Number System in Digital Electronics

A number system in digital electronics is a way of representing values using a fixed set of digits and a base (radix). Because digital hardware stores and processes information as 0s and 1s, understanding binary, octal, decimal and hexadecimal — and how to convert between them — is fundamental to every area of VLSI and embedded design. This guide covers the four common number systems and the conversions you need.

The four number systems

All four above represent the same value, eleven. Hardware works in binary; hexadecimal is a compact shorthand (one hex digit = four bits) used in registers, memory addresses and machine code.

Decimal to binary conversion

Repeatedly divide by 2 and read the remainders bottom-to-top. Convert 13₁₀:

Reading the remainders bottom-to-top gives 1101₂.

Binary to decimal conversion

Multiply each bit by its positional weight (powers of 2) and sum. For 1101₂:

1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13₁₀

Binary ↔ hexadecimal

Group binary digits into fours from the right; each group is one hex digit. 1101 0110₂ = D6₁₆. To go back, expand each hex digit into four bits. This grouping is why hexadecimal is used everywhere in digital systems — it maps cleanly onto bytes.

Binary ↔ octal

Group binary digits into threes from the right; each group is one octal digit. 001 101₂ = 15₈.

Signed numbers and complements

To represent negative values, hardware uses the one’s and two’s complement schemes. Two’s complement is dominant because addition and subtraction use the same adder hardware, with no separate sign handling.

Why number systems matter in VLSI

Register values, memory addresses, opcodes and bus data are all read and debugged in binary and hexadecimal. Designers switch between bases constantly when writing RTL, interpreting waveforms, and reading synthesis or place-and-route reports.

FAQs on number systems

What are the four types of number systems?

Binary (base 2), octal (base 8), decimal (base 10) and hexadecimal (base 16). Digital hardware uses binary internally; hexadecimal is the common human-readable shorthand.

Why is hexadecimal used in digital electronics?

One hexadecimal digit represents exactly four binary bits, so hex compactly expresses byte- and word-sized values that would be long and error-prone in binary.

How do you convert decimal to binary?

Divide the decimal number by 2 repeatedly, recording each remainder. The binary result is the remainders read from last to first (bottom to top).

What is the base of the binary number system?

Base 2 — it uses only the digits 0 and 1, matching the two voltage levels used in digital circuits.

Build on this in ChipXpert’s RTL Design course. Related: one’s & two’s complement, binary to Gray code, logic gates.