Binary Numbers and Decimal-Binary Conversion

Binary numbers are fundamental to digital systems, including computers and networking. Unlike the decimal system (base-10) we use daily, binary (base-2) uses only two digits—0 and 1—to represent all values. Understanding binary counting and conversion between these systems is crucial for grasping how computers process and store data.

Key Points

• Binary system (base-2): Uses only 0 and 1 to represent numbers.
• Decimal system (base-10): Uses digits 0 to 9, with each position representing 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., in binary, 101 = 5, not 101).
• Powers of two: Each binary position corresponds to a power of 2 (1, 2, 4, 8, 16, etc.).

How Binary Counting Works

Binary counting follows the same logic as decimal counting but with a key difference:

• In decimal, you reset to 0 and carry over 1 when reaching 9.
• In binary, you reset to 0 and carry over 1 when reaching 1.

Example:

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

Key Insight: Each time a binary number reaches all 1s (e.g., 111), adding 1 requires a new bit (e.g., 1000).

Conversion Methods

Decimal to Binary

To convert a decimal number to binary:

1. Divide the number by 2 and record the remainder.
2. Repeat the division with the quotient until it reaches 0.
3. The binary number is the remainders read bottom to top.

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

Reading remainders from bottom to top: 1101₂.

Binary to Decimal

To convert binary to decimal:

1. Identify the positions of 1 bits (starting from the right, 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

Why Bits Expand

Binary numbers grow by adding bits when the current bit limit is exceeded. Each new bit doubles the maximum representable value.

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

Formula: Maximum value = 2ⁿ - 1, where n is the number of bits.

Practical Applications

Networking: IPv4 Addresses

• An IPv4 address uses 32 bits, allowing for 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).

Common Mistakes to Avoid

• Misreading binary positions: 101 is 5, not 101.
• Assuming binary is "different": The counting logic is identical to decimal, just with a smaller base.
• Ignoring leading zeros: 0010 is the same as 10 (both = 2).
• Confusing bits and bytes: A bit is a single 0/1; a byte is 8 bits.

Visual Reference: Decimal vs. Binary

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

Learn More

• Hexadecimal (base-16): A compact way to represent binary (e.g., 1A = 00011010).
• Signed binary numbers: How computers represent negative numbers (e.g., two’s complement).
• Floating-point binary: How decimal fractions are stored in binary (e.g., 0.1 in binary is infinite).

Summary

• Binary is a base-2 system using only 0 and 1.
• Each bit represents a power of 2, and adding bits doubles the maximum value.
• Conversion between decimal and binary relies on division/remainders (decimal → binary) or positional values (binary → decimal).
• Binary is the backbone of computing, networking, and digital logic.