WEEK 2 – Video Notes
UNSIGNED BINARY
Cannot represent negative values
POSITIONAL NUMBERING SYSTEMS
The decimal system is an example of so
Called as a base-ten number system (1-10)
Digits represent different values depending on position (34 or 43)
Binary is another example
Used in basically all computer systems
Base-two number system (1 and 0 - bits)
The orders and arrangement represent a different value
Radix notation
1001010 is a decimal number
100102 is a binary number
100105 is a base-five number
Prefix notation
For binary numbers the prefix is 0b (0b01010 should be interpreted as binary)
Grouping
In decimals you group numbers in three digits (100.040.039)
Bits are grouped in 8 digits (bytes) or 4 digits (nybble)
BASE CONVERSTIONS
Subtraction approach
Division remainder approach
FRACTIONAL COMPONENT
Numbers can have one in any base
51.1 (decimal point); 1011.1101 (binary point)
In order to convert it is the same process as normal numbers but using negative exponents
Some values might have an infinite series of digits in other bases (e.g. 2/310 = 0.666…10 = 0.23)
When not possible to represent a fractional value in a specific base you have to specify the
digits of accuracy (how many decimal points)
e.g. 0.310 in binary is 0.01100 to five bits of accuracy to the right of the binary point
SIGNED-MAGNITUDE BINARY
the most significant bit of a number is going to represent the sign of the number
0 represents positive and 1 represent negative
e.g. 011012 = 1310 and 110112 = -1110
+ simple and intuitive
+ easy to convert from unsigned values
- difficult to perform arithmetic (as the sign bit has no value it needs special logic to deal with it)
- there are two representations of 0 (makes no sense mathematically)
0000 0000 = positive zero byte; 1000 0000 = negative zero byte
ONE’S COMPLEMENT BINARY
The complement of 0 is 1 and vice versa (0 -> 1; 1 -> 0)
So the complement of a number becomes the negative representation
Similarly to unsigned, when the value starts with a 1 is negative and with 0 is positive
When it’s a positive value the number is the same as if it was unsigned
+ addition is easier than in signed-magnitude (easier to perform arithmetic)
- still two representations of 0
0000 0000 positive zero byte; 1111 1111 negative zero bytes
TWO’S COMPLEMENT BINARY
Most significant bit represents a negative value
The first number value holds but it turns negative and the following numbers add up to it
To convert positive numbers is the same as unsigned but adding a 0 to the front
For negative numbers, simply convert it to one’s complement and then add ‘1’ to the value
+ no bits with special meaning, therefore simple circuits
Simpler to perform arithmetic compared to any other ones of the representations
+ single representation of 0 (0000 0000)
- no disadvantages :)
HEXADECIAMAL
A form to represent binary numbers which is simpler for humans to read
Base-16, meaning it needs 16 digits: