Binary and Computer Systems

1. Bits, bytes, and digital states

A bit is a binary digit: 0 or 1. A byte is commonly 8 bits, allowing 2⁸ = 256 possible bit patterns. A pattern does not carry meaning by itself; a format determines whether it represents a number, a character, a color component, or something else.

Digital circuits use physical signals with defined ranges interpreted as logical 0 or 1. Real systems must tolerate noise and timing limits; binary representation is an abstraction that makes reliable processing practical.

2. Positional notation

In base 10, each digit's place represents a power of 10. In base 2, each place represents a power of 2. From right to left, binary place values are 1, 2, 4, 8, 16, and so on.

Binary 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀. The subscript indicates the base and avoids ambiguity.

3. Convert decimal to binary

Repeatedly divide the positive decimal number by 2 and record each remainder. Read the remainders from bottom to top. Example: 13 ÷ 2 gives remainder 1; 6 ÷ 2 gives 0; 3 ÷ 2 gives 1; 1 ÷ 2 gives 1. Reading upward gives 1101₂.

You can check the result by expanding it: 1101₂ = 8 + 4 + 0 + 1 = 13₁₀. For zero, the representation is 0₂.

4. Binary addition

The basic rules are 0+0=0, 0+1=1, 1+0=1, and 1+1=10₂ (write 0 and carry 1). With a carry, 1+1+1=11₂ (write 1 and carry 1).

Example: 101₂ + 011₂ = 1000₂. In decimal, 5 + 3 = 8, so the result is consistent. Always account for carries from the previous column.

5. Where representations are used

Unsigned n-bit binary can represent integers from 0 through 2ⁿ−1. Four bits represent 16 patterns, from 0000 to 1111. Signed integers require a convention, such as two's complement; floating-point numbers use a different encoding. Text uses character encodings, and images use pixel/color formats.

Do not assume every bit pattern is an ordinary positive integer. The same eight bits can have different meanings under different data types and encodings.

6. Common mistakes

• Reading binary digits as if each position were a power of 10.\n• Reading division remainders from top to bottom instead of bottom to top.\n• Forgetting the carry in binary addition.\n• Assuming 1 byte always means one character in every encoding.\n• Confusing the bit pattern with the format that interprets it.

7. Practice exercises

Try each question before revealing the worked answer.

8. Quick knowledge check

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