Integers: 2's complement

2's complement (2C) may not be intuitively obvious, but it has nice properties

Negation

1. flip all bits

2. add 1 (ignoring any carry out of the msb)

Example: 11ten is 01011 in 5-bit unsigned binary.  Call this number B.

1. flip bits:

10100

~B

2. add 1:

+       1

10101

-B

What do we get if we add this to the original number?

01011

B

+

10101

+    -B

00000

0

If we negate B (-B) using 2C and negate again (--B) we get the original value B back.

1. flip bits:

01010

~(-B)

2. add 1:

+       1

01011

B

Other ways to do N-bit 2C negation:

1. Find the rightmost 1 bit, then flip all the higher bits (bits to the left).

Example:

01011

Rightmost 1 bit is b0

flip bits to the left:

10101

Why does this work?

Consider N-bit value value: B = bN-1bN-2 . . . 1 0 . . . 0

where bk is 1 and all bi are 0 for i < k

If we create the negation -B by flipping the bits to the left of bk, then

-B = ~bN-1~bN-2 . . . 1 0 . . . 0

What happens when we add these 2 numbers?

For each bi, i < k, the sum of the bits is 0 + 0 = 0

For bit k, the sum of the bits is 1 + 1 = 10, so the result is 0 with carry 1

Now, for i > k, each bit is either 0 or 1 in B and the opposite in -B

So, the result is 1 + 0 + carry bit 1 for a result of 0 and a carry 1

For the leftmost bit position N-1, the result is 0, and the final carry is ignored.

Therefore, the result consists of all 0's, which must be true for B + (-B)

2. Subtract B from value: 1 followed by N 0's.

Example:

01011

subtract from 100000:

100000

-    01011

10101

Odometer (Ferris Bueller) view:

Ferris Bueller's Day Off

Consider a 4-digit decimal odometer.

What is the largest milage it can show?

9999

What happens if we go 1 more mile?

0000

Now, think of starting at 0000 and going backward, like Ferris:

Go back 1 mile, and the odometer reads 9999, 2 miles 9998, and so forth.

N-bit 2's complement is really like a binary odometer:

Go forward from 00 . . . 0 for positive numbers.

Go backward for negative numbers.

What's really going on here mathematically?

What is the value of 1 followed by N 0's?  2N

So, N-bit 2C is really based on arithmetic modulo 2N

This is why 2C has nice properties for doing arithmetic.

Advantages of 2C representation:

Only 1 zero

Add using same hardware as unsigned binary

Subtract by taking complement and adding.

Why is there only 1 zero?

Consider the 5-bit positive zero 00000

1. flip bits:

11111

2. add 1:

+       1

00000

This is negative 0, but it has the same representation :)

Range of values:

Maximum positive number: 2N-1 - 1

Minimum negative number: - (2N-1)

Total number of values (including 0): 2N

Converting from base 10 to N-bit 2C

1. Convert to N bits UB

2. If number has minus sign, negate:

flip bits

add 1

Converting from N-bit 2C to base 10

1. If msb is 1 (negative number), negate

2. Convert result (or original value) as UB to base 10

3. If number was negative, insert minus sign

Conclusion: 2C rules!

Most computer representations of signed integers use 2C.