Base 2 or Binary

• Why do you suppose you know base 10?
• We use base 2, or binary in computing.
  • Probably because it is easy to tell if there is a positive charge or not.
  • Or if a switch is on or off.
  • As we will see, this is not the only way we could do it.
  • But it is the way we do it.
• Comparison with base 10
  • In base 10 there are 10 digits, {0,1,2,3,4,5,6,7,8,9}
  • In base 2 there are 2 digits {0,1}
  • In base 10, the radix is 10
  • In base 2, the radix is 2
  • In base 10 a number is : $d_n10^n + d_{n-1}10^{n-1} + ... + d_110+d_0$, where $d_i \in \{0,1,2,3,4,5,6,7,8,9\}$
  • In base 2 a number is : $d_n2^n + d_{n-1}2^{n-1} + ... + d_12+d_0$, where $d_i \in \{0,1\}$
• We subscript binary numbers with a 2, to indicate binary.
  • $10110_2$ is a binary number
  • 10110 is a decimal number
  • In general we do not subscript base 10 numbers.
  • But if we want $10110_{10}$ is what we would write.
• Some Terms
  • A single binary digit is called a bit.
  • A collection of 8 bits is called a byte.
  • The first bit is the most significant bit (MSB).
  • The last bit is the least significant bit (LSB).
  • The term word is hardware dependent.
    • It means the "size of data" in a computer.
• Some other terms, less frequently used
  • a nibble is four bits.
• Multiple words, bits and bytes
  • Word:
    • A half word is half of a word
    • A double word is two words
    • A quad word is four words.
    • We know these terms from programming (double, quad, ...)
  • Bits and bytes use the metric prefixes

    • value	Metric
      $10^3$	kilo (k)
      $10^6$	mega (M)
      $10^9$	giga (G)
      $10^{12}$	tera (T)
      $10^{15}$	peta (P)
      $10^{18}$	exa (E)
      $10^{21}$	zetta (Z)
      $10^{24}$	yotta (Y)
      $10^{27}$	ronna (R)
      $10^{30}$	quetta (Q)

    • In metric we use the term bit/byte so
      • 1kb (b is for bit) is 1,000 bits.
      • 1kB (B is for byte) is 1,000 bytes or 8,000 bits.
      • 1 Yb is 1,000,000,000,000,000,000,000,000 bits, and that is a lotta bits!
    • A second system by the IEC defines prefixes in powers of 2.

      value	Metric
      $2^{10} = 1024^1$	kibi (k)
      $2^{20} = 1024^2$	mibi (M)
      $2^{30} = 1024^3$	gibi (G)
      $2^{40} = 1024^4$	tebi (T)
      $2^{50} = 1024^5$	pebi (P)
      $2^{60} = 1024^6$	exbi (E)
      $2^{70} = 1024^7$	zebi (Z)
      $2^{80} = 1024^8$	yobi (Y)

    • So in this system
      • 5 kibibytes (KiB) = 5 x 2^10 bytes.
    • I am fairly certain that this system was introduced due to a lawsuit where advertisers misused the term megabyte
      • And I really don't care
      • But I think you should know up to yotta.
      • Because look here.
• What types of devices are measured in bits, bytes, and the other units?