Number Systems
Today
• Decimal
• Hexadecimal
• Binary
– Unsigned Binary
– 1’s Complement Binary
– 2’s Complement Binary
Decimal
(base 10)
( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 )
Binary
(base 2)
( 0, 1 )
Hexadecimal
(base 16)
( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F )
How do we represent numbers?
Decimal:
Binary:
100
101
102
103
104
105
20
21
22
23
24
25
26
27
28
29
=1
= 10
= 100
= 1000
= 10000
= 100000
.
.
.
.
=
=
=
=
=
=
=
=
=
=
210
220
230
240
1
2
4
8
16
32
64
128
256
512
=
=
=
=
1024 = 1 Kb
1 Mb
1 Gb
1 Tb
Example:
Decimal 1024 = (1024)10
Notes: Counting Chart
( 99 )10 + 1 = ??
( FF )16 + 1 = ??
( 100 )10
( 100 )16
Conversion
• Decimal (Base 10) --> Binary (Base 2)
Step 1 - Divide the Number by 2
Step 2 - If the result has a remainder,
--> add 1 as the current MSB
Otherwise --> add 0 as the current MSB
Step 3 - Finish when result < base
--> Add Quotient as the final MSB
Example:
Convert (1000)10 to Binary (base 2)
Conversion
• Decimal (Base 10) -> Hexadecimal (Base 16)
Step 1 - Divide the Number by 16
Step 2 - Take the remainder as the current MSB
Step 3 - Finish when result < base
--> Add Quotient as the final MSB
Example:
Convert (1000)10 to Hexadecimal (base 16)
Bits & Bytes (Side Note)
• Bit
A bit is a single binary digit, a ‘1’ or a ‘0’
• Byte
A series of 8 bits ( 8 bits = 1 Byte )
Examples:
( 1010 1010 )2
( AA )16
Conversion
• Binary (Base 2) --> Hexadecimal (Base 16)
Step 1 - Make groups of 4 bits, starting from the
LSB
Step 2 - Directly convert each group into
Hexadecimal
Example:
Convert (1111101000)2 to Hexadecimal (base 16)
Binary Addition
Example:
Add (10011011)2 and (1110)2
Signed Binary
• MSB is the sign bit
0 <-- Positive Numbers
1 <-- Negative Numbers
2’s Complement Binary
• Example: Convert (-100)10 into 2’s comp
• Example: Binary Addition
2’s Complement Binary
• Why?
– Simplifying the implementation of arithmetic on
computer hardware.
– Allows the addition of negative operands without a
subtraction circuit or a circuit that detects the sign of
a number.
– Moreover, an addition circuit can also perform
subtraction by taking the two's complement of a
number