Sign Magnitude

Objectives

1. Discuss the sign magnitude representation of negative numbers.
2. Discuss mathematics involved with this system.

Notes

• Once we learn subtraction, we naturally fall to the need to represent negative numbers.
• How do we represent negative integers in real life?
• The system we use is called sign-magnitude.
  • Think number line.
  • A sign tells us if we are to the left or right of 0
  • The magnitude tells us how far.
  • This is a natural extension of idea a positive integer is a distance on the right away from 0.
• But is this a good system?
• We need a side-trip to discuss representation
  • With 1 bit we can represent two things
    • a 0 could be false, a 1 could be true
    • A 0 could be night, a 1 could be day. ...
  • With two bits we have four different values
    • 00, 01, 10, 11
    • Therefore with two bits we can represent 4 different objects.
  • How about three bits?
  • In general, each time we add a bit, we double the number of values.
    • with n bits we have 2ⁿ values, so we can represent 2ⁿ different things.
  • Remember the inverse of exponents is logs.
  • So if we have k objects, we need ⌈ log₂k⌉ bits.
    • So if we need to represent 16 things, we need log₂ 16 = 4 bits.
    • And if we need to represent 20 things, we need ⌈log₂20⌉ = 5 bits.
  • This explains why many things we do in computing are powers of two, or multiples of powers of two.
• So if we have 8 bits to represent a number
  • We will need one of them to represent the sign.
  • And we will have seven to represent the magnitude
  • So we can represent
    • 0 through 1111111₂ or 2⁸-1
    • and -0 through -1111111 or -2⁸ -1
  • If we assume that 0 is + and 1 is -
    • what is 00000000?
    • What is 10000000?
    • Well, that is not good.
  • What do we do about -0 in real life? (we ignore it)
  • But that is harder to do with hardware.
• In general if we have n bits to represent a number
  • What is the range for unsigned numbers?
  • What is the range for signed numbers?
  • Do we lose anything by adding the sign?
• How do we do math on signed numbers?
  1. 23 + 15 , add, answer positive
  2. 23 - 15 , subtract, answer is positive
  3. 23 + -15, change to subtraction, answer positive
  4. 23 - -15, change to addition, answer positive
  5. -23 + 15, change to subtraction, answer negative
  6. -23 - 15, change to addition, answer negative
  7. -23 + -15, addition, answer negative
  8. -23 - -15, subtraction, answer negative
• What about the hard case?
  1. 15 + 23, add, answer positive.
  2. 15 - 23, , change to -23 + 15, answer negative.
  3. 15 + -23, , change to -23 + 15, answer negative.
  4. 15 - -23, , change to 23 + 15, answer positive.
  5. -15 + 23, , change to 23 - 15, answer positive.
  6. -15 - 23, , change o -23 + -15, answer negative.
  7. -15 + -23, ,change to -23 + -14, answer negative.
  8. -15 - -23, , change to 23 - 15, answer positive.
• This is bad, we have to compare signs, sometimes magnitudes and decide which computation to do based on the result.
• Looking ahead, we could do this in hardware, but it would be complex.
• We would need an adder and a subtractor.
• We might need to find a better representation.