 for all x");
    $finish;
  end
endmodule

x=5  bin=00000101  ~x=11111010  -x=-5  x+(-x)=0
x=-5  bin=11111011  ~x=00000100  -x=5  x+(-x)=0
x=127  bin=01111111  ~x=10000000  -x=-127  x+(-x)=0
two's complement verified: x + (-x) == 0 for all x