Binary Numbers
CS102
Subrina Thompson
Binary Numbers Overview
• Binary is a number system used by digital devices like computers, etc.
• Binary is a Base 2, unlike your counting system decimal which is Base 10
(denary)
• Binary has only 2 different numerals (0, and 1)
• Example of a Binary number: 10011100
Notations
• Notations used in digital systems:
4 bits
= Nibble
8 bits
= Byte
16 bits
= Word
32 bits
= Double Word
64 bits
= Quad Word (or paragraph)
Notations
• 101 is a binary number and 101 is a decimal (denary) value.
2
10
• Once we know the base then it is easy to work out the value, for example:
• 101 = 1*22 + 0*21 + 1*20 = 5 (five)
• 101 = 1 * 102 + 0*101 + 1*100 = 101 (one hundred and one)
2
10
Converting a Decimal to Binary
• Example: Convert 157 to binary.
157÷ 2 = 78 remainder 1
78 ÷ 2 = 39 remainder 0
39 ÷ 2 = 19 remainder 1
19 ÷ 2 = 9
remainder 1
9
÷ 2 = 4
remainder 1
4
÷ 2 = 2
remainder 0
2
÷ 2 = 1
remainder 0
1
÷ 2 = 0
remainder 1
Next, write down the value of the
remainder from bottom to top.
100111012 = 15710
Convert Binary to Decimal
• To convert binary into decimal, say 8 bit value 10011101, we can use a
formula like that below
128
64
32
16
8
4
2
1
1
0
0
1
1
1
0
1
• To convert, take the top row wherever there is a 1 below and then add the
values together.
• In the example above, this would give us:
128 + 16 + 8 + 4 + 1 = 157
Binary Addition
• Conceptually similar to decimal addition
• Example: Add the binary numbers 1010 and 11
(carry)
1
1 0 1 0
+
1 1
1 1 0 1
2’s Complement Notation
• 2’s complement representation – widely used in microprocessors
• Represents sign and magnitude
LSB
MSB
Sign bit (0 = + ; 1 = -)
Decimal
+7
+4
+1
0
-1
-4
-7
2’s Complement 0111
0100
0001
0000
1111
1100
1001
Convert Byte to 2’s Complement
• Positive number to Negative number
• Negative number to Positive number
• Step 1: Complement/Flip the bits (1’s
• Step 1: Complement/Flip the bits (1’s
• Step 2: Add 1 to the number
• Step 2: Add 1 to the number
complement)
• e.g. 001111112
• Step 1:
• Step 2:
 6310
11000000
11000000
complement)
• e.g. 110000012
 -6310
• Step 1:
• Step 2:
00111110
00111110
+ 00000001
+ 00000001
----------
----------
11000001  -63
00111111  -63
Convert Byte to 2’s Complement
• Note: To know what decimal number a negative byte is, find the 2’s
complement.
•
•
•
•
MAX (positive byte)  01111111
 127 = 27 - 1
MIN (negative byte)  10000000
 128 = 27
Convert Byte to 2’s Complement
•
Overflow
127
 01111111
2
 00000010
----------------129
 10000001
• Overflow – If the high bit (left-most bit result is 1 (negative) and the two numbers are positive, then we
have an OVERFLOW
• Underflow – opposite of overflow (happens when you add two negative numbers and the left-most bit is a
0 (positive))
• Note: You can’t get an overflow or underflow from adding positive and negative numbers.
Biased Notation
• To convert from biased to 2’s complement, flip the left-most bit.
• e.g. 11111101  biased
• 2’s complement  01111101