Lecture Digital logic design - Lecture 2: More number systems/complements

Lecture 2 More number systems/complements. The main contents of the chapter consist of the following: Hexadecimal numbers; related to binary and octal numbers; conversion between hexadecimal, octal and binary; value ranges of numbers; representing positive and negative numbers; creating the complement of a number.

Digital Logic Design

Lecture 2

More Number Systems/Complements

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° Hexadecimal numbers

• Related to binary and octal numbers

° Conversion between hexadecimal, octal and binary

° Value ranges of numbers

° Representing positive and negative numbers

° Creating the complement of a number

• Make a positive number negative (and vice versa)

° Why binary?

Understanding Binary Numbers

° Binary numbers are made of binary digits (bits):

° Note that each hexadecimal digit can be represented

with four bits.

• (1110) 2 = (E) 16

° Groups of four bits are called a nibble.

• (1110) 2

° Possible to convert

between the three formats

Converting Between Base 16 and Base

2

° Conversion is easy!

 Determine 4-bit value for each hex digit

° Note that there are 2 4 = 16 different values of four

bits

° Easier to read and write in hexadecimal

° Representations are equivalent!

3A9F16 = 0011 1010 1001 11112

Converting Between Base 16 and Base

8

1 Convert from Base 16 to Base 2

2 Regroup bits into groups of three starting from

right

3 Ignore leading zeros

4 Each group of three bits forms an octal digit.

Decimal 2 Binary

Number systems (Octal Numbers )

°Octal number has base 8

°Each digit is a number from 0 to 7

°Each digit represents 3 binary bits

°Was used in early computing, but was replaced by hexadecimal

Decimal to Octal Conversion:

The Division:

[359] 10 = [ ? ] 8

By using the division system:

08

5

58

44

448

Binary to Octal Conversion:

Example:-

[110101] 2 = [ ? ] 8

Here we will take 3 bits and convert it from binary to

decimal by using the decimal to binary truth table:

Octal to Binary Conversion:

Example:-

[13] 8 = [ ? ] 2

Here we will convert each decimal digit from decimal

to binary (3 bits) using the decimal to binary truth

• Converting to decimal from octal:

– Evaluate the power series

° Hexadecimal is used to simplify dealing with large

binary values:

• Base-16, or Hexadecimal, has 16 characters: 0-9, A-F

• Represent a 4-bit binary value: 0000 2 (0) to 1111 2 (F)

• Easier than using ones and zeros for large binary

Hex Values in Computers

Conversion Binary to Hexadecimal

1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0

Hexadecimals – Base 16

• Converting to decimal from hex:

– Evaluate the power series

Octal to Hex Conversion

 To convert between the Octal and Hexadecimal

numbering systems

 Convert from one system to binary first

 Then convert from binary to the new numbering system

Hex to Octal Conversion

Ex : Convert E8A16 to octal

First convert the hex to binary:

1110 1000 10102

111 010 001 010 and re-group by 3 bits

(starting on the right) Then convert the binary to octal:

7 2 1 2

So E8A = 7212

Octal to Hex Conversion

Ex : Convert 7528 to hex

First convert the octal to binary:

111 101 0102 re-group by 4 bits

000 1 1110 1010 (add leading zeros)

Then convert the binary to hex:

1 E A

So 7528 = 1EA16

Example: Hex → Octal

° First convert the hexadecimal number into its

decimal equivalent, then convert the decimal

number into its octal equivalent.

MSB

1 r

0 1 8

3 r

1 11 8

LSB

2 r

11 90 8

Fractions (Example)

Fractions (Example)

° Convert 0.513 10 to base 16 (up to 4 fractional point)

How To Represent Signed

Numbers

• Plus and minus sign used for decimal

numbers: 25 (or +25), -16, etc.

• For computers, desirable to represent

everything as bits .

• Three types of signed binary number

representations: signed magnitude, 1’s complement, 2’s complement.

• In each case: left-most bit indicates sign:

positive (0) or negative (1).

Consider signed magnitude:

000011002 = 1210 100011002 = -1210

° It has already been studied that

° subtracting one number from another is the same as making

one number negative and just adding them.

° We know how to create negative numbers in the binary

number system.

° How to perform 2’s complement process.

° How the 2’s complement process can be use to add (and

subtract) binary numbers.

° Digital electronics requires frequent addition and subtraction

of numbers You know how to design an adder, but what about

a subtracter?

° A subtracter is not needed with the 2’s complement process

The 2’s complement process allows you to easily convert a

positive number into its negative equivalent.

° Since subtracting one number from another is the same as

3-Digit Decimal

A bicycle odometer with only

three digits is an example of a

fixed-length decimal number

system.

The problem is that without a

negative sign, you cannot tell a

+998 from a -2 (also a 998) Did

you ride forward for 998 miles

or backward for 2 miles?

Note: Car odometers do not work

this way.

999 998 997

001 000 999 998

002

forward (+)

backward (-)

Negative Decimal

How do we represent negative

numbers in this 3-digit decimal

number system without using a

001 000 999 998

501 500

pos(+)

neg(-)

+499 +498 +497

+001 000 -001 -002

-499 -500

3-Digit Decimal (Examples)

3

+ 2

5

003 + 002

997

6 + (-3)

3

(-2) + (-3) (-5)

006 + 997

1 003

Disregard Overflow

998 + 997

1 995

Disregard

Complex Problem

° The previous examples demonstrate that this

process works, but how do we easily convert

a number into its negative equivalent?

° In the examples, converting the negative

numbers into the 3-digit decimal number

system was fairly easy To convert the (-3),

you simply counted backward from 1000 (i.e.,

999, 998, 997).

° This process is not as easy for large numbers

(e.g., -214 is 786) How did we determine this?

° To convert a large negative number, you can

use the 10’s Complement Process

The 10’s Complement process uses base-10

(decimal) numbers Later, when we’re working

with base-2 (binary) numbers, you will see that

the 2’s Complement process works in the same

way.

First, complement all of the digits in a number

• A digit’s complement is the number you add to the

digit to make it equal to the largest digit in the base

(i.e., 9 for decimal) The complement of 0 is 9, 1 is 8,

2 is 7, etc.

Second, add 1.

• Without this step, our number system would have

two zeroes (+0 & -0), which no number system has.

10’s Complement Examples

-003

+1

996 997 -214

One’s Complement

Representation

• The one’s complement of a binary number

involves inverting all bits.

• 1’s comp of 00110011 is 11001100

• 1’s comp of 10101010 is 01010101

• For an n bit number N the 1’s complement is

(2 n -1) – N.

• Called diminished radix complement by Mano

since 1’s complement for base (radix 2).

• To find negative of 1’s complement number

take the 1’s complement.

000011002 = 1210 111100112 = -1210

Two’s Complement

Representation

• The two’s complement of a binary number

involves inverting all bits and adding 1

• 2’s comp of 00110011 is 11001101

• 2’s comp of 10101010 is 01010110

• For an n bit number N the 2’s complement is

(2 n -1) – N + 1.

• Called radix complement by Mano since 2’s

complement for base (radix 2).

• To find negative of 2’s complement number

take the 2’s complement.

000011002 = 1210 111101002 = -1210

Two’s Complement

Shortcuts

° Algorithm 1 – Simply complement each bit and

then add 1 to the result.

• Finding the 2’s complement of (01100101) 2 and of its 2’s

complement…

N = 01100101 [N] = 10011011

10011010 01100100

+ 1 + 1

-

10011011 01100101

° Algorithm 2 – Starting with the least significant bit, copy all of the bits up to and including the first 1

bit and then complementing the remaining bits.

[N] = 1 0 0 1 1 0 1 1

° Machines that use 2’s complement arithmetic can

represent integers in the range

-2 n-1 <= N <= 2 n-1 -1 where n is the number of bits available for

representing N Note that 2 n-1 -1 = (011 11) 2 and –2 n-1 = (100 00) 2

oFor 2’s complement more negative numbers than

positive.

oFor 1’s complement two representations for zero.

oFor an n bit number in base (radix) z there are z n

different unsigned values.

(0, 1, …z n-1 )

Add carry

Final Result

Step 1: Add binary numbers

Step 2: Add carry to low-order bit

Add

1’s Complement

Subtraction

° Using 1’s complement numbers, subtracting

numbers is also easy

° For example, suppose we wish to subtract

Add carry Final

Step 1: Take 1’s complement of 2 nd operand

Step 2: Add binary numbers

Step 3: Add carry to low order bit

1’s comp Add

Step 1: Add binary numbers

Step 2: Ignore carry bit

Add

Ignore

Step 1: Take 2’s complement of 2 nd operand

Step 2: Add binary numbers

Step 3: Ignore carry bit

2’s comp Add

Ignore

2’s Complement Subtraction: Example

#2

° Let’s compute (13) 10 – (5) 10

• (13) 10 = +(1101) 2 = (01101) 2

• (-5) 10 = -(0101) 2 = (11011) 2

° Adding these two 5-bit codes…

° Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result Indeed,

2’s Complement Subtraction: Example

#3

° Let’s compute (5) 10 – (12) 10

• (-12) 10 = -(1100) 2 = (10100) 2

• (5) 10 = +(0101) 2 = (00101) 2

° Adding these two 5-bit codes…

° Here, there is no carry bit and the sign bit is 1

This indicates a negative result, which is what we expect (11001) 2 = -(7) 10

0 0 1 0 1

-1 -1 0 0 -1

1’s Complement

2’s Complement

• 1s complement with negative numbers shifted one

position clockwise

• Only one representation for 0

• One more negative number than positive number

• High-order bit can act as sign bit

° Overflow conditions

° Add two positive numbers to get a negative number

° Add two negative numbers to get a positive number

Two’s Complement Number System

0000

0001

0010

0011 0100

0101 0110

1010 1011

1100 1101

° Signed numbers represented in signed magnitude, 1’s

complement, and 2’s complement

° 2’s complement most important (only 1 representation

for zero).

° Important to understand treatment of sign bit for 1’s

and 2’s complement.