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Chapter 2

Chapter 2. Binary Values and Number Systems. - Eswari Manickam. Materials are from text book with additions and adaptations by Eswari Manickam. Chapter Goals. Distinguish among categories of numbers Describe positional notation Convert numbers in other bases to base 10

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Chapter 2

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  1. Chapter 2 Binary Values and Number Systems - Eswari Manickam Materials are from text book with additions and adaptations by Eswari Manickam

  2. Chapter Goals • Distinguish among categories of numbers • Describe positional notation • Convert numbers in other bases to base 10 • Convert base-10 numbers to numbers in other bases • Describe the relationship between bases 2 and 16 • Explain the importance to computing of bases that are powers of 2 24 6

  3. Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given base, it can only contain the digits in that base, which range from 0 up to (but not including) the base. What bases can these numbers be in? 122, 198, 178, G1A4 9

  4. Positional Notation • 642 is 600 + 40 + 2 in BASE 10 Continuing with our example… 642 in base 10 positional notation is: 6 x 102 = 6 x 100 = 600 + 4 x 101 = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10

  5. Converting Binary to Decimal What is the decimal equivalent of the binary number 1101110? 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 0 x 24 = 0 x 16 = 0 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 110 in base 10 13

  6. Converting Binary to Decimal What is the decimal equivalent of the following binary numbers? • a) 11101 • b) 1011010 • c) 10011100

  7. Bases Higher than 10 How are digits in bases higher than 10 represented? With distinct symbols for 10 and above. Hexadecimal is base 16 and has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F 10

  8. Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number 32A? 3 x 162 = 3 x 256 = 768 + 2 x 161 = 3 x 16 = 48 + A x 16º = 10 x 1 = 10 = 826 in base 10 Remember, the digits in base 16 are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

  9. Converting Hexadecimal to Decimal What is the decimal equivalent of the following hexadecimal numbers? • 87A • b) 34E • c) F000

  10. Counting in Binary/Hexadecimal/Decimal

  11. Converting Binary to Hexadecimal • Mark groups of four (from right) • Convert each group • 10101011 10101011 • A B • 10101011 is AB in base 16 18

  12. Converting Binary to Hexadecimal What is the hexadecimal equivalent of the following binary numbers? • 00001001 • 10101001 • 010111011110

  13. Converting Decimal to Other Bases Algorithm for converting number in base 10 to other bases • While (the quotient is not zero) • Divide the decimal number by the new base • Make the remainder the next digit to the left in the answer • Replace the original decimal number with the quotient 19

  14. Converting Decimal to Hexadecimal What is the hexadecimal equivalent of (3567)10 ? 222 13 0 16 3567 16 222 16 13 3216 0 36 62 13 3248 47 14 32 15 D E F 21

  15. Converting Decimal to Hexadecimal What is 356 (base 10) in base 16? What is 1135 (base 10) in base 16? What is 4759 (base 10) in base 16? Try it! 20

  16. Converting Decimal to Binary Example of converting decimal to binary What is the binary equivalent of the decimal number 35? 17 8 4 2 1 0 2 35 2 17 2 8 4 2 1 3416842 0 11 00 0 1 Adding digits to the left as we calculate: 100011

  17. Converting Decimal to Binary Easy method for converting decimal to binary What is the binary equivalent of the decimal number 35? 2 35 - 1 2 17 - 1 2 8 - 0 2 4 - 0 2 2 - 0 2 1 - 1 0 So reading from the bottom – The answer would be 100011

  18. Converting Decimal to Binary What is the binary equivalent of the following decimal integers? A) 64 B) 1066 C) 213 D) 1790

  19. Arithmetic in Binary Remember that there are only 2 digits in binary, 0 and 1 1 + 1 is 0 with a carry Carry Values 1 1 1 1 1 1 1 0 1 0 1 1 1 +1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 14

  20. Arithmetic in Binary Calculate: • 10001 + 11101 • 1110 + 1111 • 1011001 + 111010

  21. Subtracting Binary Numbers Remember borrowing? Apply that concept here: 1 2 2 0 2 1 0 1 0 1 1 1 - 1 1 1 0 1 1 0 0 1 1 1 0 0 15

  22. Subtracting Binary Numbers Calculate: • 1011011 - 10010 • 1010110 - 101010 • 1000101 - 101100

  23. Binary Numbers and Computers Computers have storage units called binary digits or bits Low Voltage = 0 High Voltage = 1 all bits have 0 or 1 22

  24. Binary and Computers • Byte • 8 bits • The number of bits in a word determines the word length of the computer, but it is usually a multiple of 8 • 32-bit machines • 64-bit machines etc. 23

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