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Tutorial: ITI1100

Tutorial: ITI1100. Dewan Tanvir Ahmed SITE, UofO Email: dahmed@site.uottawa.ca. Binary Numbers. Base (or radix) 2 example: 0110 Number base conversion example: 41 = 101001 Complements 1's complements ( 2 n - 1 ) - N 2's complements 2 n - N

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Tutorial: ITI1100

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  1. Tutorial: ITI1100 Dewan Tanvir Ahmed SITE, UofO Email: dahmed@site.uottawa.ca

  2. Binary Numbers • Base (or radix) • 2 • example: 0110 • Number base conversion • example: 41 = 101001 • Complements • 1's complements ( 2n- 1 ) - N • 2's complements 2n - N • Subtraction = addition with the 2's complement • Signed binary numbers • signed-magnitude, 10001001 • signed 1's complement, 11110110 • signed 2's complement. 11110111

  3. Binary Number System Base = 22 Digits: 0, 1 Examples: 1001b = 1 * 23 + 0 * 22 + 0 * 21 + 1 * 20 = 8 + 1 = 9  1010 1101b = 1 * 27 + 1 * 25 + 1 * 23 + 1 * 22 + 1 = 128 + 32 + 8 + 4 + 1 = 173 Note: it is common to put binary digits in groups of 4 to make it easier to read them.

  4. Ranges for Data Formats

  5. In General (binary)

  6. Signed Integers • “unsigned integers” = positive values only • Must also have a mechanism to represent “signed integers” (positive and negative values!) -1010 = ?2 • Two common schemes: • sign-magnitude and • twos complement

  7. +5: 0 0 0 1 0 1 -5: 1 0 0 1 0 1 +ve -ve 5 5 Sign-Magnitude • Extra bit on left to represent sign • 0 = positive value • 1 = negative value • 6-bit sign-magnitude representation of +5 and –5:

  8. Ranges (revisited)

  9. In General …

  10. Difficulties with Sign-Magnitude • Two representations of zero • Using 6-bit sign-magnitude… • 0: 000000 • 0: 100000 • Arithmetic is awkward!

  11. Complementary Representations • 1’s complement • 2’s complement • 9’s complement • 10’s complement

  12. Complementary Notations • What is the 3-digit 10’s complement of 207? • Answer: • What is the 4-digit 10’s complement of 15? • Answer: • 111 is a 10’s complement representation of what decimal value? • Answer:

  13. Exercises – Complementary Notations • What is the 3-digit 10’s complement of 207? • Answer:793 • What is the 4-digit 10’s complement of 15? • Answer:9985 • 111 is a 10’s complement representation of what decimal value? • Answer:889

  14. 2’s Complement • Most common scheme of representing negative numbers • natural arithmetic - no special rules! • Rule to represent a negative number in 2’s C • Decide upon the number of bits (n) • Find the binary representation of the +ve value in n-bits • Flip all the bits • Add 1

  15. 000101 +5 111010 + 1 111011 -5 2’s Complement Example • Represent -5 in binary using 2’s complement notation • Decide on the number of bits • Find the binary representation of the +ve value in 6 bits • Flip all the bits • Add 1 6 (for example) 111010

  16. +5: 0 0 0 1 0 1 -5: 1 1 1 0 1 1 +ve 5 -ve Sign Bit • In 2’s complement notation, the MSB is the sign bit (as with sign-magnitude notation) • 0 = positive value • 1 = negative value 2’s complement

  17. “Complementary” Notation • Conversions between positive and negative numbers are easy • For binary (base 2)…

  18. +5 0 0 0 1 0 1 2’s C 1 1 1 0 1 0+ 1 -5 1 1 1 0 1 1 2’s C 0 0 0 1 0 0+ 1 +5 0 0 0 1 0 1 Example 2’s C +ve -ve 2’s C

  19. Negative, sign bit = 1 Zero or positive, sign bit = 0 Range for 2’s Complement • For example, 6-bit 2’s complement notation 100000 100001 111111 000000 000001 011111 -32 -31 ... -1 0 1 ... 31

  20. Ranges

  21. In General (revisited)

  22. Sign-magnitude 2’s complement -6: 10000110+6: +00000110 10001100 -6: 11111010+6: +00000110 00000000 What is -6 plus +6? • Zero, but let’s see

  23. 2’s Complement Subtraction • Easy, no special rules • Subtract?? • Actually … addition! A – B = A + (-B) add 2’s complement of B

  24. What is 10 subtract 3? • 7, but… • Let’s do it (we’ll use 6-bit values) 10 – 3 = 10 + (-3) = 7 001010+111101 000111 +3: 000011 -3: 111101

  25. 001010+000011 001101 What is 10 subtract -3? • 13, but… • Let’s do it (we’ll use 6-bit values) 10 – (-3) = 10 + (-(-3)) = 13 -3: 111101 +3: 000011

  26. M - N • M + 2’s complement of N • M + (2n - N) = M - N + 2n • If M  N • Produce an carry, which is discarded • If M < N • results in 2n - (N - M), which is the 2’s complement of (N-M)

  27. Overflow • Carry out of the leading digit • If we add two positive numbers and we get a carry into the sign bit we have a problem • If we add two negative numbers and we get a carry into the sign bit we have a problem • If we add a positive and a negative number we won't have a problem • Assume 4 bit numbers (+7 : -8)

  28. N = 4 Number Represented Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Unsigned 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Signed Mag 0 1 2 3 4 5 6 7 -0 -1 -2 -3 -4 -5 -6 -7 1's Comp 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 -0 2's Comp 0 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -1

  29. Overflow • If we add two positive numbers and we get a carry into the sign bit we have a problem 3 0011 4 0100 3 0011 4 0100 6 0110 8 1000

  30. Overflow -5 1011 -4 1100 -3 1101 -5 1011 -8 11000 -9 10111 • If we add two negative numbers and we get a carry into the sign bit we have a problem

  31. Overflow • If we add a positive and a negative number we won't have a problem 5 0101 -4 1100 -3 1101 5 0101 2 10010 1 10001

  32. Overflow • If we add two positive numbers and we get a carry into the sign bit we have a problem 3 0011 4 0100 3 0011 4 0100 6 0110 8 1000 carry in 0 carry out 0 carry in 1 carry out 0

  33. Overflow • If we add two negative numbers and we get a carry into the sign bit we have a problem -5 1011 -4 1100 -3 1101 -5 1011 -8 11000 -9 10111 carry in 1 carry out 1 carry in 0 carry out 1

  34. Overflow • If we add a positive and a negative number we won't have a problem 5 0101 -4 1100 -3 1101 5 0101 2 10010 1 10001 carry in 1 carry out 1 carry in 1 carry out 1

  35. Binary Codes • n-bit binary code • 2n distinct combinations • BCD – Binary Coded Decimal (4-bits) • 0 0000 • 1 0001 • … … • 9 1001 • BCD addition • Get the binary sum • If the sum > 9, add 6 to the sum • Obtain the correct BCD digit sum and a carry

  36. Binary Codes • ASCII code • American Standard Code for Information Interchange • alphanumeric characters, printable characters (symbol), control characters • Error-detection code • one parity bit - an even numbered error is undetected • “A” 41:100|0001 - - > • 0100|0001 (even), • 1100|0001 (odd)

  37. Binary Logic • Boolean algebra • Binary variables: X, Y • two discrete values (true or false) • Logical operations • AND, OR, NOT • Truth tables

  38. Logic Gates • Logic circuits • circuits = logical manipulation paths • Computations and controls • combinations of logic circuits • Logic Gates

  39. Timing diagram

  40. Thank You!

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