2’s Complement and Floating-Point

2¡¯s Complement and

Floating-Point

CSE 351 Section 3

Two¡¯s Complement

?An n-bit, two¡¯s complement

number can represent the

range [?2 ??1 , 2 ??1 ? 1].

?

Note the asymmetry of this

range about 0 ¨C there¡¯s one more

negative number than positive

?Note what happens when

you overflow

4-bit two¡¯s complement range

Understanding Two¡¯s Complement

?

An easier way to find the decimal value of a two¡¯s complement number:

~x + 1 = -x

?

We can rewrite this as x = ~(-x - 1), i.e. subtract 1 from the given number,

and flip the bits to get the positive portion of the number.

?

Example: 0b11010110

?

Subtract 1: 0b11010110 - 1 = 0b11010101

?

Flip the bits: ~0b11010101 = 0b00101010

?

Convert to decimal: 0b00101010 = (32+8+2)10 = 4210

?

Multiply by negative one, Answer: -4210.

Two¡¯s Complement: Operations

1. Convert numbers to binary

int

int

int

int

x = 0xAB;

y = 17;

z = 5;

result = ~(x | y) + z;

What is the value of

result in decimal?

?

0xAB = 0b10101011

?

1710 = 0b00010001

2. Compute x | y

0000 ... 1010 1011

| 0000 ... 0001 0001

-------------------0000 ... 1011 1011

3. Compute ~(x | y) (flip the bits)

~ 0000 0000 0000 0000 0000 0000 1011 1011

----------------------------------------1111 1111 1111 1111 1111 1111 0100 0100

Two¡¯s Complement: Operations

4. Compute ~(x | y) + z

int

int

int

int

x = 0xAB;

y = 17;

z = 5;

result = ~(x | y) + z;

What is the value of

result in decimal?

?

You can either convert to decimal before or after

adding z

?

1710 = 0b00010001

?

Convert

111111111111111111111111010001002 to

decimal

(hint: -x = ~x + 1)

?

~111111111111111111111111010001002 =

0000000000000000000000000101110112

?

~0000¡­¡­101110112 + 1 = 0000¡­¡­101111002

?

-x = 0000¡­¡­101111002

?

x = 1000¡­¡­101111002

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