Data Rep & Number Systems

Question 1

1 byte = ? bits

Answer 1

8 bits

Question 2

N bits can represent up to how many values?

Answer 2

2^N values

Question 3

Unsigned vs Signed

Answer 3

Unsigned = only non-negative values
Signed = all values (positive & negative)

Question 4

What are the 3 common representations for signed binary numbers

Answer 4

1. Sign-and-magnitude
2. 1s Complement
3. 2s Complement

Question 5

What is the range for an 8-bit for the 3 common representations?

Answer 5

Sign-and-Magnitude:
-127 to +127, +0 and -0

1s complement:
same as ^ (-(2^(n-1) - 1) to 2^(n-1)-1)

2s complement:
0 is associated with the positive side, which means positive side get to represent 1 less.
-(2^(n - 1)) to 2^(n-1) - 1

Question 6

What are the shortcuts for deriving the complements?

Answer 6

1s complement:
Invert all the bits

2s complement:
Invert all the bits then add 1

Question 7

What is the algorithm for addition/subtraction using both complement systems?

Answer 7

• Perform addition for both
• Check for overflow:
  1. + and + = -
  2. - and - = +

1s complement:
Add back the carry out of the MSB

2s complement:
Ignore the carry out of the MSB

Question 8

What is the purpose of excess representation?

Answer 8

Allows the range of values to be distributed evenly between the positive and negative values.

It shifts all the values by Excess-N amount.
The MSB denotes whether is it negative/positive:
0 -> negative
1 -> positive

Usually N is determined by the taking 2^(number of bits - 1) or 2^(number of bits - 2).

Question 9

What are the different components in an IEEE 754 Floating-Point rep?

Answer 9

• Convert the decimal into binary
• Normalise the binary

1. Sign:
   0 for positive
   1 for negative
2. Exponent:
   Uses excess-127 representation
   127 + the power derived from normalising, then convert to binary
3. Mantissa
   Normalised with an implicit leading bit 1
   Copy over the values in the fraction portion