Integers. Integer Storage Since Binary consists only of 0s and 1s, we can’t use a negative sign ( - ) for integers. Instead, the Most Significant Bit.

Integers

Integer Storage

Since Binary consists only of 0s and 1s, we can’t use a negative sign ( - ) for integers.

Instead, the Most Significant Bit is used to represent the sign.

This way, half the combinations in a fixed length of bits can be used to represent negative values.

But which value of the sign bit (0 or 1) will represent a negative number?

Integers

2’s Complement Notation

2’s Complement Notation(examples in 8 bits to save space)

Fixed length notation system.

Uses 1 to represent negative values.

Since 1 is always greater than 0, the largest non-negative value: 01111111 the smallest non-negative value: 00000000 the largest negative value: 11111111 the smallest negative value: 10000000

2’s Complement Notation(examples in 8 bits to save space)

What is the decimal equivalent of these?

The largest non-negative value: 01111111

The smallest non-negative value: 00000000

The largest negative value: 11111111

The smallest negative value: 10000000

2’s Complement Notation(examples in 8 bits to save space)

What is the decimal equivalent of these?

The largest non-negative value: 01111111

+127 The smallest non-negative value: 00000000

The largest negative value: 11111111

The smallest negative value: 10000000

2’s Complement Notation(examples in 8 bits to save space)

What is the decimal equivalent of these?

The largest non-negative value: 01111111

+127 The smallest non-negative value: 00000000

+0 The largest negative value: 11111111

The smallest negative value: 10000000

2’s Complement Notation(examples in 8 bits to save space)

What is the decimal equivalent of these?

The largest non-negative value: 01111111

+127 The smallest non-negative value: 00000000

+0 The largest negative value: 11111111

-1 The smallest negative value: 10000000

2’s Complement Notation(examples in 8 bits to save space)

What is the decimal equivalent of these?

The largest non-negative value: 01111111

+127 The smallest non-negative value: 00000000

+0 The largest negative value: 11111111

-1 The smallest negative value: 10000000

-128

2’s Complement Notation

The representations of non-negative integers in 2’s Complement look the same as they do for Natural numbers.

However, negative values look VERY different than we might expect.

2’s Complement Notation

Complementary numbers sum to 0. Decimal is a Signed Magnitude system so

complements have the same magnitude but different signs: 5 and -5, for example.

2’s Complement is a Fixed Length system. There are no signs, so to find a number’s complement, another technique is needed.

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length 00000101

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length 00000101

“flip the bits” (1 → 0, 0 → 1)

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length 00000101

“flip the bits” (1 → 0, 0 → 1) 11111010

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length 00000101

“flip the bits” (1 → 0, 0 → 1) 11111010

add 1 to the new pattern +1

2’s Complement Notation

One such technique is to simply change each bit to its opposite, and then add 1.

To find the 2’s complement notation for -5:

Represent +5 in fixed length 00000101

“flip the bits” (1 → 0, 0 → 1) 11111010

add 1 to the new pattern +1

to produce -5 11111011

2’s Complement Notation

Complementary numbers sum to 0.

2’s Complement Notation

Complementary numbers sum to 0.

So if to +5 00000101

2’s Complement Notation

Complementary numbers sum to 0.

So if to +5

we add -5

00000101

+11111011

2’s Complement Notation

Complementary numbers sum to 0.

So if to +5

we add -5

we should get

00000101

+11111011

1 00000000discard the carry bit to retain the fixed length