Computer Number Formats

Endianness §

• Endianness pertains to the manner in which we store a sequence of bytes in digital data in memory .
• A Big Endian system stores the most significant byte at the smallest address. The memory layout ends up storing the number backwards
• A Little Endian system stores the least significant byte at the smallest memory address. The memory layout ends up storing the number in reading order.

Number Representations §

NaN §

• A Not a Number results denotes an entity which cannot be particular format (i.e., because of indeterminate forms)
  • A signaling NaN is a special form of NaN which should raise an invalid operation exception when consumed by most operations.
  • A quiet NaN is a NaN which does not raise any additional exceptions as it propagates through must operations.

Extensions §

• Extension pertains to the process of representing a number that was in bits to another number in bits, where .

  We have two possible definitions of extensions

  • Signed Extension involves repeatedly adding leading ’s if the number is positive and leading ’s if the number is negative.
  • Zero extension involves adding leading s.

Coded Decimals §

• Unpacked Binary Coded Decimal encodes the number by representing the digit with its equivalent -bit number in binary.

• Packed BCD removes the first four MSb.

  Each digit’s representation is shown in the table below

  digit	UBCD	Packed BCD
  0	0000 0000	0000
  1	0000 0001	0001
  2	0000 0010	0010
  3	0000 0011	0011
  4	0000 0100	0100
  5	0000 0101	0101
  6	0000 0110	0110
  7	0000 0111	0111
  8	0000 1000	1000
  9	0000 1001	1001

Floating Point Standard §

• Defines the standard for representing floating point numbers.

• Let be the sign bit be the stored exponent be the mantissa. be the number of bits to be used - the number of bits for the exponent the number of bits for the mantissa.

• The format is given as

 1   |    b_e   |   b_m
sign | exponent | mantissa

• The real number is to be represented as

  must be normalized to ensure canonical representation. Thus, we have an implied

• The sign bit represents the sign of . is positive, is negative

• The stored exponent is calculated as

• The mantissa represents the remaining bits and store the significand .

Special Values §

• Note that and denote numbers arbitrarily close to but not .

Infinity §

• (all one’s)

• Infinity denotes the largest magnitude number representable within the format

Subnormal Numbers §

These numbers are of the form , where in binary

Not a Number §

• is any value.
• (all one’s)
• begins with for Signaling NaN
• begins with for Quiet NaN

Links §

How floating point works