Transcript Representation2

Review Two’s complement
Negate a Binary Number in 2C
•
•
•
Negation means to find the corresponding negative
number with the same absolute value (in decimal
the Negation of 10 is –10)
The method is the same whether you negate a
positive to a negative or vice versa and doing it
twice should give you the same result: -(-a) = a
Method:
1. Invert the binary number (“flip all bits”)
Inverse (01001101) = 10110010
2. Add to the result binary 1 (append 0’s on the left)
10110010 + 1=10110010+00000001=10110011
Convert Decimal to 2C
• Decimal to Binary (word length k = 8):
– If positive: Fraction number into a sum of
powers of 2
bin(13)=8+4+1=23+22+20 = 00001101
– If Negative: Fraction the negated number and
negate binary number (invert and add 1)
bin(-13) = Negate(bin(13)) = Negate(00001101)
= Invert(00001101)+00000001=11110010+00000001
= 11110011
Convert 2C to Decimal
• Binary to Decimal (word length k=8):
– If most significant bit (left most) is 0 the number is positive: Sum
the powers of 2
Dec(00101101)=0*27 +0*26+1*25 +0*24 +1*23 +1*22 +0*21 +1*20
=1*128+0*64+1*32+0*16+1*8+1*4+0*2+1*0
=173
– If most significant bit (left most) is 1 the number is negative, that
means you can not just add the powers of 2
– Find the decimal value of the negated binary number and negate it
back in decimal (putting a – in front)
Dec(10101101) = - (Dec(Negate(10101101))
= - (Dec(Invert(10101101)+0000001))
= - (Dec(01010010+0000001))
= - (Dec(01010011))
- now it is a positive number and we can sum the powers of 2!
= -(64+16+2+1) = - 83
Operations on 2C
• Addition: bitwise including carry as in
decimal
• Negation: Invert and add 1
• Subtraction: Addition of Negated operand:
A – B =A + (– B)
• Multiplication by 2: Shift left
00101101 * 2 = 01011010
• Division by 2: Shift right
00101101 ÷ 2 = 00010110
Issues: Binary Numbers
• Signed Extension:
– Sometimes we are lazy and don’t write out all bits. In 2C you
can only add 0’s to the left if the number is positive,
otherwise you make a negative number positive:
-3 = 1101
for k=4
= 11111101 for k=8
• For negative number you have to add 1’sto the left!
• Overflow:
– If the result of an addition/subtraction is bigger than the
biggest or smaller than the smallest possible number of word
length k
– Can never happen if you add a positive and a negative
number
Hexadecimal Numbers
• Same principle as decimal and binary but
with a base of 16
• Number has a little x to the side: xF5
• Motivation: Word size is either 32 or 64,
that are very long binary numbers, in Hex
you can save space:
• Conversion from Hex to binary is simple
in blocks of 4(this works because 16=24):
x5B2F = 0101 1011 0010 1111
To decimal (if positive):
= 5*163 + 11*162 + 2*161 + 15*160
= 20480 + 2816 + 32 + 15
= 23343
Decimal
Hex
Binary
0
0
0000
1
1
0001
2
2
0010
3
3
0011
4
4
0100
5
5
0101
6
6
0110
7
7
0111
8
8
1000
9
9
1001
10
A
1010
11
B
1011
12
C
1100
13
D
1101
14
E
1110
15
F
1111
LOGICAL OPERATORS
A B
0 0
0 1
1 0
1 1
AND
0
0
0
1
A
0
1
NOT
1
0
A B
0 0
0 1
1 0
1 1
OR
0
1
1
1
A B
0 0
0 1
1 0
1 1
XOR
0
1
1
0
AND: is only 1 if all inputs are 1
OR: is 1 if any of the inputs is 1
XOR: is 1 if either input is 1
NOT: is the inverse (“flip the bit”)
Logic on Words and Multiple Input
00101101
AND 11101010
00101000
00101101
10011000
AND 11101010
00001000
OR
OR
00101101
11101010
11101111
00101101
10011000
11101010
11111111
Text Data: ASCII Code
• ASCII is a 7 bit encoding
scheme for all keyboard
symbols and some
additional ones
• Digits: x30 to x39
• Uppercase: x41 to x5A
• Lowercase: x61 to x7A
• Conversion: Table
• Question: How can you
mathematically convert a
lowercase letter into
uppercase?