Table of Contents

Description

• Bits are base-2 numbers.
  • A single bit can either be 0 or 1.
• A binary number consists of several bits. E.g. 010101011.
• Bit manipulation involves processing a binary number and performing some operation on its individual bits, then outputting a new binary number.

Bits and Bytes

Bits and Space

• A 32-bit number has $2^{32}$ possible values.
  • An unsigned 32-bit number ranges from 0 to $2^{32}-1$
  • A signed 32-bit number ranges from $-2^{31}$ to $2^{31}-1$ (even split).
  • Storing every possible 32-bit number takes $2^{32}$ bits.
• Increasing bit size.
  • Adding 1 bit (digit) doubles the space usage and number of possible binary numbers.
    • $2^{30}$ is 1Gbits. $2^{31}$ is 2Gbits.
  • Multiplying bits by 10 is roughly the same as multiplying space by 1000.
    • $2^1$ ≈ 1bit
    • $2^{10}$ ≈ 1Kbits.
    • $2^{20}$ ≈ 1Mbits
• Bytes
  • 1 (20) byte = 8 (23) bits
    • To convert from bits to bytes, lower the exponent by 3.
  • Size properties are the same.
    • $2^{30}$ Bytes = 1GB.
    • $2^{31}$ = 2GB
    • $2^{40}$ = 1TB

Interpreting Binary Numbers

• A decimal number can be seen as a sum of powers of 10.
  • $956 = (9)10^2 + (5)10^1 + (6)10^0$
    • $= 900 + 50 + 6$
• A binary number can be seen as a sum of powers of 2.
  • $1010 = (1)2^3 + (0)2^2 + (1)2^1 + (0)2^0$
• Shortcut: 1111 is $2^4 - 1 = 15$

Negative Binary Numbers

• Usually, the leftmost bit in a binary number (the most significant bit) is the signed bit.
  • The signed bit represents the sign of the number.
  • If it’s 1, the number is negative. If it’s 0, it’s positive.
• $1010_2$
  • Negative number
  • Magnitude = $010_2$
• $0010_2$
  • Positive number.
  • Same magnitude, $010_2$

Converting Binary-Decimal