Programming for GCSE

Topic 2.3: Converting Binary and Decimal

Teaching London Computing

William MarshSchool of Electronic Engineering and Computer Science

Queen Mary University of London

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

Teaching Issue

• So far, used counting … • … now more mathematics.

• Still, minimise notation and terms

CONVERTING BETWEEN BINARY AND DECIMAL

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

Base 10 Table

110100

1 2 3

mostsignificant digit

leastsignificant digit

123 = 1 x 100 + 2 x 10 + 3 x 1

Base 2 Table

124816

1 0 1 0 0

mostsignificant bit

leastsignificant bit

10100 = 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 110100 = 16 + 4 = 20

Conversion to Binary

• To convert a decimal integer to binary:• Odd 1, Even 0• Divide by 2• Stop when result of the division is 0

123 61 30 15 7 3 1 0

MostSignificantbit01 1 1 1 1 1

12310 = 1 1 1 1 0 1 12

LeastSignificant

bit

Quiz

• Convert 00112 to decimal

• Convert 11112 to decimal

POWERS AND EXPONENTS

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

Base 2 Table

2021222324

1 0 1 0 0

mostsignificant bit

leastsignificant 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

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?

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

Quiz

• Which is larger• 232 or• the number of people in the world?

HOW MANY NUMBERS

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

How Many Binary Numbers?

bits max binary max base10 how many

1 1 1 22 11 3 43 111 7 84 1111 15 165 11111 31 326 111111 63 647 1111111 127 1288 11111111 255 256

Quiz

• 161 student in this class 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?

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 READ0111 WRITE1111 STOP

Overflow

Arithmetic with Fixed Number of Digits

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

+

Fixed Bit Arithmetic

• 4-bit0000

0001

0010

0011

0100

0101

0110

011110001001

1010

1011

1100

1101

1110

1111

01

2

3

4

5

6

78

910

11

12

13

1415

Fixed Bit Arithmetic

• 4-bit

• Add 4

00000001

0010

0011

0100

0101

0110

011110001001

1010

1011

1100

1101

1110

1111

0 0 0 0 0 1 0 0 0 1 0 0

+

Fixed Bit Arithmetic

• 4-bit

• Add 4

00000001

0010

0011

0100

0101

0110

011110001001

1010

1011

1100

1101

1110

1111

0 1 1 1 0 1 0 0 1 0 1 1

+

Fixed Bit Arithmetic

• 4-bit

• Add 4

00000001

0010

0011

0100

0101

0110

011110001001

1010

1011

1100

1101

1110

1111

1 1 1 0 0 1 0 0 1 0 0 1 0

+

Overflow (Unsigned)

• When you passed the read line

• E.g.• 14 + 4 = 2

00000001

0010

0011

0100

0101

0110

011110001001

1010

1011

1100

1101

1110

1111

01

2

3

4

5

6

78

910

11

12

13

1415

IMAGES

Two Ideas – Images

• Pixels and resolution• Image is an array of pixels

• Number of bits per pixel• ‘Colour’ of each pixel is a number

Original

• Red – 8 bits• Green – 8 bits• Blue – 8 bits• ‘Million’

colours

508 × 578 pixels24 bit RGB colour

Fewer Pixels

• m100 × 114 pixels24 bit RGB colour

50 × 57 pixels24 bit RGB colour

Fewer Colours

508 × 578 pixels24 bit RGB colour

508 × 578 pixels256 colours (indexed)

SYLLABUS

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

Summary

• Understand powers of 2• How many bits 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)