Download - Introduction to Computer Systems and Software Lecture 2 of 2 Simon Coupland [email protected]
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Representation of Data Within the Computer Contents:
Decimal and Binary Integer Numbers Binary Addition Signed Binary Numbers Overflow Hexadecimal Numbers Number Conversion Real Numbers Character Encoding
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Introduction
A few terms: A bit – a single Binary digIT, 0 or 1 A byte – eight bits A word – one or more bytes Integer – whole number Real number – a number with decimal points Binary – Base 2 numbers Octal – Base 8 numbers Decimal – Base 10 numbers (everyday numbers) Hexadecimal – Base 16 numbers
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Decimal and Binary Numbers
We all use decimal numbers Base 10 numbers Example: 124
Digit 7 6 5 4 3 2 1 0
Digit Value 107 106 105 104 103 102 101 100
Digit Value 10M 1M 100k 10k 1k 100 10 1
Example 0 0 0 0 0 1 2 4
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Decimal and Binary Numbers
Computers use binary numbers Base 2 numbers: Example: 124
Bit 7 6 5 4 3 2 1 0
Bit Value 27 26 25 24 23 22 21 20
Bit Value 128 64 32 16 8 4 2 1
Example 0 1 1 1 1 1 0 0
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Decimal and Binary Numbers
More binary numbers:
00010011 = 16 + 2 + 1
= 19
01011101 = 64 + 16 + 8 + 4 + 1
= 93
11111111 = 255
00000000 = 0
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Binary Addition
When adding binary numbers we use binary logic Binary Addition Truth Table 1:
A B A + B Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
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Binary Addition Binary Addition Truth Table 2:
A B Carry (in) A + B Carry (out)
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
=
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 0
1
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 10
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 110
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 1110
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 01110
1
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 101110
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 0101110
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Binary Addition
Binary Addition Example:
00010001
+ 00011101
= 00101110
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Binary Addition
Binary Addition Q1:
00101101
+ 00011001
=
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Binary Addition
Binary Addition Q1:
00101101
+ 00011001
= 01000110
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Binary Addition
Binary Addition Q2:
00001110
+ 00001111
=
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Binary Addition
Binary Addition Q2:
00001110
+ 00001111
= 00011101
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Binary Addition
Binary Addition Q3:
01011001
+ 00111011
=
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Binary Addition
Binary Addition Q3:
01011001
+ 00111011
= 10010011
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Negative Binary Numbers
Sign-true Magnitude Left most bit holds sign Example: -10
Bit 7 6 5 4 3 2 1 0
Bit Value sign 26 25 24 23 22 21 20
Bit Value +/- 64 32 16 8 4 2 1
Example 1 0 0 0 1 0 1 0
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Negative Binary Numbers
Ones complement All 1’s and 0’s are switched When negative, result = -255 + value Example: -10
Bit 7 6 5 4 3 2 1 0
Bit Value 27 26 25 24 23 22 21 20
Bit Value 128 64 32 16 8 4 2 1
Example 1 1 1 1 0 1 0 1
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Negative Binary Numbers
Twos complement Left most bit holds sign When negative, result = -128 + result Example: -10
Bit 7 6 5 4 3 2 1 0
Bit Value Sign 26 25 24 23 22 21 20
Bit Value +/-128 64 32 16 8 4 2 1
Example 1 1 1 1 0 1 1 0
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Negative Binary Numbers
Conversion to Twos complement Ones complement the byte/word Add 1
Example: 00001010 +10
11110101 Ones complement + 00000001 = 11110110
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Why Use Twos Complement?
Because addition rules still work:
00010110 22
+ 10001000 + -120
= 10011110 = -98
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Overflow
Overflow is when the number of bits is too small to store the result of an arithmetic operation
Example (twos complement) :
01011001 89
+ 01111011 + 123
= 11010011 = -45
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Overflow
Overflow can be easily detected for signed binary numbers
Errors can then be corrected Adding two positive numbers should give a
positive result Adding two negative numbers should give a
negative result Adding a positive and negative number
together can never result in overflow. Why?
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Hexadecimal Numbers
Writing code with long binary numbers would be cumbersome and error prone
A hexadecimal digit can take 16 values (0-F) One hex digit can represent a four bit word Examples:
Decimal Hexadecimal Binary
4 4 0100
10 A 1010
74 4A 01001010
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Number Conversion
Binary to Hexadecimal: From the least significant (rightmost) bit split the
binary number into groups of four bits. Each 4 bits has a hexadecimal equivalent Example:0001001110011110
0001 0011 1001 1110
1 3 9 E
139E
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Number Conversion
Hexadecimal to Binary: Convert each hexadecimal digit into its 4 bit binary
equivalent Join the all 4 bit words together Example:A42F
A 4 2 F
1010 0100 0010 1111
1010010000101111
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Number Conversion
Question Hexadecimal to Binary:
Convert D36B into binary
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Number Conversion
Question Hexadecimal to Binary:
Convert D36B into binaryD 3 6 B
1101 0011 0110 1011
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Number Conversion
Question Binary to Hexadecimal:
Convert 1001101001001010 into hexadecimal
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Number Conversion
Question Binary to Hexadecimal:
Convert 1001101001001010 into hexadecimal1001 1010 0100 1010
9 A 8 A
9A8A
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Number Conversion
Decimal to Binary Use the following algorithm, begin with LSB
int dec_value = some_number;
int next_bit;
while(dec_value > 0)
{
next_bit = dec_value % 2;
dec_value = dec_value / 2;
}
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Number Conversion
Decimal to Binary Convert 42 to binary:
dec_value /2 %2 Result
42 21 0 0
21 10 1 10
10 5 0 010
5 2 1 1010
2 1 0 01010
1 0 1 101010
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Number Conversion
Question Convert 39 to binary:
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Number Conversion
Question Convert 39 to binary:
dec_value /2 %2 Result
39 19 1 1
19 9 1 11
9 4 1 111
4 2 0 0111
2 1 0 00111
1 0 1 100111
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Number Conversion
Binary to Decimal Use the following algorithm, begin with MSB
int dec_value = 0;
int bit_value;
int bit_index = MSB_index;
while(bit_index >= 0)
{
bit_value = word[bit_index];
dec_value = dec_value * 2 + bit_value;
bit_index--;
}
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Number Conversion
Binary to Decimal Convert 011010 to decimal:
bit_index bit_value dec_value
5 0 0
4 1 1
3 1 3
2 0 6
1 1 13
0 0 26
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Number Conversion
Binary to Decimal Question, convert 101110 to decimal:
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Number Conversion
Binary to Decimal Question, convert 101110 to decimal:
bit_index bit_value dec_value
5 1 1
4 0 2
3 1 5
2 1 11
1 1 23
0 0 46
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Number Conversion
Decimal to Hexadecimal Use the following algorithm, begin with LSB
int dec_value = some_number;
char hex_digit;
while(dec_value > 0)
{
hex_digit = to_hex(dec_value % 16);
dec_value = dec_value / 16;
}
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Number Conversion
Decimal to Hexadecimal Convert 1863 to hexadecimal
dec_value dec_value / 16 dec_value % 16 hex_digit
1863 116 7 7
116 7 4 4
7 0 7 7
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Number Conversion
Decimal to Hexadecimal Question, convert 1437 to hexadecimal
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Number Conversion
Decimal to Hexadecimal Question, convert 1437 to hexadecimal
dec_value dec_value / 16 dec_value % 16 hex_digit
1437 89 13 D
89 5 9 9
5 0 5 5
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Number Conversion
Hexadecimal to Decimal Use the following algorithm, begin with MSB
int dec_value = 0;
char digit_value;
while(!word.off)
{
digit_value = to_dec(hex_digit);
dec_value = dec_value * 16 + digit_value;
}
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Number Conversion
Hexadecimal to Decimal Convert F2AC to decimal
Index hex_value dec_value result
3 F 15 15
2 2 2 242
1 A 10 3882
0 C 12 62124
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Number Conversion
Hexadecimal to Decimal Convert E59A to decimal
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Number Conversion
Hexadecimal to Decimal Convert E59A to decimal
Index hex_value dec_value result
3 E 14 14
2 5 5 229
1 9 9 3673
0 A 10 58778
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Real Numbers
A number n is a real number if n % 1 != 0 Take a lot of processing power/time Often math co-processors are used to
perform arithmetic on these numbers
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Fixed Point Real Numbers
Similar to integer representation 8 bit example:
Bit 8 7 6 5 4 3 2 1
value -/+ 23 22 21 20 2-1 2-2 2-3
value -/+ 8 4 2 1 0.5 0.25 0.125
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Fixed Point Real Numbers
Maximum number is small Accuracy is limited Not used because of these reasons
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Floating Point Real Numbers
Floating point numbers are held in two parts Mantissa Exponent
In 0.3876 104
The mantissa is 0.3876 The exponent is 4
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Floating Point Real Numbers
Example (not IEEE float):
00000000000000110010100111110011mantissa exponent
Mantissa = 809 Exponent = -13 809 10-13
0.0000000000809
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Character Encoding
ASCII EBCDIC UNICODE ISO-8859-1
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Character Encoding
ASCII American Standard Code for Information Interchange Characters are represented by integers Table in your notes
A = 65 B = 66 a = 96 z = 123 @ = 64
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Character Encoding
ASCII Special functions characters are also encoded Examples:
Carriage return = 13 Escape = 27 Space = 32
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Recap
Binary numbers can be signed of unsigned Twos complement gives simple addition Overflow - when the result of a arithmetic
operation is too large to be represented Hexadecimal numbers – base 16 Real numbers use floating point Characters are represented as intergers