Section 2.3 - Introduction to computer arithmetic

1. x < y is not the same as x - y < 0
2. x + 1 < x
3. x + y is not the same as y + x
4. x + (y + z) is not the same as (x + y) + z

• Closure: for all integers x and y, x + y is an integer
• Commutativity: for all integers x and y, x + y = y + x
• Associativity: for all integers x, y, and z, (x + y) + z = x + (y + z)
• Identity: for any integer, x, x + 0 = x.
• Inverse: for any integer, x, this is an integer, y satisfying x + y = 0

Section 2.3.1 - Unsigned addition

• w bits, maximum value is 2^w - 1.
• It might take as many as w+1 bits to represent the value of x + y
• Addition is done modulo 2^w
• Unsigned addition is an abelian group.
• When x + y does not produce the correct result, we say that overflow has occurred.
• In C, overflow is not detected.
• How can you test whether adding x and y will produce an overflow?

1. What is the maximum (mathematical) sum of two w-bit unsigned integers?
2. What are the min and max values on the right side of the figure?
3. What does 2^w -1 get mapped to?
4. What does 2^w get mapped to?
5. What does the maximum (mathematical) sum get mapped to?
6. Write an expression that sums two unsigned values and gives the largest possible result.
   Note: UINT_MAX + UINT_MAX does not work!

• An arithmetic operation on unsigned integers produces an unsigned integer.
• If x and y are unsigned, x - y will be unsigned.
• The result will be correct only when x >= y.
• What is the output of the following program: Example 2.3.1 Subtraction
  Example 2.3.1 Subtraction Output
• How would the output change if w, x, y, and z were declared as int?
  Example 2.3.1 Subtraction Output Using Ints

Section 2.3.2 - Two's complement addition

• Two's complement addition is done by the same hardware as unsigned addition.
• This means that the arguments are treated as unsigned, added as unsigned values (modulo 2^w), and the result is treated as signed.
• This is illustrated by the following formula:
  x +^t_w y = U2T_w(T2U_w(x) +^u_w T2U_w(y))

1. What is the maximum (mathematical) sum of two w-bit signed integers?
2. What is the minimum (mathematical) sum of two w-bit signed integers?
3. What are the min and max values on the right side of the figure?
4. What does 2^w-1 -1 get mapped to?
5. What does -2^w-1 get mapped to?
6. What does the maximum (mathematical) sum get mapped to?

Section 2.3.3 - Two's complement negation

• Every two's complement number has a negative
• The only special case is the smallest two's complement value which is negative and is the negative of itself.
• Three ways of finding the two's complement negative, x:
  1. Calculate 2^w - x as an unsigned quantity
  2. Take the ones' complement and add 1
  3. Find the position of the rightmost 1, and complement all bits to the left of it.
• Suppose x is an int and x == -x is true? What is x?

Section 2.3.4 - Unsigned multiplication

Section 2.3.5 - Two's complement multiplication

• Two's complement multiplication is done using the same hardware as unsigned multiplication.
• x *^t_w y = U2T((x•y) mod 2^w)
• It will give the correct result if the mathematical product fits in a w-bit two's complement value.

Section 2.3.6 - Multiplying by constants

• A left shift by k bits is equivalent to multiplying by 2^k.
• Works for unsigned numbers as long as no 1-bit is shifted out.
• Works for two's complement numbers as long as a the sign bit does not change (treating the shift as one bit at a time).
• If a positive constant has a small number of 1 bits, it can be faster to multiply by shifting.
  Example: 49 = [110001] = 2⁵ + 2⁴ + 2⁰
  49*x = (x << 5) + (x << 4) + x
• If a postive constant has a large number of 1 bits in a row we can use subtraction.
  Example: 78 = [1001110] = 2⁶ + 2⁴ - 2;
  78*x = (x << 6) + (x << 4) - (x << 1)

Section 2.3.7 - Dividing by powers of 2

• A logical right shift of an unsigned number by k bits is equivalent to dividing by 2^k.
• An arithmetic right shift of a two's complement number by k bits is equivalent to dividing by 2^k but may not round it correctly.
• Recall that C does not require that right shifts of signed values by done arithmetically.

Section 2.3.8 - Final Thoughts