topic 2.2: converting binary and decimal
TRANSCRIPT
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Topic 2.2: Converting Binary and Decimal
Teaching London Computing
William Marsh School of Electronic Engineering and Computer Science
Queen Mary University of London
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Aims
• Understand powers of 2 • Number of numbers
• Convert between binary and decimal • Using powers of 2
• Do binary arithmetic with more interpretation • How fixed width leads to overflow in addition
• Look at some pictures • Notes on the syllabus
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Teaching Issue
• So far, used counting … • … now more mathematics.
• Still, minimise notation and terms
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CONVERTING BETWEEN BINARY AND DECIMAL
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Decimal – base 10
• Base 10 • 10 numerals • ‘0’, ‘1’, ‘2’, … , ‘9’
• What does ‘123’ mean?
123 = 1 x 100 + 2 x 10 + 3 x 1
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Base 10 Table
1 10 100
1 2 3
most significant digit
least significant digit
123 = 1 x 100 + 2 x 10 + 3 x 1
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Base 2 Table
1 2 4 8 16
1 0 1 0 0
most significant bit
least significant bit
10100 = 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1
10100 = 16 + 4 = 20
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Conversion to Binary
• To convert a decimal integer to binary: • Odd à 1, Even à 0 • Divide by 2 • Stop when result of the division is 0
123 à 61 à 30 à 15 à 7 à 3 à 1 à 0
Most Significant bit 0 1 1 1 1 1 1
12310 = 1 1 1 1 0 1 12!
Least Significant
bit
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Quiz
• Convert 00112 to decimal • Convert 11112 to decimal
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POWERS AND EXPONENTS
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Powers and Exponents • 10N Power of 10
‘N’ is an exponent
• 100 = 1 • 10(X+Y) = 10X x 10Y
100 = 1 101 = 10 102 = 10 x 10 103 = 10 x 10 x 10
21 = 1 21 = 2 22 = 2 x 2 23 = 2 x 2 x 2
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Base 2 Table
20 21 22 23 24
1 0 1 0 0
most significant bit
least significant bit
10100 = 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 0 x 20
= 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1
= 16 + 4 = 20
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Quiz
• Write out powers of 2, up to 28 (then 216) • Convert 101000112 to decimal • Convert 011111112 to decimal
• What is the next number after (the 10 digit number) 11111111112 in base 10?
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K and 210 and 103
• 210 = 1024, approximately equal to 103
• 210 abbreviated by ‘K’ • 1 KByte is 1024 Bytes • 220 = 210 x 210 ≈ 103 x 103 = 106 • 220 abbreviated ‘M’ • 1 MByte = 220 Bytes ≈ 106 Bytes
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Quiz
• Which is larger • 232 or • the number of people in the world?
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HOW MANY NUMBERS
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How Many Numbers?
• How many decimal numbers less that 100? • 2-digit numbers : NN • 0 .. 99 • 100 different numbers
• General rule: • 10n n-digits (decimal) numbers • 2n n-digit (binary) numbers
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How Many Binary Numbers?
bits max binary max base10 how many
1 1 1 2 2 11 3 4 3 111 7 8 4 1111 15 16 5 11111 31 32 6 111111 63 64 7 1111111 127 128 8 11111111 255 256
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Quiz
• 161 student in this class w How many bits to represent each student with a
unique binary number?
• A computer can execute 9 different machine
instructions: ADD, SUB, MUL, DIV, JUMP, LOAD, READ, WRITE, STOP. " How many bits do we need to give each instruction
a different code? " What could these codes be?
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Quiz – Answers
• 7 bits?? • NO! With 7 bits we can only represent 27 = 128
patterns. • We need 8 bits. 8 bits can represent up to 28 = 256
patterns • To represent 9 bit patterns we need 4 bits: 24 =
16 0000 ADD 0001 SUB 0010 MUL 0011 DIV 0100 JUMP
0101 LOAD 0110 READ 0111 WRITE 1111 STOP
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Overflow
Arithmetic with Fixed Number of Digits
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Overflow
• The addition of 8 bit numbers may overflow 8 bits
• Computer arithmetic has a limited number
of bits
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
+
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Fixed Bit Arithmetic
• 4-bit
0000 0001
0010
0011
0100
0101
0110
0111 1000 1001
1010
1011
1100
1101
1110
1111
0 1
2
3
4
5
6
7 8
9 10
11
12
13
14 15
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Fixed Bit Arithmetic
• 4-bit • Add 4
0000 0001
0010
0011
0100
0101
0110
0111 1000 1001
1010
1011
1100
1101
1110
1111
0 0 0 0 0 1 0 0 0 1 0 0
+
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Fixed Bit Arithmetic
• 4-bit • Add 4
0000 0001
0010
0011
0100
0101
0110
0111 1000 1001
1010
1011
1100
1101
1110
1111
0 1 1 1 0 1 0 0 1 0 1 1
+
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Fixed Bit Arithmetic
• 4-bit • Add 4
0000 0001
0010
0011
0100
0101
0110
0111 1000 1001
1010
1011
1100
1101
1110
1111
1 1 1 0 0 1 0 0 1 0 0 1 0
+
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Overflow (Unsigned)
• When you passed the read line • E.g. • 14 + 4 = 2
0000 0001
0010
0011
0100
0101
0110
0111 1000 1001
1010
1011
1100
1101
1110
1111
0 1
2
3
4
5
6
7 8
9 10
11
12
13
14 15
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IMAGES
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Two Ideas – Images
• Pixels and resolution • Image is an array of pixels
• Number of bits per pixel • ‘Colour’ of each pixel is a number
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Original
• Red – 8 bits • Green – 8 bits • Blue – 8 bits • ‘Million’ colours
508 × 578 pixels 24 bit RGB colour
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Fewer Pixels
• m 100 × 114 pixels 24 bit RGB colour
50 × 57 pixels 24 bit RGB colour
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Fewer Colours 508 × 578 pixels 24 bit RGB colour
508 × 578 pixels 256 colours (indexed)
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SYLLABUS
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Syllabus – Binary
• GCSE (OCR) • Conversion between binary and decimal • Hexadecimal • Binary addition
• AS/A2 (AQA) • (AS) Negative numbers - two’s complement • (AS) More arithmetic • (A2) Real (floating point) numbers
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Summary
• Understand powers of 2 • How many bits à binary representation • Arithmetic with fixed number of bits leads to
overflow
• Images • Pixels • Bits per pixel
• Anything can be represented by numbers (i.e. digitally)