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Binary and Number Systems
Binary, hex, octal, and how computers represent numbers
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Computers store and process information using number systems built from symbols and place value. Binary matters because digital circuits naturally represent two stable states, usually called 0 and 1. Understanding binary, decimal, octal, and hexadecimal helps students see how data, memory addresses, colors, instructions, and files are represented inside a computer.
Each number system uses a base, which tells how many digits are available and what each position is worth. In binary, place values are powers of 2, while decimal uses powers of 10, octal uses powers of 8, and hexadecimal uses powers of 16. Conversions work by expanding a number into place values or by grouping binary digits into sets of 3 for octal and sets of 4 for hexadecimal.
Key Facts
- Binary uses base 2, so its digits are only 0 and 1.
- Decimal uses base 10, so place values are 1, 10, 100, 1000, and so on.
- A binary number converts to decimal by adding powers of 2: 1011₂ = 1⋅8 + 0⋅4 + 1⋅2 + 1⋅1 = 11₁₀.
- In any base b, place values are powers of b: b^0, b^1, b^2, b^3, and so on.
- One hexadecimal digit equals 4 binary bits: 1111₂ = F₁₆.
- One octal digit equals 3 binary bits: 101₂ = 5₈.
Vocabulary
- Bit
- A bit is the smallest unit of digital information and can have the value 0 or 1.
- Base
- The base of a number system is the number of different digit symbols it uses.
- Place value
- Place value is the value a digit has because of its position in a number.
- Hexadecimal
- Hexadecimal is a base 16 number system that uses digits 0 through 9 and letters A through F.
- Octal
- Octal is a base 8 number system that uses digits 0 through 7.
Common Mistakes to Avoid
- Treating binary numbers like decimal numbers is wrong because 101₂ means 5₁₀, not one hundred one.
- Forgetting the rightmost place is 2^0 is wrong because the first place value is 1, not 2.
- Using invalid digits in a base is wrong because each base has a limited digit set, such as no digit 8 in octal and no digit 2 in binary.
- Grouping binary digits from the left when converting to hex or octal is wrong because groups should start from the right, with leading zeros added on the left if needed.
Practice Questions
- 1 Convert 110101₂ to decimal.
- 2 Convert 3A₁₆ to decimal, then write the same value in binary.
- 3 Explain why hexadecimal is more convenient than binary for writing long memory addresses.