Notes

We will NOT ask you to do arithmetic or conversions on sign-magnitude or 1's complement numbers

Unsigned Integer

Unsigned Integer is a 32-bit datum that encodes a nonnegative integer in the range [0 to 4294967295].

Binary to Decimal Conversion

• $0100 0100_2 = 68_{10}$

  128	64	32	16	8	4	2	1
  0	1	0	0	0	1	0	0

• $0011 1111_2 = 63_{10}$

  128	64	32	16	8	4	2	1
  0	0	1	1	1	1	1	1

Binary Addition and Subtraction

Carry in if the bit value is > 1. Borrow if < 0.

• $00001111_2+00010001_2=00100000_2$
  • 15+17=32

• $110000_2-000011_2=101101_2$
  • 48-3=45

Signed Integer

Signed Integer is a 32-bit datum that encodes an integer in the range [-2147483648 to 2147483647] —> Can represent negative numbers.

• Table representation of the three main methods

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

Signed Magnitude

Flip the leading 0 to 1 to represent the negative sign.

1's Complement

Flip all the bits to represent the corresponding negative number.

2's Complement

Flip all the bits + 1 to represent the corresponding negative number

$-n=\backsim n + 1$

• What is 2's compliment representation for -13?

  $+13_{10} = 01101_{2}$

  $\backsim 01101_{2}=10010_{2}$

  $10010_{2}+1{2}=10011_{2}=-13_{10}$

  Binary Addition:

  $01101$

  +$10011$

  =$00000$

• Used in most computers today

• No need for a hardware subtractor, just a bitwise complement

• There exist only one 0

• Addition still works as expected from binary arithmetic (if you don’t wrap)

• The first bit still indicates the sign (as long as you will tolerate zero being positive)

• Easy calculation(Easy to find compliment of a number)
  • Since finding the complement is easy, there’s no need for subtraction: just complement and add

Sign Extension

• Sign-extension is performed in order to be able to operate on bit patterns of different lengths. It does not affect the values of the numbers being represented.

• Positive Integer: +5 can be represented as 000101 or 0000000000000101

• Negative Integer: -5 can be represented as 11101 or 1111111111111011

• Example: +5 + -5 = 0
  • Incorrect: - not flipping leading 0s when representing negative integers

    $0000000000000101$

    +$0000000000111011$

    =$0000000001000000$

  • Correct: Flipping leading 0s when representing negative integers.

    $0000000000000101$

    +$1111111111111011$

    =$0000000000000000$

• More Sign Extension Examples
  • Add 3 and 64 (0011 and 01000000)
  • $0011+01000000=01000011$
  • Add -3 and 64 (1101 and 01000000)
    • $11111101+01000000=00111101$
      • Throw away the carried out 1

Fractional Binary Numbers

$1101_{\land}1100=10.75_{10}$

Questions & Answers

Conversions

• What's the formula for negating a binary number in 2's complement?

  Invert all the bits and add one to the result

• Convert the 2's complement binary number 0b11101010 into decimal

• Convert the decimal number -72 into 8-bit 2's complement binary

• How many bits would we need to represent the decimal number -1000 in 2's complement binary?

In 6-bit binary, what’s the range of possible values you can represent with:

• An unsigned number?

• A sign-magnitude number?

• A 1’s complement number?

• A 2’s complement number?

In n-bit binary, what’s the range of possible values you can represent with:

• An unsigned number?

• A sign-magnitude number?

• A 1’s complement number?

• A 2’s complement number?

On a conceptual level, why do we prefer 2’s complement over the other signed representations?

• Used in almost all computers manufactured today

• Allows for low cost, high performance circuitry to do math operations

• There’s no need for a hardware subtractor, just a bitwise complement

• Useful Properties:
  • There is only one zero
  • Addition still works as expected from binary arithmetic (if you don’t wrap)
  • The first bit still indicates the sign (as long as you will tolerate zero being positive)
  • Finding the complement (additive inverse) of a number is easy
  • Since finding the complement is easy, there’s no need for subtraction: just complement and add

• Useful new pattern:
  • You can find the two’s complement of any number by flipping the bits and adding 1.
  • And a two’s complement conveniently represents the additive inverse of the originally represented number!

How does one compute the additive inverse of a two’s complement number?

Take the boolean NOT of each bit and add 1