Unit 4
e.g. denary 751
|
binary |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
|
hex |
2 |
E |
F |
|
oct |
1 |
3 |
5 |
7 |
Two's complement
Copy digits from the LSB towards the MSB, up to and including the first "1" digit and then invert the rest.
|
5 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
|
-5 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
|
14 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
|
-14 |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
OR add one to the one's complement
|
e.g. |
5 |
= |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
|
1 |
1 |
1 |
1 |
1 |
0 |
1 |
0 |
||
|
+ |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
|
|
-5 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
Fixed point notation
e.g. (taking radix point between 4th and 5th bits)
|
5CHEX |
= |
0 |
1 |
0 |
1 |
|
1 |
1 |
0 |
0 |
|
= |
+ |
5 ¾ |
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Floating point
|
MANTISSA |
EXPONENT |
||||||
|
-1 |
1 /2 |
1 /4 |
1 /8 |
1 /16 |
-4 |
2 |
1 |
|
^ |
|||||||
e.g.
|
73HEX |
= |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
1 |
|
^ |
|||||||||
|
= |
7 /8 x 23 |
= |
7 |
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Normalisation
In normalised form the first two bits are distinct
Overflow - when a number is too large for the exponent to be expressed (positive exponent)
Underflow - when a number is too near zero for the exponent to be expressed (negative exponent)
|
e.g. normalise: |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
|||
|
= |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
|
Exponent = 2 |
|||||||
|
^ |
|||||||||||||||||||
|
= |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
|
Exponent = 2 - 3 |
|||||||
|
^ |
|||||||||||||||||||
|
= |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
|||
|
^ |
|||||||||||||||||||
Adding two normalised floating point numbers
|
e.g. |
2.6 x 102 |
+ |
8.9 x 103 |
|
= |
2.6 x 102 |
+ |
89 x 102 |
|
= |
91.6 x 102 |
||
|
= |
9.16 x 103 |
||
Multiplying two normalised floating point numbers
|
e.g. |
2.6 x 102 |
x |
8.9 x 103 |
|
= |
23.14 x 105 |
||
|
= |
2.314 x 106 |
||