Computer Number Systems

Number Representation in Computers

Decimal System vs. Binary System

• Decimal System:
  • Uses base 10.
  • Columns represent powers of 10 (e.g., ones, tens, hundreds).
  • Ten possible digits (0-9) for each placeholder.
• Binary System:
  • Used by computers.
  • Uses base 2.
  • Based on transistors, which can be either off (0) or on (1).
  • Columns represent powers of 2 (e.g., ones, twos, fours).

Place Values in Binary

• Decimal system columns represent powers of 10: $10^0, 10^1, 10^2$, etc.
• Binary system columns represent powers of 2: $2^0, 2^1, 2^2$, etc.
• Example:
  • $2^0 = 1$ (ones column)
  • $2^1 = 2$ (twos column)
  • $2^2 = 4$ (fours column)
  • $2^3 = 8$ (eights column)
  • $2^4 = 16$ (sixteens column)
  • $2^5 = 32$ (thirty-twos column)
  • $2^6 = 64$ (sixty-fours column)
  • $2^7 = 128$ (one-hundred-twenty-eights column)

Converting Binary to Decimal

• Multiply the digit in each binary column by its corresponding power of 2.
• Add the results to get the decimal equivalent.
• Example: Binary 000
  • $(4 \times 0) + (2 \times 0) + (1 \times 0) = 0$
• Example: Binary 001
  • $(4 \times 0) + (2 \times 0) + (1 \times 1) = 1$
• Example: Binary 010
  • $(4 \times 0) + (2 \times 1) + (1 \times 0) = 2$
• Example: Binary 011
  • $(4 \times 0) + (2 \times 1) + (1 \times 1) = 3$
• Example: Binary 100
  • $(4 \times 1) + (2 \times 0) + (1 \times 0) = 4$
• Example: Binary 101
  • $(4 \times 1) + (2 \times 0) + (1 \times 1) = 5$
• Example: Binary 110
  • $(4 \times 1) + (2 \times 1) + (1 \times 0) = 6$
• Example: Binary 111
  • $(4 \times 1) + (2 \times 1) + (1 \times 1) = 7$

Adding More Bits

• To count higher, add more bits (transistors).
• Each bit doubles the counting capacity.
• Example: Four bits
  • 1000 in binary represents 8 in decimal ( $(8 \times 1) + (4 \times 0) + (2 \times 0) + (1 \times 0) = 8$ ).

Bytes

• A byte is a unit of 8 bits.
• More practical than using only a few bits.
• A byte can represent numbers from 0 to 255.

Byte Representation Example

• Eight bits (one byte): 00000000 (decimal 0)

Maximum Value in a Byte

• With eight bits, the highest number you can count to is 255.
• All bits set to one: 11111111
• $(128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255)$
• Including zero, there are 256 total possibilities (0-255).

Significance of Powers of Two

• Numbers like 255 and 256 are common in computer systems because they relate to the units of measure (bytes) that systems use.