Number Systems & Their Conversion Used in Digital Electronics

Definition

A number system is a way of representing numbers using a set of symbols and rules based on a fixed base or radix. In digital electronics, the most commonly used number systems are:

Decimal (base 10)

• : Uses digits 0 to 9

Binary (base 2)

• : Uses digits 0 and 1

Octal (base 8)

• : Uses digits 0 to 7

Hexadecimal (base 16)

• : Uses digits 0 to 9 and A to F

Number system conversion is the process of changing a number from one base to another without changing its value.

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Main Content

1. Positional Number Systems and Their Bases

• In a positional number system, the value of each digit depends on its position and the base of the system. The rightmost digit has the lowest positional value, and moving left increases the power of the base.
• The general form of a positional number can be written as:
  $$(d_n d_{n-1} ... d_1 d_0)_b = \sum d_i \times b^i$$ where b is the base or radix.

Common number systems in digital electronics:

Decimal system (base 10)

Uses 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Example:
$$347_{10} = 3 \times 10^2 + 4 \times 10^1 + 7 \times 10^0$$

Binary system (base 2)

Uses 2 symbols: 0 and 1
Example:
$$1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}$$

Octal system (base 8)

Uses 8 symbols: 0 to 7
Example:
$$725_8 = 7 \times 8^2 + 2 \times 8^1 + 5 \times 8^0$$

Hexadecimal system (base 16)

Uses 16 symbols: 0 to 9 and A to F, where A=10, B=11, ..., F=15
Example:
$$2F_{16} = 2 \times 16^1 + 15 \times 16^0 = 47_{10}$$

Why positional systems are important:

• They allow compact representation of numbers.
• They make arithmetic operations systematic.
• They are the foundation of digital computation.

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2. Binary, Octal, and Hexadecimal Representation in Digital Electronics

Binary is the most fundamental number system in digital electronics

• because electronic circuits have two stable states: ON/OFF, HIGH/LOW, TRUE/FALSE. These states are directly represented by 1 and 0.
• A binary number is made of bits, where each bit represents a power of 2. Groups of bits are frequently used to represent larger values:
• 4 bits = nibble
• 8 bits = byte
• 16 bits = word (commonly, though exact size may vary)

Binary representation example:

• $$110101_2$$ = $1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$ = $32 + 16 + 4 + 1 = 53_{10}$

Octal representation:

• Octal is often used as a compact form of binary because each octal digit corresponds exactly to 3 binary bits.
• Example:
  $$110101_2 = 65_8$$ because
  $$110\ 101 = 6\ 5$$

Hexadecimal representation:

• Hexadecimal is widely used in computing because each hex digit corresponds exactly to 4 binary bits.
• It reduces long binary strings into shorter and more readable forms.
• Example:
  $$111101101011_2 = F6B_{16}$$ because
  $$1111\ 0110\ 1011 = F\ 6\ B$$

Importance in digital systems:

• Binary is used in logic gates, memory, and registers.
• Octal and hexadecimal are used to simplify large binary numbers.
• Hexadecimal is commonly used in memory addresses, machine code, color codes, and debugging.

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3. Number System Conversion Methods

• Converting from one number system to another is essential in digital electronics for communication between humans and machines.
• The conversion depends on whether the number is an integer, fraction, or binary-coded representation.

A. Decimal to Binary

• Divide the decimal number by 2 repeatedly.
• Record the remainders in reverse order.

Example: Convert $25_{10}$ to binary

• 25 ÷ 2 = 12 remainder 1
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

So,
$$25_{10} = 11001_2$$

B. Binary to Decimal

• Multiply each bit by its positional power of 2 and add the results.

Example: Convert $10110_2$ to decimal

$$1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$$ $$= 16 + 0 + 4 + 2 + 0 = 22_{10}$$

C. Binary to Octal

• Group binary digits into sets of 3 starting from the right.
• Convert each group to its octal equivalent.

Example: Convert $101110_2$ to octal

• Group: $101\ 110$
• $101_2 = 5_8$
• $110_2 = 6_8$

So,
$$101110_2 = 56_8$$

D. Octal to Binary

• Replace each octal digit with its 3-bit binary equivalent.

Example: Convert $47_8$ to binary

• $4_8 = 100_2$
• $7_8 = 111_2$

So,
$$47_8 = 100111_2$$

E. Binary to Hexadecimal

• Group binary digits into sets of 4 starting from the right.
• Convert each group to hexadecimal.

Example: Convert $11011110_2$ to hex

• Group: $1101\ 1110$
• $1101_2 = D_{16}$
• $1110_2 = E_{16}$

So,
$$11011110_2 = DE_{16}$$

F. Hexadecimal to Binary

• Replace each hex digit with its 4-bit binary form.

Example: Convert $3A_{16}$ to binary

• $3_{16} = 0011_2$
• $A_{16} = 1010_2$

So,
$$3A_{16} = 00111010_2$$

G. Decimal to Octal / Hexadecimal

• Use repeated division by 8 or 16 respectively.
• The remainders are read in reverse.

Example:
$$83_{10} = 123_8$$ $$255_{10} = FF_{16}$$

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Working / Process

1. Identify the source and target number systems

First, determine from which base the number must be converted and into which base it should be changed. This is important because the method depends on the base. For example, binary to hexadecimal uses grouping, while decimal to binary uses repeated division.

2. Apply the correct conversion technique

Use the suitable method:

• Repeated division for decimal to binary/octal/hexadecimal
• Positional expansion for binary/octal/hexadecimal to decimal
• Grouping bits in 3s for binary to octal or 4s for binary to hexadecimal
• Direct digit replacement for octal/binary or hexadecimal/binary conversions

3. Verify the result by reconversion or checking positional value

After conversion, confirm the answer by converting it back or by evaluating the positional weights. This helps avoid mistakes, especially in long binary numbers and fractional values.

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Advantages / Applications

• Number systems make it easy to represent and process data in digital circuits and computer systems.
• Binary simplifies hardware design because electronic devices naturally support two states.
• Octal and hexadecimal reduce the length of binary numbers, making them easier to read, write, and debug.
• Number system conversion is used in microprocessors, memory addressing, programming, embedded systems, and digital design analysis.
• Hexadecimal is especially useful in machine language, address maps, and color encoding in computer graphics.
• Understanding number systems helps in Boolean algebra, logic gate analysis, and arithmetic circuit design.
• It improves accuracy and efficiency in digital communication and data representation.

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Summary

• Number systems are the foundation of digital electronics and are used to represent data in different bases.
• Binary is the core system in digital circuits, while octal and hexadecimal are compact forms used for convenience.
• Conversion between number systems is done using repeated division, positional expansion, and bit grouping methods.
• These concepts are essential for understanding how computers, processors, and digital devices store and manipulate information.