Binary Numbers and Decimal-Binary Conversion

Computers and digital systems rely on binary numbers—a base-2 system using only 0 and 1—to process and store data. Unlike the familiar decimal (base-10) system, binary simplifies hardware design by representing all values with two states. Mastering binary counting and conversion between decimal and binary is essential for understanding computing fundamentals, networking, and digital logic.

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Key Concepts

Binary vs. Decimal Systems

• Binary (base-2): Uses only 0 and 1. Each digit represents a power of 2.
• Decimal (base-10): Uses digits 0 to 9. Each position represents a power of 10.
• Bit: The smallest unit of data in computing, representing a single binary digit (0 or 1).
• Positional notation: The value of a digit depends on its position (e.g., binary 101 = decimal 5).

Key Insight: Binary and decimal counting follow the same logic—reset to 0 and carry over 1 when the base limit is reached.

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How Binary Counting Works

Binary counting increments like decimal but resets after 1 instead of 9. Each new bit doubles the maximum representable value.

Example: Counting from 7 to 10

Decimal:  7 → 8 → 9 → 10
Binary:   0111 → 1000 → 1001 → 1010

Note: Adding 1 to 111 (7) requires a new bit: 1000 (8).

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Conversion Methods

Decimal to Binary

1. Divide the decimal number by 2 and record the remainder.
2. Repeat with the quotient until it reaches 0.
3. Read the remainders bottom to top to get the binary number.

Example: Convert 13 to binary

13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Result: 1101₂

Binary to Decimal

1. Identify the positions of 1 bits (starting from the right at position 0).
2. Multiply each 1 by 2ⁿ (where n is its position).
3. Sum the results.

Example: Convert 1010₂ to decimal

1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10₁₀

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Bit Growth and Maximum Values

Each additional bit doubles the maximum representable value. The formula for n bits is: Maximum value = 2ⁿ - 1

Bits	Maximum Value (2ⁿ - 1)	Example Binary
1	1	1
2	3	11
3	7	111
4	15	1111
5	31	11111

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Practical Applications

Networking: IPv4 Addresses

• IPv4 uses 32 bits, enabling 4,294,967,296 unique addresses (2³²).
• Example: 192.168.1.1 is a 32-bit binary number split into four 8-bit segments (octets).

Memory and Storage

• 1 byte = 8 bits (e.g., 01001101).
• 1 kilobyte (KB) = 1024 bytes (2¹⁰ bytes).

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Common Mistakes to Avoid

• Misreading binary positions (e.g., 101 is 5, not 101).
• Ignoring leading zeros (e.g., 0010 = 10 = 2).
• Confusing bits (single 0/1) with bytes (8 bits).
• Assuming binary counting is "different"—it follows the same logic as decimal, just with a smaller base.

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Decimal vs. Binary Reference Table

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
16	10000

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Learn More

• Hexadecimal (base-16): A compact representation of binary (e.g., 1A = 00011010).
• Signed binary numbers: Two’s complement for negative values.
• Floating-point binary: How decimal fractions (e.g., 0.1) are stored in binary.

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Summary

• Binary is a base-2 system using 0 and 1 to represent all values.
• Each bit corresponds to a power of 2, and adding bits doubles the maximum value.
• Convert decimal to binary via division/remainders and binary to decimal via positional values.
• Binary underpins computing, networking, and digital logic (e.g., IPv4 addresses, memory storage).