Understanding Different Number Systems: A Complete Guide

In the digital world, numbers are everything. Computers, smartphones, and every digital device you use rely on different number systems to process and store information. Today, we’ll dive deep into number systems, what base or radix means, how to convert numbers between different systems, and what direct conversion is all about.

What is a Number System?

A number system is a way to represent numbers using a set of symbols or digits. Every number system is based on a particular base (also called radix), which tells us how many unique digits are used in that system.

What is Base or Radix?

Base (or Radix) is the number of different digits or symbols a number system uses to represent numbers.

Example:

  • Decimal System (Base 10) uses 10 digits: 0–9.
  • Binary System (Base 2) uses 2 digits: 0 and 1.
  • Octal System (Base 8) uses 8 digits: 0–7.
  • Hexadecimal System (Base 16) uses 16 digits: 0–9 and A–F (where A=10, B=11, etc.).

How to Convert Decimal to Other Number Systems?

Decimal numbers can be converted into other systems by different methods for integer and fractional parts.

1. Integer Part Conversion

  • Method: Divide the decimal number by the base you want to convert to and record the remainder.

  • Steps:

    1. Divide the number by the new base.

    2. Write down the remainder.

    3. Repeat the process with the quotient until it becomes 0.

    4. Read the remainders from bottom to top.

Example: Convert 45 to binary.

So, 45 in binary = 101101.

2. Fractional Part Conversion

  • Method: Multiply the fractional part by the base and record the integer part.

  • Steps:

    1. Multiply the fraction by the new base.

    2. Write down the integer part.

    3. Repeat the process with the fractional part.

    4. Stop after a few digits or when the fraction becomes 0.

Example: Convert 0.625 to binary.

        0.625 × 2 = 1.25  → 1

        0.25 × 2 = 0.5      → 0

        0.5 × 2 = 1.0        → 1

So, 0.625 in binary = 0.101.

How to Convert Other Systems to Decimal?

The general method is positional multiplication:

Integer Part:

Multiply each digit with the base raised to the position's power (starting from right to left with position 0).

Example 1: Convert 1011 (binary) to decimal.

Example 2: Convert 2F (hexadecimal) to decimal.

Fractional Part

The idea is very similar, but instead of positive powers, you use negative powers of the base. Multiply each digit after the point (.) by the base raised to a negative power:

Example 1: Convert (0.1011)₂ (binary) to decimal.

Now add them up:

0.5 + 0 + 0.125 + 0.0625 = 0.6875

So, (0.1011)₂ = (0.6875)₁₀

Example 2: Convert (0.3A)₁₆ (hexadecimal) to decimal.

Now add them:

0.1875 + 0.0390625 = 0.2265625

Thus, (0.3A)₁₆ = (0.2265625)₁₀

What is Direct Conversion?

Sometimes you can convert between two systems directly, without going through decimal.

  • Binary to Octal: Group binary digits into sets of 3 (from right) and convert each group.

  • Binary to Hexadecimal: Group binary digits into sets of 4 (from right) and convert.

Example: Convert 11010111 (binary) to hexadecimal.

Group: (0001) (1010) (1111) → 1A7

Note: Always pad with extra zeros if needed to make complete groups.

Understanding number systems is the foundation of learning digital electronics, computer science, and networking. Master these conversions, and you’ll open doors to deeper and more complex topics easily!

Coming up next in the Digital Series: Data Representation - stay tuned to hobitronics.blog

Comments

  1. ChocoCherub28 April 2025 at 02:02

    When should we use direct conversion?

    ReplyDelete
    Replies
    1. Hobitronics28 April 2025 at 03:00

      We use direct conversion when the two number systems are related by powers, such as binary to octal (2³ = 8) or binary to hexadecimal (2⁴ = 16), making the conversion faster and easier without needing to go through decimal. In practice, direct conversion is used when the two number systems are related by powers of 2 — mainly between binary, octal, and hexadecimal — because binary is the base language of digital electronics. Direct conversion can happen between any related bases (like base-3 and base-9), but in electronics, we focus only on powers of 2.

      Delete

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