Data Representation as Bits

• lecture

#lecture note based on 15-213 Introduction to Computer Systems

Either 0 or 1? That’s abstraction of electronic implementation. Votage isn’t discrete.

Representations - Bits are just bits, but we can represent int, long, float…

Boolean Algebra -> In 2’s complement, flip bits and plus one flips sign

Addition for signed and unsigned done the same way on hardware. Only interpreted differently.

Typical size (in byte) for C data types

• 0 is false, everything else is true.
• All operators return 0 or 1
• Has early termination
  • 0x69 || 0x55 => 0x01
  • p && *p possible, but not recommended

• << always add 0 to right
• >>
  • unsigned - logical - add 0 to left
  • 2’s complement - arithmatic - extend left-most bit
• Shift $< 0$ or $\geq \text{size}$ are undefined behaviour

• Between signed and unsigned: maintain bit pattern and reinterpret
• From shorter to longer signed: make enough copies of sign bit (i.e. left most bit)
• From shorter to longer unsigned: add 0s to left
• From longer to shorter signed/unsigned: truncate top bits

• -x == ~x + 1
• constant integer like 42 defaults to signed. to get unsigned do 42U
• if signed and unsigned in same expression, signed implicitely cast to unsigned
• !!x != x, ~~x == x
• x << k = $2^k x$, both signed and unsigned
• When doing comparison ops on mix of signed and unsigned, things implicitely cast to unsigned

• Addition - carry bit stored somewhere on hardware but discarded
• Multiplication - sometimes we need $2w$ bits for muitiplying $w$ bits integers. The extra bits are discarded