Basics of Computers - Number System

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The Number System in computers is a mathematical system used to represent data and perform calculations. Computers primarily use binary numbers, but other number systems like decimal, octal, and hexadecimal are also important in computing.

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Types of Number Systems

1. Decimal Number System (Base-10):

   • Most commonly used by humans.
   • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
   • Each position represents a power of 10.
   • Example: $452$ = $(4 \times 10^2) + (5 \times 10^1) + (2 \times 10^0)$

2. Binary Number System (Base-2):

   • Used by computers to represent data and perform calculations.
   • Digits: 0, 1
   • Each position represents a power of 2.
   • Example: $1011$ = $(1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 11$ (Decimal).

3. Octal Number System (Base-8):

   • Used in computing as a shorthand representation of binary numbers.
   • Digits: 0, 1, 2, 3, 4, 5, 6, 7
   • Each position represents a power of 8.
   • Example: $57_8$ = $(5 \times 8^1) + (7 \times 8^0) = 47$ (Decimal).

4. Hexadecimal Number System (Base-16):

   • Commonly used in programming and digital electronics.
   • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (A-F represent 10-15 in decimal).
   • Each position represents a power of 16.
   • Example: $1A3_{16}$ = $(1 \times 16^2) + (10 \times 16^1) + (3 \times 16^0) = 419$ (Decimal).

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Conversions Between Number Systems

1. Binary to Decimal:

• Multiply each binary digit by $2^{\text{position}}$ and sum the results.
• Example: $1010_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 10_{10}$.

2. Decimal to Binary:

• Divide the decimal number by 2 and record the remainders.
• Example: $10_{10}$:
  $10 \div 2 = 5$ R $0$
  $5 \div 2 = 2$ R $1$
  $2 \div 2 = 1$ R $0$
  $1 \div 2 = 0$ R $1$
  Result: $1010_2$.

3. Binary to Hexadecimal:

• Group binary digits in sets of 4 (from right to left) and convert each group to its hexadecimal equivalent.
• Example: $10101111_2 = 1010 \, 1111 = A \, F = AF_{16}$.

4. Hexadecimal to Binary:

• Convert each hexadecimal digit to its 4-bit binary equivalent.
• Example: $AF_{16} = A(1010) \, F(1111) = 10101111_2$.

5. Decimal to Hexadecimal:

• Divide the decimal number by 16 and record the remainders.
• Example: $419_{10}$:
  $419 \div 16 = 26$ R $3$
  $26 \div 16 = 1$ R $10 (A)$
  $1 \div 16 = 0$ R $1$
  Result: $1A3_{16}$.

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Importance of Number Systems in Computers

1. Binary System:
   • Fundamental for computers since they operate using electrical signals (on/off, 1/0).
2. Hexadecimal System:
   • Simplifies binary representation, especially in programming and memory addresses.
3. Octal System:
   • Used less frequently now but historically used in older computing systems.

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Would you like detailed practice problems, or assistance with specific conversions?

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