Number Systems and Number Systems and

Unit 01 2

Unit 1 Unit 1

Introduction Introduction

Number Systems and Number Systems and

Conversion

Conversion

Unit 01 3

Unit 1 Unit 1

zz

Introduction Introduction

Number Systems and Conversion Number Systems and Conversion

‹‹1.1 Digital Systems and Switching Circuits1.1 Digital Systems and Switching Circuits

‹‹1.2 Number Systems and Conversion1.2 Number Systems and Conversion

‹‹1.3 Binary Arithmetic1.3 Binary Arithmetic

‹‹1.4 Representation of Negative Numbers1.4 Representation of Negative Numbers

9Addition of 29Addition of 2’’s Complement Numberss Complement Numbers 99Addition of 1Addition of 1’’s Complement Numberss Complement Numbers

‹‹1.5 Binary Codes1.5 Binary Codes

Unit 01 4

Digital Systems and Switching Circuits Digital Systems and Switching Circuits

zz

Digital system Digital system

‹‹The physical quantities or signals can The physical quantities or signals can assume only

assume only discretediscrete valuesvalues

‹‹Greater accuracyGreater accuracy zz

Analog system Analog system

‹‹The physical quantities or signals may vary The physical quantities or signals may vary continuously

continuously over a specified rangeover a specified range

Unit 01 5

Unit 01 6

Digital Systems and Switching Circuits Digital Systems and Switching Circuits

zz Design of digital systemsDesign of digital systems

‹‹ System designSystem design

99Breaking the overall system into subsystemsBreaking the overall system into subsystems 99Specifying the characteristics of each subsystemSpecifying the characteristics of each subsystem

9 E.g. digital computer : 9E.g. digital computer : memory units, arithmetic unit, I/O memory units, arithmetic unit, I/O devices, control unit

devices, control unit

‹‹ Logic designLogic design

99Determining how to interconnect basic logic building blocks to Determining how to interconnect basic logic building blocks to perform a specific function

perform a specific function

9 E.g. arithmetic unit : binary addition: 9E.g. arithmetic unit : binary addition: logic gates, Fliplogic gates, Flip--Flops, Flops, interconnections

interconnections

‹‹ Circuit designCircuit design

99Specifying the interconnection of specific components such as Specifying the interconnection of specific components such as resistors, diodes, and transistors to form a gate, flip

resistors, diodes, and transistors to form a gate, flip--flop or other flop or other logic building block

logic building block

9 E.g. Flip9E.g. Flip--Flop: Flop: resistors, diodes, transistorsresistors, diodes, transistors

Unit 01 7

Digital Systems and Switching Circuits Digital Systems and Switching Circuits

zz Many of subsystems of a digital system take the Many of subsystems of a digital system take the form of a switching network

form of a switching network

‹‹ Switching NetworksSwitching Networks

99Combinational NetworksCombinational Networks

•• No memoryNo memory

99Sequential NetworksSequential Networks

•• Combinational Circuits + MemoryCombinational Circuits + Memory

Unit 01 8

Number Systems and Conversion Number Systems and Conversion

zz

Positional notation Positional notation Base 10:

Base 10:

Base 2:

Base 2:

Unit 01 9

Number Systems and Conversion Number Systems and Conversion

zz

Base Base R R : :

Any positive integer

Any positive integer R R ( ( R R >1) can be chosen as >1) can be chosen as the radix or base of a number system.

the radix or base of a number system.

where . where .^{0 ≤ ai ≤ R − 1}

Unit 01 10

Number Systems and Conversion Number Systems and Conversion

Example:

Example:

For bases greater than 10, more than 10 symbols are needed For bases greater than 10, more than 10 symbols are needed to represent the digits. In hexadecimal (base 16),

to represent the digits. In hexadecimal (base 16), AA presents presents 1010₁₀₁₀, B, B presents 11presents 11₁₀₁₀, C , C presents 12presents 12₁₀₁₀, D, D presents 13presents 13₁₀₁₀, E, E

presents 14

presents 14₁₀₁₀, F, F presents 15presents 15₁₀₁₀..

Unit 01 11

Number Systems and Conversion Number Systems and Conversion

zz

Convert a decimal Convert a decimal integer integer to base to base R R

This process is continued until we finally This process is continued until we finally obtain

obtain a a

2 3

3 1

4 4

1 2 3

1 2

2 1

3 3

1 1 2

0 1

1 1

2 2

1 1

remaider

,

remaider

,

remaider

,

a Q

a R

a R

a R

R a Q

a Q

a R

a R

a R

R a Q

a Q

a R

a R

a R

R a N

n n

n n

n n

n n

n n

n n

= +

+ +

+

=

= +

+ +

+

=

= +

+ +

+

=

−

−

−

−

−

−

−

−

−

L L L

Unit 01 12

Number Systems and Conversion Number Systems and Conversion

zz Example : Convert 53Example : Convert 53₁₀₁₀ to binary.to binary.

Unit 01 13

Number Systems and Conversion Number Systems and Conversion

zz

Convert a decimal fraction to base Convert a decimal fraction to base R R

This process is continued until we have This process is continued until we have obtained a

obtained a sufficient sufficient number of digits. number of digits.

Unit 01 14

Number Systems and Conversion Number Systems and Conversion

zz

Example: Convert 0.625 Example: Convert 0.625

to binary. to binary.

Unit 01 15

Number Systems and Conversion Number Systems and Conversion

zz Example: Convert 0.7Example: Convert 0.7_{10 10} to binary.to binary.

Unit 01 16

Number Systems and Conversion Number Systems and Conversion

zz Example: Convert 231.3Example: Convert 231.3₄₄ to base 7.to base 7.

Unit 01 17

Number Systems and Conversion Number Systems and Conversion

zz

Conversion from binary to hexadecimal Conversion from binary to hexadecimal ( and conversely)

( and conversely)

‹‹One hexadecimal digit corresponds to four One hexadecimal digit corresponds to four bibinary digitnary digits (bits (bits)s)

D D

6 0

3

) 1101 0110

0000 0011

( )

. 306

( ₁₆ = ⋅ ₂

Unit 01 18

Binary Arithmetic Binary Arithmetic

zz AdditionAddition

‹‹ Example: Add 13Example: Add 13₁₀₁₀ and 11and 11₁₀₁₀ in binary.in binary.

Unit 01 19

Binary Arithmetic Binary Arithmetic

zz SubtractionSubtraction

‹‹ Examples:Examples:

Unit 01 20

Binary Arithmetic Binary Arithmetic

zz

Multiplication Multiplication

‹‹Example:Example:

Unit 01 21

Binary Arithmetic Binary Arithmetic

zz

Division Division

‹‹Example: 145/11=13 Example: 145/11=13 --- 22

Unit 01 22

Representation of Negative Numbers Representation of Negative Numbers

zz

2 2 ’ ’ s Complement Number System s Complement Number System

‹‹Positive Number Positive Number N N

99 N N is represented by a 0 followed by the is represented by a 0 followed by the magnitude.

magnitude.

‹‹Negative Number Negative Number ––NN

99––N N is represented by its 2is represented by its 2’’s complement, s complement, NN*. If the word length is n,*. If the word length is n,

NN*=2*=2ⁿⁿ--NN

Unit 01 23

Representation of Negative Numbers Representation of Negative Numbers

zz 11’’s Complement Number Systems Complement Number System

‹‹ Positive Number Positive Number N N

99 N N is represented by a 0 followed by the is represented by a 0 followed by the magnitude

magnitude

‹‹ Negative Number Negative Number ––NN

9–9–N N is represented by its 1is represented by its 1’’s complement . s complement . If the word length is

If the word length is nn,,

N*=2N*=2ⁿⁿ-N-N=(2=(2ⁿⁿ--11--N)+1= +1N)+1= +1N

N N = ( 2

− 1 ) −

N

Unit 01 24

Representation of Negative Numbers Representation of Negative Numbers

zz

Sign and Magnitude Binary Numbers Sign and Magnitude Binary Numbers

Unit 01 25

Addition of 2

Addition of 2 ’ ’ s Complement Numbers s Complement Numbers

zz Addition of Addition of nn--bit signedbit signed binary numbersbinary numbers

‹‹ Any carry from the sign position is ignored.Any carry from the sign position is ignored.

‹

‹ n=4n=4

Unit 01 26

Addition of 2

Addition of 2 ’ ’ s Complement Numbers s Complement Numbers

).

( is result the

so

, 2 g subtractin to

equivalent is

carry last away the Throwing

carry) , (

2 ) (

2

) 2

(

n

*

A B

A B A

B

B A B

A B A

n n

n

−

>

≥

− +

=

+

−

= +

= +

−

Unit 01 27

Addition of 2

Addition of 2 ’ ’ s Complement Numbers s Complement Numbers

).

( of tion representa correct

the is which

, ) (

) (

2 yields carry last the Discarding

carry) ,

2 (

2 ) (

2 2

) 2 ( ) 2 (

* 1

*

*

B A

B A B A B

A B

A

B A

B A B A

n n n

n n

n n

+

−

+

= +

−

≤ +

≥ +

− +

=

− +

−

= +

=

−

−

−

Unit 01 28

Addition of 2

Addition of 2 ’ ’ s Complement Numbers s Complement Numbers

Example: Add

Example: Add --8 and +19 in 28 and +19 in 2’’s complements complement for a word length of n=8.

for a word length of n=8.

Unit 01 29

Addition of 1

Addition of 1 ’ ’ s Complement Numbers s Complement Numbers

zz Addition of Addition of nn--bit signedbit signed binary numbersbinary numbers

‹‹ Add the Add the last carrylast carry ( ( endend--around carryaround carry) to the n-) to the n- bit sum in the position furthest to the right.

bit sum in the position furthest to the right.

‹‹ n = 4

Unit 01 30

Addition of 1

Addition of 1 ’ ’ s Complement Numbers s Complement Numbers

).

( is result the

so

, 1 adding and

2 g subtractin to

equivalent is

carry around

- end The

carry) , (

2 1 ) (

2

) 1 2 (

n

A B

A B A

B

B A B

A B A

n n

n

−

>

≥

−

− +

=

+

−

−

= +

= +

−

Unit 01 31

Addition of 1

Addition of 1 ’ ’ s Complement Numbers s Complement Numbers

).

( of tion representa correct

the is which

, ) (

) (

1 2 yields carry last the Discarding

carry) ,

2 (

2 1 )]

( 1 2 [ 2

) 1 2 ( ) 1 2 (

1

B A

B A B

A B A B

A

B A

B A B A

n

n n

n n

n n

+

−

+

= +

−

−

<

+

≥

− +

−

− +

=

−

− +

−

−

= +

=

−

−

−

Unit 01 32

Addition of 1

Addition of 1 ’ ’ s Complement Numbers s Complement Numbers

Example: Add

Example: Add --11 and -11 and -20 in 120 in 1’’s complement for s complement for a word length of n=8.

a word length of n=8.

Unit 01 33

Binary Codes Binary Codes

zz Weighted codeWeighted code

ww₃₃-w-w₂₂-w-w₁₁-w-w₀₀ weighted code weighted code aa₃₃aa₂₂aa₁₁aa₀₀ aa₃₃aa₂₂aa₁₁aa₀₀= w= w₃₃aa₃₃+w+w₂₂aa₂₂+w+w₁₁aa₁₁+w+w₀₀aa₀₀

‹‹ Binary-Binary-Coded-Coded-Decimal, BCD; 8Decimal, BCD; 8--44--22--1BCD code1BCD code 0101=0

0101=0^..8+18+1^..4+04+0^..2+12+1^..1=51=5

‹‹ 6-6-3-3-11--1 code1 code 0101=0

0101=0^..6+16+1^..3+03+0^..1+11+1^..1=41=4

zz Excess-Excess-3 code3 code

‹‹ 8-8-4-4-22--1 code + 00111 code + 0011

‹‹ The code of iThe code of i is the 1’is the 1’s complement of code s complement of code 99--ii

z

z 2-2-outout--ofof--5 code5 code

‹‹ Exactly 2 out of 5 bits are 1Exactly 2 out of 5 bits are 1

‹‹ Error-Error-checking propertieschecking properties zz Gray codeGray code

‹

‹ The codes for successive decimal digits differ in exactly on bit.The codes for successive decimal digits differ in exactly on bit. zz ASCII codeASCII code

‹

‹ American American SStandard tandard CCode forode forIInformation Information Interchangenterchange

Unit 01 34

Binary Codes

Binary Codes

Unit 01 35

Binary Codes

Binary Codes

Unit 01 36

Homework Homework

z z 1.1 (b); 1.1 (b);

z z 1.2 (b); 1.2 (b);

z z 1.4 (c); 1.4 (c);

z z 1.5 (b); 1.5 (b);

z z 1.7 (a), (c); 1.7 (a), (c);

z z 1.9 1.9

z z 1.16 (b); 1.16 (b);

z z 1.17 (b); 1.17 (b);

z z 1.24 1.24

z z 1.25 (a); 1.25 (a);

Unit 2 Unit 2

Boolean

Boolean Algebra Algebra

Unit 02 2

Unit 2 Unit 2

zz

Boolean Algebra Boolean Algebra

‹‹2.1 Introduction2.1 Introduction

‹‹2.2 Basic Operations2.2 Basic Operations

‹‹2.3 Boolean Expressions and Truth Tables2.3 Boolean Expressions and Truth Tables

‹‹2.4 Basic Theorems2.4 Basic Theorems

‹‹2.5 Commutative, Associative, and 2.5 Commutative, Associative, and Distributive Laws

Distributive Laws

‹‹2.6 Simplification Theorems2.6 Simplification Theorems

‹‹2.7 Multiplying Out and Factoring2.7 Multiplying Out and Factoring

‹‹2.8 2.8 DemorganDemorgan’’ss LawsLaws

Unit 02 3

Introduction Introduction

zz George George BooleBoole developed Boolean algebra developed Boolean algebra in 1847 and used it to solve problems in in 1847 and used it to solve problems in

mathematical logic.

mathematical logic.

zz Boolean AlgebraBoolean Algebra

A Boolean algebraA Boolean algebra is an algebrais an algebra (B; . , + ,

(B; . , + ,’’ ;0, 1);0, 1) consisting of a set consisting of a set BB (which (which contains at least two elements

contains at least two elements 00 and and 11) ) together with

together with threethree operations,operations,

the the ANDAND (Boolean product(Boolean product) operation ) operation .. ,, the the OROR (Boolean sum(Boolean sum) operation ) operation ++, and, and

the the NOTNOT (complement(complement) operation ) operation ’’ , defined , defined on the set, such that

on the set, such that

Unit 02 4

Introduction Introduction

‹‹ A0. ClosureA0. Closure: For any : For any x,yx,y, and z of B, , and z of B, xx^..yy, , xx++yy, x, x’’ are in Bare in B

‹‹ A1. IdempotentA1. Idempotent: : xx^..xx=x =x x+xx+x=x=x

‹‹ A2. CommutativeA2. Commutative: : xx^..yy==yy^..xx xyxy==yxyx

‹‹ A3. AssociativeA3. Associative: :

xx^.. ((yy^..zz)=()=(xx^..yy) ) ^..z=x z=x ^.. y y ^.. z z xx++ ((yy++zz)=()=(xx++yy) ) ++z=x z=x ++ y y ++ zz

‹‹ A4. AbsorptiveA4. Absorptive: x : x ^.. (x (x ++ y)=x x y)=x x ++(x(x^.. y)=xy)=x

‹‹ A5. DistributiveA5. Distributive::

x x ^.. (y (y ++ z)=(x z)=(x ^.. y) y) ++(x (x ^.. z)z) x x ++(y (y ^.. z)=(x z)=(x ++ y) y) ^.. (x (x ++ z)z)

‹‹ A6. ZeroA6. Zero( null, smallest), ( null, smallest), 00 , and One, and One( universal, largest), ( universal, largest), 11 , , elements are in B

elements are in B

x x ^.. 11==11 ^.. x=x x x=x x ++ 00==0 0 ++ x=xx=x

‹‹ A7. ComplementA7. Complement

For every x in B, there exists a unique x

For every x in B, there exists a unique x’’ in B such thatin B such that x x ^.. xx’’==00, x , x ++ xx’’==11

Unit 02 5

Introduction Introduction

zz Example:Example: TwoTwo--element Boolean Algebraelement Boolean Algebra (Switching Algebra)

(Switching Algebra) BB₂₂=({0,1}; =({0,1}; ^{. .} ,+, ,+, ’’ ; 0,1); 0,1)

AND OR NOT AND OR NOT

11 00

11

00 00

00

11 00

..

11 11

11

11 00

00

11 00

++

00 11

11 00

’’

Unit 02 6

Introduction Introduction

zz Example:Example: FourFour--element Boolean Algebraelement Boolean Algebra BB₄₄=({0,a,b,1} ; =({0,a,b,1} ; ^{. .} ,+, ,+, ’’ ; 0,1) ; 0,1)

AND OR NOT AND OR NOT

11 bb aa 00 11

bb bb 00 00 bb

aa 00 aa 00 aa

00 00 00 00 00

11 bb aa 00 ..

11 11 11 11 11

11 bb 11 bb bb

11 11 aa aa aa

11 bb aa 00 00

11 bb aa 00 ++

00 11

aa bb

bb aa

11 00

’’

Unit 02 7

Introduction Introduction

zz An element of a Boolean algebra B is An element of a Boolean algebra B is called a

called a constant constant on B.on B.

e.g. 0,a,b,1 in B e.g. 0,a,b,1 in B₄₄..

zz A symbol that may represent any one of A symbol that may represent any one of element of B is called a

element of B is called a (Boolean) (Boolean) variable

variable on B.on B.

e.g.

e.g. x,y,zx,y,z,,……

Unit 02 8

Introduction Introduction

zz A Boolean expressionA Boolean expression over an algebra over an algebra system (B;

system (B; ^{. .} ,+, ,+, ’’ ; 0,1) is defined as ; 0,1) is defined as follows:

follows:

1.1. Any element of B ( constantAny element of B ( constant) is a Boolean expression.) is a Boolean expression.

2.2. Any variableAny variable name is a Boolean expression.name is a Boolean expression.

3.3. If eIf e₁₁ and eand e_{2 2} are Boolean expression, then eare Boolean expression, then e₁₁’, e’, e₂₂’, e’, e₁₁+e+e₂₂, , ee₁₁.e.e₂₂ are Boolean expressions.are Boolean expressions.

4.4. Any expression that can be constructed by a finite Any expression that can be constructed by a finite number of applications of the above rules, and

number of applications of the above rules, and onlyonly such a expression is a Boolean expression.

such a expression is a Boolean expression.

Unit 02 9

Introduction Introduction

zz A A function f(xfunction f(x₁₁,x,x₂₂,,……,,xx_nn)),, from

from BBⁿⁿ to B is called a Boolean functionto B is called a Boolean function if it if it can be specified by a Boolean expression of can be specified by a Boolean expression of n variables x

n variables x₁₁,x,x₂₂,,……,,xx_nn .. f(a,b,c

f(a,b,c)=)=abab’’c+ac+a’’b+bb+b’’cc’’

zz Each appearance of Each appearance of a variable or its a variable or its complement

complement in an expression is referred to in an expression is referred to as a

as a literalliteral.. f(a,b,c

f(a,b,c)=)=abab’’c+ac+a’’b+bb+b’’c’c’ has 3 variables,

has 3 variables, a,ba,b,, and and cc, 7 , 7 literals(literals(aa, b, b’’ , c, , c, aa’’, b, b, b, b’’, c, c’’ ).).

B B

f :

→

Unit 02 10

Basic Operations Basic Operations

zz

The basic operations of Boolean algebra are The basic operations of Boolean algebra are AND, OR, and NOT (complement, or

AND, OR, and NOT (complement, or inverse)

inverse) . .

‹‹NOT (Complement)NOT (Complement)

‹‹InverterInverter 1

0′ = 1′ = 0

1

0

and

0

1 = ′ = =

′ = if X X if X

X

Unit 02 11

Basic Operations Basic Operations

‹‹AND OperationAND Operation

9Omit the symbol 9 Omit the symbol “ “

” ” , A , A

B=AB B=AB

AND Gate AND Gate

Unit 02 12

Basic Operations Basic Operations

‹‹OR operationOR operation

OR Gate OR Gate

Unit 02 13

Boolean Expressions and Truth Tables Boolean Expressions and Truth Tables

zz

Order in which the operations are perform Order in which the operations are perform Parentheses

Parentheses Æ Æ Complentation Complentation Æ Æ AND AND Æ Æ OR OR

zz

Circuits for expressions Circuits for expressions

C B

A ′ +

BE D

C

A( + )]′ + [

Unit 02 14

Truth Table Truth Table

zz If an expression has If an expression has nn variables, the number of different variables, the number of different combinations of values of the variables is

combinations of values of the variables is 22ⁿⁿ. Therefore, a truth . Therefore, a truth table for n

table for n--variable expression will have variable expression will have 22ⁿⁿ rows.rows.

zz There are functions of There are functions of 2⁽²ⁿ⁾ nn variablesvariables. .

Unit 02 15

Truth Table Truth Table

zz AB′ + C = (A + C)(B′ + C)

Unit 02 16

Basic Theorems Basic Theorems

zz

Unit 02 17

Basic Theorems Basic Theorems

zz

Unit 02 18

Basic Theorems Basic Theorems

zz

Switching Circuits Switching Circuits

‹‹ = =

‹‹

‹‹

‹‹

Unit 02 19

Commutative, Associative, and Commutative, Associative, and

Distributive Laws Distributive Laws

zz

Commutative Laws Commutative Laws

zz

Associative Laws Associative Laws

zz

Distributive Laws Distributive Laws

Unit 02 20

Multiple

Multiple - - Input Gates Input Gates

zz Associative Laws for AND Associative Laws for AND andand OR OR operations.

operations.

(AB)C=A(BC)=ABC (AB)C=A(BC)=ABC

(A+B)+C=A+(B+C)=A+B+C (A+B)+C=A+(B+C)=A+B+C

Unit 02 21

Proof of Boolean Theorems Proof of Boolean Theorems

zz By Truth TableBy Truth Table

‹‹ Example: X+YZ=(X+Y)(X+Z)Example: X+YZ=(X+Y)(X+Z)

11 11

11 11

11 11

11 11

11 11

11 11

00 00

11 11

11 11

11 11

00 11

00 11

11 11

11 11

00 00

00 11

11 11

11 11

11 11

11 00

00 00

11 00

00 00

11 00

00 11

00 00

00 11

00 00

00 00

00 00

00 00

00 00

(X+Y)(X+Z) (X+Y)(X+Z) X+ZX+Z

X+YX+Y X+YZX+YZ

YZYZ ZZ

YY XX

Unit 02 22

Proof of Boolean Theorems Proof of Boolean Theorems

zz

By basic laws: By basic laws:

‹‹Example:Example: X+YZ=(X+Y)(X+Z)X+YZ=(X+Y)(X+Z)

Unit 02 23

Simplification Theorems Simplification Theorems

zz

Useful Simplification Theorems Useful Simplification Theorems

zz

Proof Proof

Unit 02 24

Simplification Examples Simplification Examples

zz Example: SimplifyExample: Simplify Sol:

Sol:

zz Example: Simplify Example: Simplify

Sol:

Sol:

] ) (

][

[ + ′ + + + ′ + + ′

= A BC D EF A BC D EF Z

C B A

X

Z = = + ′

) (

) )(

( + ′ + ′ ′ + + ′

= AB C B D C E AB C

Z

) (

) (

) )(

(

then

, )

( Let

+ ′

′+ + ′

= ′ +

=

′ +

=

+ ′

′ + + ′

+ ′

=

′ = + ′

= ′ + ′

C AB E

C D B Y X Y X Y

C AB E

C D B C AB Z

X E

C D B Y C

AB

Unit 02 25

Simplification Examples Simplification Examples

zz Example:Example: Find the output Find the output YY of the following of the following circuit and design a simpler circuit having the circuit and design a simpler circuit having the

same output.

same output.

Sol:

Sol:

The resulting circuit contains only

The resulting circuit contains only one ORone OR gate.gate.

A B

A B

B B A A

B)B B

(A

A B))B

(AB B

(A Y

+

= +

′ +

= +

′+

=

+ +

′ +

=

Unit 02 26

Multiplying Out and Factoring Multiplying Out and Factoring

zz SumSum--ofof--products (SOP)products (SOP)

An expression is said to be in

An expression is said to be in sumsum--ofof--productsproducts form when all form when all products are the

products are the products of only single variables.products of only single variables.

zz ProductProduct--ofof--sums (POS)sums (POS)

An expression is said to be in

An expression is said to be in productproduct--ofof--sumssums form when all form when all sums are the

sums are the sums of only single variablessums of only single variables..

form.

SOP in (

form, SOP

in

,

not is EF B)CD

A

are E D C B A E

C A E D C B A

+ +

+ ′

′+

′ +

′ +

′+

form.

POS in

form, POS

in )

( ,

) )(

)(

(

EF is not D)

B)(C (A

are E

D C B A E

C A E D

C B A

+ +

+

′+

′ + ′

+ ′

′+

′ + +

Unit 02 27

Multiplying Out and Factoring Multiplying Out and Factoring

zz

Using the ordinary distributive law Using the ordinary distributive law X(Y+Z)=XY+XZ

X(Y+Z)=XY+XZ

to multiply out to multiply out an expression to obtain a an expression to obtain a sum sum - - of of - - products form. products form.

zz

Using the second distributive law Using the second distributive law X+YZ=(X+Y)(X+Z)

X+YZ=(X+Y)(X+Z)

to factor to factor an expression to obtain a an expression to obtain a product

product - - of of - - sums form. sums form.

Unit 02 28

Multiplying Out and Factoring Multiplying Out and Factoring

zz

Example: Multiply out (A+BC)(A+D+E). Example: Multiply out (A+BC)(A+D+E).

Sol Sol - - 1: 1:

and then eliminating redundant terms:

(A+BC)(A+D+E)=A+AD+AE+ABC+BCD+BCE (A+BC)(A+D+E)=A+AD+AE+ABC+BCD+BCE

=A(1+D+E+BC)+BCD+BCE

=A(1+D+E+BC)+BCD+BCE

=A+BCD+BCE

=A+BCD+BCE

Sol Sol - - 2: 2:

(A+BC)(A+D+E)=A+BC(D+E)=A+BCD+BCE (A+BC)(A+D+E)=A+BC(D+E)=A+BCD+BCE