Exercise 1
==========

#. An :math:`\mathsf{N}`-bit binary register can hold
   :math:`\mathsf{2^N}` unique bit patterns. To represent an unsigned
   integer, the bit patterns are interpreted as the base-2 numbers 0 to
   :math:`\mathsf{2^N-1}`. There are several possible ways to represent
   signed integers. Nonetheless, on all modern computers this is
   accomplished using the *two's complement* scheme (we'll
   denote it as base :math:`\mathsf{\bar{2}}`), in which the the numbers
   :math:`\mathsf{-2^{N-1}}` to :math:`\mathsf{+2^{N-1}-1}` are
   represented by the same bit patterns reinterpreted as
   :math:`\mathsf{0}` to :math:`\mathsf{2^{N-1}-1}` and then
   :math:`\mathsf{-2^{N-1}}` to :math:`\mathsf{-1}` in lexical order.

   The table below illustrates the unsigned (left) and signed (right)
   values for :math:`\mathsf{N=8}`:

   .. list-table:: 

      * - 127
        - 0 1 1 1 1 1 1 1 
        - 127
      * - 126
        - 0 1 1 1 1 1 1 0 
        - 126
      * - 2
        - 0 0 0 0 0 0 1 0 
        - 2
      * - 1
        - 0 0 0 0 0 0 0 1
        - 1
      * - 0
        - 0 0 0 0 0 0 0 0 
        - 0
      * - 255
        - 1 1 1 1 1 1 1 1 
        - −1 
      * - 254 
        - 1 1 1 1 1 1 1 0 
        - −2 
      * - 253 
        - 1 1 1 1 1 1 0 1 
        - −3 
      * - 129 
        - 1 0 0 0 0 0 0 1 
        - −127
      * - 128
        - 1 0 0 0 0 0 0 0
        - −128

   a. The advantage of two's complement numbers is that they can be
      added, in the mechanistic sense, exactly the same way as 
      conventional binary numbers:

      .. math::

         \begin{aligned}
         &\texttt{\phantom{0}\phantom{0}{\color{red}1}{\color{red}1}\phantom{0}\phantom{0}{\color{red}1}} \\  
         &\texttt{01011001} \\ 
         +&\texttt{10011101} \\ \hline 
         &\texttt{11110110}
         \end{aligned}

      Here, red ones represent the carry bits.  

      Explain the two ways the computation above can be interpreted---as
      either unsigned binary or two's complement.

   b. Determine the 8-bit unsigned binary representation of 116.  Does
      the two's complement representation of that number have the same bit
      pattern?

   c. In the :math:`\mathsf{N}`-bit two's complement scheme, negation
      (i.e., mapping a number :math:`x \to -x`) is accomplished by all
      flipping all :math:`N` bits and then adding one (modulo :math:`2^N`,
      of course).  Negate the 8-bit two's complement number :math:`116`
      from part b.

   d. One advantage of two's complement is that it encodes a unique zero
      (rather than both :math:`0` and :math:`-0`). But there is one nonzero
      value that also behaves as :math:`x \to x` under negation.  What is
      it?

   e. There are four possible values for the two leftmost carry
      bits (marked with dots): 00, 01, 10, 11. Which combinations
      denote an overflow for unsigned binary and which for signed
      (two's complement) binary?

#. Change the octal numbers :math:`\mathsf{60177756_8}`,
   :math:`\mathsf{3006012_8}`, :math:`\mathsf{3726755_8}`,
   :math:`\mathsf{76545336_8}`, :math:`\mathsf{5655_8}`, and
   :math:`\mathsf{67545336_8}` to hexadecimal notation using the digits
   :math:`\mathsf{0,1,\ldots,F}`. *Hint:* Doing this by hand is easiest
   if you use the base-2 bit patterns as an intermediate step.

#. We can generalize a positional number system to negative bases by
   writing

   .. math::

      \mathsf{(\cdots a_3a_2a_1a_0.a_{-1}a_{-2}\cdots)_{b} =
      \cdots + a_2b^2 + a_1 b + a_0 + a_{-1}b^{-1} + \cdots}

   in the usual way but requiring that the digits be in the range 
   :math:`0 \le a_k < | b |`.  For example, 

   .. math::

      \begin{aligned}
      \mathsf{1322_{-4}} &= \mathsf{-22}\\
      \mathsf{133.21_{-5}} &= \mathsf{12.64}\\
      \mathsf{2050.23_{-6}} &= \mathsf{-462.25}\\
      \mathsf{6231.46_{-8}} &= \mathsf{-2967.40625}
      \end{aligned}

   a. Negate each of the identities above.

   b. Is there a unique negative base representation for each number?

   c. Is it ever necessary to have a leading minus sign, 
      e.g., :math:`\mathsf{-54_{-3}}`, when the base is negative?

#. Consider a 6-digit *factoradic* system

   .. math::
 
         \mathsf{(a_6 \ldots a_3a_2a_1) = a_6\cdot 6! + \cdots + a_3\cdot 3!
         + a_2 \cdot 2! + a_1, \quad \textsf{where}\ 0 \le a_k \le k}

   a. What are the largest and smallest numbers representable? 

   b. Is every number in between representable?