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Operations on Bits

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Chapter 4

Operations

on Bits

主讲：李艳

云南农业大学

Carry------------

1

1

Number of 1s------------

None

One

Two

Three

Result------------

0

1

0

1

ARITHMETIC OPERATIONS ON INTEGERS

- Addition in Two’s Complement
Adding numbers in two’ s complement is like adding the numbers in decimal; you add column by column, and if there is a carry, it is added to the next column. When you add 2 bits, the result is 0 or 1.

Rule of Adding Integers in Two’s Complement

Add 2 bits and propagate the carry to the next column. If there is a final carry after the leftmost column addition, discard it.

Example 1

Solution

Add two numbers in two’s complement representation: (+17) + (+22) (+39)

Carry 1

0 0 0 1 0 0 0 1 +0 0 0 1 0 1 1 0

-------------------------------------Result0 0 1 0 0 1 1 1 39

Example 2

Solution

Add two numbers in two’s complement representation: (+24) + (-17) (+7)

Carry 1 1 1 1 1

0 0 0 1 1 0 0 0 +1 1 1 0 1 1 1 1

-------------------------------------Result0 0 0 0 0 1 1 1 +7

Example 3

Solution

Add two numbers in two’s complement representation: (-35) + (+20) (-15)

Carry 1 1 1

1 1 0 1 1 1 0 1 +0 0 0 1 0 1 0 0

----------------------------------Result1 1 1 1 0 0 0 1 -15

Example 4

Solution

Add two numbers in two’s complement representation: (+127) + (+3) (+130)

Carry 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1 + 0 0 0 0 0 0 1 1

----------------------------------Result 1 0 0 0 0 0 1 0 -126(Error)An overflow has occurred.

Overflow

Overflow is an error that occurs when you try to store a number that is not within the range defined by the allocation. When you add numbers in two’s complement using N bits, make sure that each number and the result are in the range defined for two’s complement representation.

Range of numbers in two’s complementrepresentation

- (2N-1) ---------- 0 ----------- +(2N-1 –1)

Two’s complement numbers visualization

When you do arithmetic operations onnumbers in a computer, remember thateach number and the result should bein the range defined by the bit allocation.

- Subtraction in Two’s Complement
One of the advantages of two’s complement representation is that there is no difference between addition and subtraction. To subtract, negate (two’s complement) the second number and add. In other words, the following two statements are the same:

Number 1 – Number 2 Number 1 + (-Number 2)

Example 5

Solution

Subtract 62 from 101 in two’s complement: (+101) - (+62) (+101) + (-62)

Carry 1 1

0 1 1 0 0 1 0 1 +1 1 0 0 0 0 1 0

----------------------------------Result0 0 1 0 0 1 1 1 39The leftmost carry is discarded.

A single bit can be either 0 or1.You can interpret 0 as the logical value false and 1 as the logical value true.

A logical operation can accept 1 or 2 bits to create only 1 bit. If the operation is applied to only one input, it is a unary operation, if it is applied to 2 bits, it is a binary operation.

We will discuss one unary operator (NOT) and three binary operators (AND, OR, XOR).

TRUTH TABLES

One way to show the result of a logical operation is with a truth table. A truth table lists all the possible input combinations with the corresponding output.

Example 7

Solution

UNARY OPERATOR

NOT Operator

The NOT operator has one input (a bit pattern). It inverts bits; that is, it changes 0 to1 and 1 to 0.

Use the NOT operator on the bit pattern 10011000.

Target 1 0 0 1 1 0 0 0NOT ------------------Result 0 1 1 0 0 1 1 1

Example 8

Solution

BINARY OPERATORS

AND Operator

It takes two inputs (two bit patterns) and creates one output (bit pattern). For each pair of input bits, the result is 1 if and only if both bits are 1; otherwise, it is 0.

Use the AND operator on bit patterns 10011000 and 00110101.

Target 1 0 0 1 1 0 0 0 AND 0 0 1 1 0 1 0 1 ------------------Result 0 0 0 1 0 0 0 0

Inherent Rule of the AND Operator

One interesting point about the AND operator is that if a bit in one input is 0, you do not have to check the corresponding bit in the other input, you can quickly conclude that the result is 0.

Example 9

Solution

OR Operator

It takes two inputs (two bit patterns) and creates one output (bit pattern). For each pair of input bits, the result is 0 if and only if both bits are 0; otherwise, it is 1.

Use the OR operator on bit patterns 10011000 and 00110101.

Target 1 0 0 1 1 0 0 0OR0 0 1 1 0 1 0 1-------------------Result 1 0 1 1 1 1 0 1

Inherent rule of the OR Operator

One interesting point about the OR operator is that if a bit in one input is 1, you do not have to check the corresponding bit in the other input; you can quickly conclude that the result is 1.

Example 10

Solution

XOR Operator

It takes two inputs (two bit patterns) and creates one output (bit pattern). For each pair of input bits, the result is 0 if and only if both bits are equal; otherwise, it is 1.

Use the XOR operator on bit patterns 10011000 and 00110101.

Target 1 0 0 1 1 0 0 0XOR0 0 1 1 0 1 0 1-------------------Result 1 0 1 0 1 1 0 1

Inherent Rule of the XOR Operator

One interesting point about the XOR operator is that if a bit in one input is1, the result is the inverse of the corresponding bit in the other input.

APPLICATIONS

The three logical binary operations can be used to modify a bit pattern. They can unset, set, or reverse specific bits. The bit pattern to be modified is ANDed, ORed, or XORed with a second bit pattern, which is called the mask. The mask is used to modify another bit pattern.

Unsetting Specific Bits

One of the applications of the AND operator is to unset (force to 0) specific bits in a bit pattern. To do so, use an unsetting mask with the same bit length.

The rules for constructing an unsetting mask can be summarized as follows:

1. To unset a bit in the target bit pattern, use 0 for the corresponding bit in the mask.

2. To leave a bit in the target bit pattern unchanged, use 1 for the corresponding bit in the mask.

Example 11

Solution

Use a mask to unset (clear) the 5 leftmost bits of a pattern. Test the mask with the pattern 10100110.

The mask is 00000111.

Target 1 0 1 0 0 1 1 0ANDMask 0 0 0 0 0 1 1 1-------------------Result 0 0 0 0 0 1 1 0

Example 12

Solution

Imagine a power plant that pumps water to a city using eight pumps. The state of the pumps (on or off) can be represented by an 8-bit pattern. For example, the pattern 11000111 shows that pumps 1 to 3 (from the right), 7 and 8 are on while pumps 4, 5, and 6 are off. Now assume pump 7 shuts down. How can a mask show this situation?

Use the mask 10111111 to AND with the target pattern. The only 0 bit (bit 7) in the mask turns off the seventh bit in the target.

Target 1 1 0 0 0 1 1 1ANDMask1 0 1 1 1 1 1 1-------------------Result 1 0 0 0 0 1 1 1

Setting Specific Bits

One of the applications of the OR operator is to set (Force to 1) specific bits in a bit pattern. To do so, use a setting mask with the same bit length.

The rules for constructing a setting mask can be summarized as follows:

1. To set a bit in the target pattern, use 1 for the corresponding bit in the mask.

2. To leave a bit in the target bit pattern unchanged, use 0 for the corresponding bit in the mask.

Example 13

Solution

Use a mask to set the 5 leftmost bits of a pattern. Test the mask with the pattern 10100110.

The mask is 11111000.

Target 1 0 1 0 0 1 1 0 ORMask 1 1 1 1 1 0 0 0------------------Result1 1 1 1 11 1 0

Example 14

Solution

Using the power plant example, how can you use a mask to to show that pump 6 is now turned on?

Use the mask 00100000.

Target 1 0 0 0 0 1 1 1 ORMask0 0 1 0 0 0 0 0 ------------------Result 1 0 1 0 0 1 1 1

Flipping Specific Bits

One of the applications of XOR is to flip bits, which means to change the value of specific bits from 0s to 1s, and vice versa.

The rules for constructing an XOR mask can be summarized as follows:

1. To flip a bit in the target bit pattern, use 1 for the corresponding bit in the mask.

2.To leave a bit in the target bit pattern unchanged, use 0 for the corresponding bit in the mask.

Example 15

Solution

Use a mask to flip the 5 leftmost bits of a pattern. Test the mask with the pattern 10100110.

Target 1 0 1 0 0 1 1 0XORMask 1 1 1 1 1 0 0 0 -------------------Result 0 1 0 1 1 1 1 0

Another common operation on bit patterns is the shift operation. A bit pattern can be shifted to the right or to the left. The right-shift operation discards the rightmost bit, shift every bit to the right, and inserts 0 as the leftmost bit. The left-shift operation discards the leftmost bit, shifts every bit to the left, and inserts 0 as the rightmost bit.

Shift operations can only be used when a pattern represents an unsigned number.

Example 16

Solution

Show how you can divide or multiply a number by 2 using shift operations.

If a bit pattern represents an unsigned number, a right-shift operation divides the number by two. The pattern 00111011 represents 59. When you shift the number to the right, you get 00011101, which is 29. If you shift the original number to the left, you get 01110110, which is 118.

Example 17

Solution

Use a combination of logical and shift operations to find the value (0 or 1) of the fourth bit (from the right).

Use the mask 00001000 to AND with the target to keep the fourth bit and clear the rest of the bits.

Target a b c d e f g hANDMask 0 0 0 0 1 0 0 0------------------Result 0 0 0 0e 0 0 0

Shift the new pattern three times to the right 0000e000 00000e00 000000e0 0000000e

Now it is easy to test the value of the new pattern as an unsigned integer. If the value is 1, the original bit was 1; otherwise the original bit was 0.

SUMMARY

- You can perform arithmetic or logical operations on bits.
- Most computers use the two’s complement method of integer representation.
- If there is a carry after addition of the leftmost digits, the carry is discarded.
- To subtract in two’s complement, just negate the number to be subtracted and add.
- Numbers to be added must be within the range defined by the bit allocation.
- The term overflow describes a condition in which a number is not within the range defined by the bit allocation.
- Logical operation on bits can be unary (one input) or binary (two inputs).
- The unary NOT operator inverts its input.

SUMMARY (continued)

- The result of the binary AND operation is true only if both inputs are true.
- The result of the binary OR operation is false only if both inputs are false.
- The result of the binary XOR operation is false only if both inputs are the same.
- A mask is a bit pattern that is applied to a target bit pattern to achieve a specific result.
- To unset (clear) a bit in a target bit pattern, set the corresponding mask bit to 0 and use the AND operator.
- To set a bit in a target bit pattern, set the corresponding mask bit to 1 and use the OR operator.
- To flip a bit in a target bit pattern, set the corresponding mask bit to 1 and use the XOR operator.

Thanks！