Chapter 6 Digital Arithmetic:Operations and Circuits

ECE 221 Intro. Digital SystemsFall Semester 2002

ECE Department, UMASS-AmherstProf. HillProf. Gong

Objectives

• Addition, subtraction, multiplication, and division of two binary numbers.

• Addition and subtraction of hexadecimal numbers• Difference between binary addition and OR addition.• Comparison among three different systems for

representing signed numbers.• Manipulate signed binary numbers using the 2’s –

complement system.• BCD adder circuit and addition process.• Basic operation of an arithmetic/logic unit

6-1 Binary Addition• The addition of two binary numbers is performed in

exactly the same manner as the addition of decimal numbers.

• Least-significant-digit first.• “Carry” of 1 into the next position may be needed.• 4 different cases for binary addition

positionnext into 1 ofcarry 111111

positionnext into 1 ofcarry 01011

101

000

•The operations of subtraction, multiplication, and divisionactually use only addition as their basic operation

Example

1001(9)

110(6)

(3) 011

)24(11000

1111(15)

1001(9)

)125.6(001.110

50)10.110(2.7

75)11.011(3.3

Review Question

• Add the following pairs of binary numbers.– 10110+00111– 011.101+010.010– 10001111+00000001

Representing signed numbers (6.2)Considerations: representing both positive and negative numbers; efficient computation.

Sign-magnitude system

Sign-magnitude system: represents + and – numbers, but we need to design special circuits to add positive and negative numbers.

ex: 5 + (-5) = 0

use 8 bit signed-binary numbers and add using full-adder circuits :

0 0 0 0 0 1 0 1 ( 5) + 1 0 0 0 0 1 0 1 (-5)_____________________

1 0 0 0 1 0 1 0 (-10)

1’s complement system

– Change each 0 to 1, and each 1 to 0.– Example

(45) 1 0 1 1 0 1 original binary number

(-45) 0 1 0 0 1 0 complement each bit

1 0 1 1 0 1 (45)+ 0 1 0 0 1 0 (-45)____________________ 1 1 1 1 1 1

Add one to this result, get zero.

2’s complement of a binary number:

– Take the 1’s complement of the number– Add 1 to the least-significant-bit position

numberbinary original of complement s2' 010011

complement s2' form to1 add 1

complement s1' form bit toeach complement 010010

45 of equivalentbinary 101101

Representing signed numbers using 2’s complement form

• If the number is positive, the magnitude is represented in its positional-weighted binary form, and a sign bit of 0 is placed in front of the MSB.

• If the number is negative, the magnitude is represented in its 2’s complement form, and a sign bit of 1 is placed in front of the MSB.

example

Example

• Represent each of the following signed decimal numbers as a signed binary number in the 2’s-complement system. Use a total of five bits including the sign bit.

(a) +13 (b) –9 (c) +3 (d) –2 (e) -8

Negation

• Negation is the operation of converting a postive number to its negative equivalent or a negative number to its positive equivalent.

• We negate a signed binary number by 2’s-complementing it.

• Example– Each of the following numbers is a signed binary

number in the 2’s-complement system. Determine the decimal value in each case:

(a)01100 (b) 11010 (c)10001

Special case in 2’s-complement representation

• Whenever a signed number has a 1 in the sign bit and all 0s for the magnitude bits, its decimal equivalent is –2N, where N is the number of bits in the magnitude.

• The complete range of values that can be represented in the 2’s-complement system having N magnitude bits is –2N to +(2N - 1).

• What is the range of unsigned decimal values that can be represented in a byte?

• What is the range of signed decimal values that can be represented in a byte?

Review Questions

• Represent each of the following values as an eight-bit signed number in the 2’s-complement system(a)+13 (b) –7 (c)-128

• Each of the following is a signed binary number in the 2’s-complement system. Determine the decimal equivalent for each.(a)100011 (b)1000000 (c)01111110

• What range of signed decimal values can be represented in 12 bits(including the sign bit)?

• How many bits are required to represent decimal values ranging from –50 to +50?

• What is the largest negative decimal value that can be represented by a two-byte number?

• Perform the 2’s-complement operation on each of the following.(a)10000 (b)10000000 ©1000

• Define the negation operation

6-3 Addition in the 2’s-complement system

• Case I: Two Postive Numbers.

+9 0 1001 (augend)+4 0 0100 (addend)

0 1101 (sum = +13)

Sign bits

Addition, cont.

• Case II: Positive Number and Smaller Negative Number

+9 0 1001 (augend)-4 1 1100 (addend) 1 0 0101

Sign bits

This carry is disregarded; the result is 01001(sum=+5)

Addition, cont.

• Case III: Positive Number and Larger Negative Number

-9 10111+4 00100

11011 (sum = -5)

Negative sign bit

Addition, cont.

• Case IV: two negative Numbers

-9 10111-4 11100 1 10011

Sign bit

This carry is disregarded; the result is 10011(sum =-13)

Addition, cont.

• Case V: Equal and Opposite Numbers

-9 10111+9 010010 100000

Disregard; the result is 00000(sum = +0)

Review Questions

• Assume the 2’s complement system for both questions– True or False: Whenever the sum of two signed

binary numbers has a sign bit of 1, the magnitude of the sum is in 2’s complement form.

– Add the following pairs of signed numbers. Express the sum as a signed binary number and as a decimal number• (a) 100111+111011• (b) 100111+011001

6-4 Subtraction in the 2’s-complement System

• The procedure for subtracting one binary number(the subtrahend) from another binary number(the minuend)– Negate the subtrahend. This will change the

subtrahend to its equivalent value of opposite sign.

– Add this to the minuend. The result of this addition will represent the difference between the subtrahend and the minuend.

Arithmetic Overflow

• When two positive or two negative numbers are being added, an overflow could occur if there is a carry happening to the sign-bit position.

• Overflow can occur when the minuend and subtrahend have different signs.

Review Questions

• Perform the subtraction on the following pairs of signed numbers using the 2’s-complement system. Express the results as signed binary numbers and as decimal values.

(a)01001-11010 (b)10010-10011• How can arithmetic overflow be detected when signed

numbers are being added? Subtracted?

6-5 Multiplication of Binary numbers

• The same manner as the multiplication of decimal numbers.

1001 multiplicand = 910

1011 multiplier=1110

1001

1001

0000

1001

1100011 final product = 9910

Multiplication in the 2’s-Complement System

• If the two numbers to be multiplied are positive, they are already in true binary form and are multiplied as they are.

• When the two numbers are negative, they will be in 2’s-complement form. Each is converted to a positive number, and then the two numbers are multiplied. The product is kept as a positive number and is given a sign bit of 0.

• When one of the number is positive and the other is negative, the negative number is first converted to a positive magnitude by taking its 2’s complement. The product will be in true-magnitude form, should be changed to 2’s complement form and given a sign bit of 1.

Review Question

• Multiply the unsigned numbers 0111 and 1110

6-6 Binary Division

• The same as for decimal numbers---long division

0

11

0011

011

1001

0010 11

0

100

100

100

1.00100.1010100

The division of signed numbers is handled in the same way as multiplication.

6-7 BCD ADDITION

• Sum Equals 9 or Less

9for BCD 1001 9

4for BCD 0100 4

5for BCD 0101 5

78for BCD 1000 0111 78

33for BCD 1001 0011 33

45for BCD 0101 0100 45

6-8 Hexadecimal Arithmetic

• Hex Addition– Add the two hex digits in decimal, mentally

inserting the decimal equivalent for those digits larger than 9.

– If the sum is 15 or less, it can be directly expressed as a hex digit.

– If the sum is greater than or equal to 16, subtract 16 and carry a 1 to the next digit position.

• Examples– Add the hex numbers 58 and 24– Add the hex numbers 58 and 4B– Add 3AF and 23C

Hex subtraction

• How to get the 2’s-complement of a hex number?• Convert to binary, take the 2’s complement of the binary equivalent,

and then convert back to hex.

73A

0111 0011 1010

1000 1100 0110

8C6

• A quicker procedure: subtract each hex digit from F; then add 1.

F F F

-7 -3 -A

8 C 5

+1

8 C 6

} Subtract each digit from F

Add 1

Hex equivalent of 2’s complement

Example

• Subtract 3A516 from 59216.

Review Questions

• Add 67F+2A4• Subtract 67F-2A4• Which of the following hex numbers represent

positive values: 2F, 77EC, C000, 6D, FFFF?

6-9 Arithmetic Circuits

6-10 Parallel binary adder

Parallel adder

Review Questions

• How many inputs does a full adder have? How many outputs?

• Assume the following input levels in Parallel Adder:– A4A3A2A1A0=01001; B4B3B2B1B0=00111; C0=0

(a)What are the logic levels at the outputs of FA#2?

(b)What is the logic level at the C5 output?

6-11 Design of a Full Adder

•Truth table for a full-adder circuit

K Mappings for the full-adder outputs

Complete circuitry for a full adder

6-12 Complete parallel adder with registers

Sequence of operations

• [A]=0000• [M]-> [B]• [S]*->[A]• [M]->[B]• [S]->[A]

Review Questions

• Suppose that four different four-bit numbers are to be taken from memory and added by the circuit in the last figure, how many CLEAR pulses will be needed? How many TRANSFER pulses? How many LOAD pulses?

• Determine the contents of the A register after the following sequence of operations:– [A]=0000, [0110]->[B], [S]->[A]

[1110]->[B], [S]->[A]

6-14 Integrated-Circuit Parallel Adder

Example

• Determine the logic levels at the inputs and outputs of the eight-bit adder in Figure 6-10(b) when 7210 is added to 13710.

Review Questions

• How many 74HC283 chips are needed to add two 20-bit numbers?

• If a 74HC283 has a maximum propagation delay of 30 ns from C0 to C4, what will be the total propagation delay of a 32-bit adder constructed from 74HC283s?

• What will be the logic level at C4 in Last Example?

6-15 2’s Complement system

• The operations of addition and subtraction of signed numbers can be performed using only the addition operation if we use the 2’s complement form to represent negative numbers.

Addition

Subtraction

Combined Addition and Subtraction

Review Questions

• Why does C0 have to be a 1 in order to use the adder circuit in Figure 6-12 as a subtractor?

• Assume that [A]=0011 and [B]=0010 in Figure 6-13. If ADD=1 and SUB=0, determine the logic levels at the OR gate outputs.

• Repeat question 2 for ADD=0, SUB=1• Ture or False: When the adder/Subtractor circuit is

used for subtraction, the 2’s complement of the subtrahend appears at the input of the adder.

6-16 BCD ADDER

• Add the BCD code groups for each decimal digit position; use ordinary binary addition.

• For those positions where the sum is 9 or less, the sum is in proper BCD form and no correction is needed

• When the sum of two digits is greater than 9, a correction of 0110 should be added to that sum to produce the proper BCD result. This will produce a carry to be added to the next decimal position.– A3A2A1A0 BCD code group

– B3B2B1B0 BCD code group

S4S3S2S1S0 straight binary sum

Truth table

S4 S3 S2 S1 S0

0 1 0 1 0 (10)

0 1 0 1 1 (11)

0 1 1 0 0 (12)

0 1 1 0 1 (13)

0 1 1 1 0 (14)

0 1 1 1 1 (15)

1 0 0 0 0 (16)

1 0 0 0 1 (17)

1 0 0 1 0 (18)

X=S4+S3(S2+S1)

A BCD adder

Example

• Determine the inputs and outputs when the below circuit is used to add 53810 to 24710. Assume CARRY IN=0.

Review Questions

• What are the three basic parts of a BCD adder circuit?

• Describe how the BCD adder circuit detects the need for a correction and executes it.

6-17 ALU Integrated Circuits

Operations

• CLEAR• ADD• SUBTRACT• XOR• OR• AND• PRESET

Example

• Determine the 74HC382 outputs for the following inputs: S2S1S0=010, A3A2A1A0=0100, B3B2B1B0=0001, and CN=1.

• Change the select code to 011 and repeat.

Expanding the ALU

Example

• How would the arrangement in the last figure have to be changed in order to perform the subtraction(B-A)?

Review Questions

• Apply the following inputs to the ALU of Figure 6-16, and determine the outputs: S2S1S0=001, A3A2A1A0=1110, B3B2B1B0=1001, CN=1.

• Change the select code to 011 and CN to 0, and repeat question 1.

• Change the select code to 110 and repeat question 1.• Apply the following inputs to the circuit of Figure

6-17, and determine the outputs: B=01010011, A=00011000.

• Change the select code to 111, and repeat question 4.

• How many 74HC382s are needed to add two 32-bit numbers?