Sign Magnitude

Objectives

We would like to :

Discuss the sign magnitude representation of negative numbers.
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.
  - If a^b = c
  - log_a(a^b) = b
  - or log_a(c) = b
  - By the way, log_b(a) = log_c(a)/log_c(b)
    - Frequently we don't have access to log₂(n)
    - So we will take $\frac{log_{10}(n)}{log_{10}(2)}$
    - Or we will take $\frac{ln(n)}{ln(2)}$
    - Remember $ln(n) = log_e(n)$
    - We will need this in algorithms, so some day I might prove it to you.
- Floors and ceilings
  - remember ⌈ n ⌉ is the integer greater than or equal to n
    - So ⌈ 5 ⌉ = 5
    - So ⌈ 6.0001 ⌉ = 7
  - remember ⌊ n ⌋ is the integer less than or equal to n
    - So ⌊ 6.0001 ⌋ = 6
    - So ⌊ 2 ⌋ = 2
  - Floor means truncate
- 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⌉ ≈ ⌈4.3219⌉ = 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.