Transcript Topic 2 - Storing Data (Part 1)

Computing Science
Software Design and
Development
Storing Data
Learning Objectives
•
By the end of this topic you will be able to:
• understand why computers store numbers as
binary code;
• convert between binary and decimal;
• understand how computers store integers using
two’s complement notation;
• convert between a decimal integer and two’s
complement notation;
• explain how computers store real numbers using
floating point notation;
•
Why Binary?
Computers store data in billions of electronic
switches called transistors
Switches can either be on or off (two states)
The easiest way to represent this is as a series of
0s and 1s
Human (Decimal) system
104
103
102
101
100
Ten
Thousands
Thousands
Hundreds
Tens
Units
0
1
3
5
6
1 x 1000 + 3 x 100 + 5 x 10 + 6 x 1 = 1356
Binary system
27
26
25
24
23
22
21
20
128s 64s
32s
16s
8s
4s
2s
Units
0
0
0
0
0
1
1
0
2+1 =3
Binary system
27
26
25
24
23
22
21
20
128s 64s
32s
16s
8s
4s
2s
Units
0
0
0
1
0
1
0
0
8 + 2 = 10
Binary system
27
26
25
24
23
22
21
20
128s 64s
32s
16s
8s
4s
2s
Units
0
1
0
0
0
1
1
0
32 + 2 + 1= 35
Binary system
27
26
25
24
23
22
21
20
128s 64s
32s
16s
8s
4s
2s
Units
1
1
1
1
1
1
1
1
128 + 64 + 32 + 16 + 8 + 4 + 2 + 1= 255
(28 = 256)
Converting Decimal to Binary
Find the largest power of two which is less
than the number to convert.
Subtract it and repeat with the remainder.
Converting Decimal to Binary
Convert 113 to binary
26 = 64
128 64 32 16 8
113-64 = 49
25=32
1 1 1 0
49-32 = 17
24=16
17-16 =1
4
2
U
0
0
1
Positive Integers
000
001
010
011
100
101
110
111
0
1
2
3
4
5
6
7
1000
1001
1010
1011
1100
1101
1110
1111
10000
8
9
10
11
12
13
14
15
16
Negative Integers (Using a sign-bit)
1001
1010
1011
1100
1101
1110
1111
-1
-2
-3
-4
-5
-6
-7
• There are two problems
with this method
• 1000 and 0000 both
mean zero
• Adding a positive
number to a negative
number gives the wrong
answer
Why adding integers using the sign-bit
method doesn’t work
1011
0100
1111
-3
+4
-7
+
Two’s Complement (8 bits)
Make the most significant bit negative
27
26
25
24
23
22
21
20
-128s 64s
32s
16s
8s
4s
2s
Units
0
0
0
0
0
1
1
0
2+1 =3
Two’s Complement (8 bits)
27
26
25
24
23
22
21
20
-128s 64s
32s
16s
8s
4s
2s
Units
1
0
0
0
0
1
1
0
-128 + 2 + 1 = -125
Two’s Complement (8 bits)
27
26
25
24
23
22
21
20
-128s 64s
32s
16s
8s
4s
2s
Units
1
0
0
1
0
1
0
0
-128 + 8 + 2 = -118
Storing numbers using
Two’s Complement (3 bits)
In this example the 4s column is the sign bit
010 +2
001 +1
000 0
111 -1
110 -2
101 -3
Addition now works and there is only
one code for zero
011
100
111
3
-4
-1
+
Converting a negative decimal number
1.
2.
3.
4.
Establish the bit length required
Convert positive number to binary
Complement the binary number
Add 1
-3 (8 bits)
1. +3 =
2. Complement
00000011
11111100
3. Add 1
+1
4. Result
11111101
-15 (8 bits)
1. +15 =
2. Complement
00001111
11110000
3. Add 1
+1
4. Result
11110001
-35 (8 bits)
1. +35 =
2. Complement
00100011
11011100
3. Add 1
+1
4. Result
11011101
Real Numbers
• Real numbers are stored using
Floating Point notation
• A Floating Point number takes up
more memory than a two’s
complement number and may be
less accurate
Scientific Notation
You will be familiar with how numbers like
this are stored in the Decimal system:
1.34 X 103
Mantissa
= 1340
Exponent
Scientific Notation
The exponent can be positive or negative
Exponent
1.34 X 10 -3
Mantissa
= 0.00134
Floating Point Notation
Computers use a similar system called floating
point notation
For example 18.75 in binary is 10010.11
10010.11 = .10010 x 2
Mantissa
101
Exponent
The Exponent is a power of two stored in Two’s
Complement notation because it can be positive or negative
Floating Point Notation
• The number of bits allocated to the Exponent
determine the range of numbers you can store
• The number of bits allocated to the Mantissa
determine the accuracy of the numbers you
can store.
Range
The exponent is an integer stored as a 2’s
complement number
8 bits allocated to the exponent means it can
represent a range of numbers between
2127 and 2-128
Accuracy
• The Mantissa is an integer stored as an
unsigned (positive) number
• The larger the number of bits allocated to the
mantissa, the more accurate the number will
be
Floating Point Notation
• Floating Point Notation will always be a compromise
between how accurately you can store a number and
how wide a range of numbers you wish to store.
• For example if your floating point numbers as stored in
5 bytes (40 bits) you could allocate 32 bits to the
mantissa and 8 bits to the exponent.
• If you wanted a more accurate number you would have
to increase the number of bits allocated to the
mantissa which would reduce the number available to
the exponent