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Programming for GCSE Topic 2.3: Converting Binary and Decimal

T eaching L ondon C omputing. Programming for GCSE Topic 2.3: Converting Binary and Decimal. William Marsh School of Electronic Engineering and Computer Science Queen Mary University of London. Aims. Understand powers of 2 Number of numbers Convert between binary and decimal

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Programming for GCSE Topic 2.3: Converting Binary and Decimal

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  1. TeachingLondon Computing Programming for GCSETopic 2.3: Converting Binary and Decimal William Marsh School of Electronic Engineering and Computer Science Queen Mary University of London

  2. 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

  3. Teaching Issue So far, used counting … … now more mathematics. Still, minimise notation and terms

  4. Converting Between Binary and Decimal

  5. 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

  6. Base 10 Table 123 = 1 x 100 + 2 x 10 + 3 x 1 100 10 1 1 2 3 most significant digit least significant digit

  7. Base 2 Table 16 8 4 2 1 1 0 1 0 0 least significant bit most significant bit 10100 = 1 x 16 +0 x 8+ 1x 4 + 0 x 2 + 0 x 1 10100 = 16 +4 = 20

  8. 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 Least Significant bit Most Significant bit 1 1 1 0 1 1 1 12310 = 1 1 1 1 0 1 12

  9. Quiz Convert 00112to decimal Convert 11112to decimal

  10. Powers and Exponents

  11. Powers and Exponents 10NPower 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= 2x 2 23= 2x 2x 2

  12. Base 2 Table 24 23 22 21 20 1 0 1 0 0 least significant bit most significant bit 10100 = 1 x 24+ 0 x 23+ 1 x 22+ 0 x 21+ 0 x 20 = 1 x 16 +0 x 8+ 1x 4 + 0 x 2 + 0 x 1 = 16 + 4 = 20

  13. 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?

  14. 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

  15. Quiz Which is larger 232 or the number of people in the world?

  16. How Many Numbers

  17. 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

  18. 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

  19. 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?

  20. 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

  21. Arithmetic with Fixed Number of Digits Overflow

  22. 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 1 111 1 1 1 1 0 0 0 0 0 0 0 0 +

  23. Fixed Bit Arithmetic 4-bit 0000 1111 0001 0010 1110 0011 1101 1100 0100 0101 1011 1010 0110 1001 0111 1000 0 15 1 14 2 13 3 12 4 11 5 6 10 9 7 8

  24. Fixed Bit Arithmetic 4-bit Add 4 0000 1111 0001 0010 1110 0011 1101 1100 0100 0101 1011 1010 0110 1001 0111 1000 0 0 0 0 0 1 0 0 0 1 0 0 +

  25. Fixed Bit Arithmetic 4-bit Add 4 0000 1111 0001 0010 1110 0011 1101 1100 0100 0101 1011 1010 0110 1001 0111 1000 0 1 1 1 0 1 0 0 1 0 1 1 +

  26. Fixed Bit Arithmetic 4-bit Add 4 0000 1111 0001 0010 1110 0011 1101 1100 0100 0101 1011 1010 0110 1001 0111 1000 1 1 1 0 0 1 0 0 1 0 0 1 0 +

  27. Overflow (Unsigned) When you passed the read line E.g. 14 + 4 = 2 0000 1111 0001 0010 1110 0011 1101 1100 0100 0101 1011 1010 0110 1001 0111 1000 0 15 1 14 2 13 3 12 4 11 5 6 10 9 7 8

  28. Images

  29. Two Ideas – Images • Pixels and resolution • Image is an array of pixels • Number of bits per pixel • ‘Colour’ of each pixel is a number

  30. Original • Red – 8 bits • Green – 8 bits • Blue – 8 bits • ‘Million’ colours 508 × 578 pixels 24 bit RGB colour

  31. Fewer Pixels 100 × 114 pixels 24 bit RGB colour 50 × 57 pixels 24 bit RGB colour • m

  32. Fewer Colours 508 × 578 pixels 24 bit RGB colour 508 × 578 pixels 256 colours (indexed)

  33. Syllabus

  34. 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

  35. 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)

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