Arithmetics and Data Storage
1
INFOL1001 Introduction to Algorithm
10 November 2010
Goals (continued)
• Understand how to do simply binary
arithmetic
• How computers encode data
• How computers manipulate data using
the elemental commands
• How the Fetch & Execute Cycle works
2
INFOL1001 Introduction to Algorithm
10 November 2010
Binary Arithmetic
• Computer processing requires a little more
capability in binary math, notably
arithmetic.
• Let’s consider addition and subtraction …
3
INFOL1001 Introduction to Algorithm
10 November 2010
Binary Addition
• Just a few combinations are
possible:
– 02 + 0 2 = 02
– 02 + 1 2 = 12
– 12 + 0 2 = 1 2
• What about 12 + 12?
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INFOL1001 Introduction to Algorithm
10 November 2010
Binary Addition
12 + 12 = 102
• Remember what that 10 means: it means a
zero in the right column, and a one in the left
column
• When you carry in base two, you are bringing
a power of two to the left…
5
INFOL1001 Introduction to Algorithm
10 November 2010
Binary Addition
1+ 1+
6
11102
+10112
=1410
110012
=2510
=1110
INFOL1001 Introduction to Algorithm
10 November 2010
Binary Subtraction
• Subtraction in base two is similar to
addition
– 112 – 112 = 002
– 112 – 002 = 112
– 102 – 002 = 102
• What about 102 - 12?
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INFOL1001 Introduction to Algorithm
10 November 2010
Binary Subtraction
102 - 12 = 12
• Think about borrowing in decimal – you are really
borrowing a power of your base …
• In Base-10, we borrow 10 from the left column and
give it to the next column to the right. So …
• In Base-2, we borrow 2 from the left column and
give it to the next column to the right.
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INFOL1001 Introduction to Algorithm
10 November 2010
Binary Subtraction
0
1
0
1
1
001100112
-000101102
0 0 0 1 1 1 01 2
9
INFOL1001 Introduction to Algorithm
=5110
=2210
=2910
10 November 2010
Binary Borrowing: A Little Trick
• Many students find it convenient to just mentally convert
the binary number to decimal, complete the subtraction,
and then convert the answer back to binary.
• Consider 102 – 12 = ?
– A one in binary is still a one in decimal
– When you borrow for the zero, you bring over a power of two, which is
equal to two
– The decimal conversion for the borrowing is 2-1, which is 1
– And 110 = 12
• Be sure to add your answer back up as a check
10
INFOL1001 Introduction to Algorithm
10 November 2010
Negative Numbers
Subtract by adding
73
-35
38
10’s complement
73
+65
138
Ignore carry
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INFOL1001 Introduction to Algorithm
10 November 2010
Negative Numbers
Subtract by adding
73
-35
38
01001001
73
01001001
1 10111 01
35
00100011
00100110
2’s comp
11011100 flip
+1
-----------35
12
INFOL1001 Introduction to Algorithm
11011101
10 November 2010
Table 2.2
Positive and Negative Binary Numbers
Signed decimal
-128
-127
-126
…
…
…
-3
-2
-1
0
1
2
3
…
…
125
126
127
13
Hex
80
81
82
…
…
…
FD
FE
FF
00
01
02
03
…
…
7D
7E
7F
Binary
10000000
10000001
10000010
…
…
…
11111101
11111110
11111111
00000000
00000001
00000010
00000011
…
…
…
01111101
01111110
01111111
INFOL1001 Introduction to Algorithm
Unsigned decimal
128
129
130
…
…
…
253
254
255
0
1
2
3
…
…
…
125
126
127
10 November 2010
Signed Numbers
4-bit:
8H = -8 to 7H = +7
1000 to 0111
8-bit: 80H = -128 to 7F = +127
16-bit: 8000H = -32,768 to
7FFFH = +32,767
32-bit: 80000000H = -2,147,483,648 to
7FFFFFFFH = +2,147,483,647
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INFOL1001 Introduction to Algorithm
10 November 2010
Questions
What is the two’s complement of
00101100?
What hex number represents the
decimal number -40?
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INFOL1001 Introduction to Algorithm
10 November 2010
Computer Data
• Computers are very good
at representing data in
binary (1s and 0s)
• However, computers can
also deal with other data
types:
–
–
–
–
–
Text
Numbers
Graphics
Music
Video
• How?
16
INFOL1001 Introduction to Algorithm
10 November 2010
Storing Integer Numbers
• Convert the integer to binary
• Add a switch for positive/negative
-1910 = -101012
17
1
0
1
INFOL1001 Introduction to Algorithm
0
1
10 November 2010
Storing Floats (Real Numbers)
• Uses two memory positions:
– Store the number, without the decimal point, in one slot
– Store the number of digits to the LEFT of the decimal point in
another slot
123.456710
1234567
18
3
INFOL1001 Introduction to Algorithm
10 November 2010
Storing Strings
• Need to convert strings to ASCII (American
Standard Code for Information Interchange)
code first
• ASCII code is then converted to Base-2
equivalent:
CHAR
A
B
C
D
19
DEC
65
66
67
68
BIN
1000001
1000010
1000011
1000100
INFOL1001 Introduction to Algorithm
10 November 2010
ASCII Properties
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INFOL1001 Introduction to Algorithm
10 November 2010
Questions?
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INFOL1001 Introduction to Algorithm
10 November 2010