Digital Integrated Circuit Design Using Verilog and Systemverilog

Appendix C

Number systems

Digital circuits all work in binary, where each digit can only take the values of zero or one. Despite this seeming limitation, any arithmetic problem can be solved to any arbitrary level of precision, if enough resources are brought to bear.

In binary numbers, each digit represents a power of two. For a four-digit number, the maximum value will be 15, as the value will be d₀ × 2⁰ + d₁ × 2¹ + d₂ × 2² + d₃ × 2³ for digits d₀ through d₃. If all four are set (logic 1), then the value will be 1 + 2 + 4 + 8, or 15.

Because binary numbers can only take two values per digit, it takes more digits tog represent numbers than are necessary in the decimal system. The largest number that can be represented in n binary bits is ...