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KembaraXtra-Computer Science-Number Systems
1. Introduction
1. Introduction
- Computers are digital machines that operate using 0s and 1s.
- Understanding number systems is essential to understanding how computers work with numbers.
- The number system humans commonly use.
- Uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
- Employs place-value notation
- Each position in a number represents a different power of 10 (base 10).
- Example: 275
- 5 is in the ones place (10⁰ = 1), so its value is 5 * 1 = 5.
- 7 is in the tens place (10¹ = 10), so its value is 7 * 10 = 70.
- 2 is in the hundreds place (10² = 100), so its value is 2 * 100 = 200.
- Total value: 5 + 70 + 200 = 275.
- Each place is a power of 10.
- Rightmost place: 10⁰ = 1 (ones place)
- Next place: 10¹ = 10 (tens place)
- Next place: 10² = 100 (hundreds place)
- And so on (10³ = 1000, etc.)
- Decimal uses ten symbols (0-9) because each place can represent ten different values.
- Any ten unique symbols could be used, but 0-9 is the convention.
- The number system used by computers (two symbols)
- Uses only two symbols: 0 and 1.
- Still a place-value system, but each place represents a power of 2.
- Each position is a power of 2 (base 2).
- Example: 101 (binary) which equals 5 (decimal)
- Rightmost place: 2⁰ = 1
- Next place: 2¹ = 2
- Next place: 2² = 4
- Total value: (1 * 1) + (0 * 2) + (1 * 4) = 1 + 0 + 4 = 5
- Multiply the symbol in each place by the place-value weight (power of 2) and sum the results.
- Find the largest power of 2 that is less than or equal to the decimal number.
- Subtract that power of 2 from the decimal number and repeat the process with the remainder.
- Put a 1 in the places corresponding to the powers of 2 that were used and a 0 in the other places.
- To avoid confusion, binary numbers are often written with a 0b prefix (e.g., 0b10 represents 2 in binary).
- Numbers without a prefix are assumed to be decimal.
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