Practice floating point problems

Format for toy computer: 8 bit storage:
   S - 1 bit - sign: 0 for positive, 1 for negative (bit 7)
   E - 4 bits - bias 3  (AKA "excess 3")  (bits 6 - 3)
   m - 3 bits, implied 1  (bits 2 - 0)
   
   7       6   5   4   3       2   1   0   
|------|-------------------|---------------|
|  S   |   exponent        |   mantissa    |
|------|-------------------|---------------|
   
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(1) What number does the pattern 1100 1100 (AKA 0xCC) represent?

S = 1 --> -

exponent = 1001 = 9
  9 - 3 = 6
        ---> 6
        
mantissa = 100
  fraction = .1100

So the base two number being represented is:

- 0.1100 * 2 ^ 6

= - 110000

= - (1*2^5 + 1*2^4) = -(32+16) = - 48

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(2) What number does the pattern 0000 1001  (aka 0x09)

S = 0  --> +

exponent = 0001 = 1
  1 - 3 = -2
  
mantissa = 001
  fraction = .1001
  
So, the base two number being represented is:

+ 0.1001 * 2^(-2)


Convert to base 10:

0.1001 * 2^(-2) = 1001 * 2^(-6) = 9 * 1/(2^6) = 9/64 = 0.0.140625

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(3) What is the representation of  1 1/3 = 1.33..

Only 4 bits worth of fraction can be stored:
(1.3333)b10 ~= (1.0101011)b2  ~= (1.010)b2

Normalize: 1.010 = 0.1010 * 2^(1)

Sign = + ==> 0

exponent = 1; add bias 1 + 3 --> 4

fraction = .1010  --> mantissa =  010

Pattern:
  0 0100 010
  
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(4) What is actually begin represented by 0010 0010? (i.e., the answer to (3) )

S = 0 --> +

exp = 0100 = 4; remove bias 4 - 3 = 1.

mantissa = 010; Fraction = .1010

The base two number is + .1010 * 2^1  = 
    (1*2^(-1)+ 1*2^(-3)) *2^1 = (1*2^0) + 1*2^(-2) = 1 + .25 = 1.25