Ch02a Numbering Sytems

NUMBER SYSTEMS, OPERATION AND CODES

Lecture 2

Digital Design

Dr. PO Kimtho

Department of Computer Sciences

Norton University (NU)

Numbers & Codes

Numbering Systems

Decimal numbering system (Base 10)

Binary numbering system (Base 2)

Hexadecimal numbering system (Base 16)

Octal numbering system (Base 8)

Number Conversion

Binary Arithmetic

1s and 2s Complements of Binary Numbers

Numbers & Codes (cont..)

Signed Numbers

Arithmetic Operations with Signed Numbers

Other Number Codes

Binary-Coded-Decimal (BCD)

ASCII codes

Gray codes

Digital Codes & Parity

Numbering Systems

0 ~ 9

0 ~ 1

0 ~ 7

0 ~ F

Decimal (base 10)

Binary (base 2)

Octal (base 8)

Hexadecimal (base 16)

Num. Systems (Characteristics)

The digits are consecutive.

The number of digits is equal to the size of the base.

Zero is always the first digit.

When 1 is added to the largest digit, a sum of zero and a carry of one results.

Numeric values determined by the implicit positional values of the digits.

00000000

00000001

00000010

00000011

00000100

00000101

00000110

00000111

00001000

00001001

00001010

00001011

00001100

00001101

00001110

00001111

000

001

002

003

004

005

006

007

010

011

012

013

014

015

016

017

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Binary Octal Hex Dec

Numbering Systems (Cont.)

Numbering System (Decimal)

Also called the Base 10 system

Have 10 digits : 0 9

The position for each digit in the decimal number indicates the magnitude of the quantity represented and can be assigned a weight

Numbering System (Decimal)

The weight for whole numbers are positive powers of ten that increase from right to left

105 104 103 102 101 100

For fractional numbers, the weights are negative powers of ten that decrease from left to right

102 101 100 . 10-1 10-2 10-3.

Decimal point

Significant Digits

Binary: 11101101

Hexadecimal: 1D63A72A

Most significant digit Least significant digit

Question: How many bits does the numbers represent?

Decimal: 19635724

Bsince binary digits are properly known as bits, therefore

the term Most Significant Bit (MSB) and Least Significant

Bit (LSB) are widely used.

Numbering System (Binary)

Also called the Base 2 system

The binary number system is used to model the series

of electrical signals computers use to represent

information

0 represents the no voltage or an off state

1 represents the presence of voltage or an on state

Number Conversion

Any Radix (base) to Decimal Conversion

Binary to Decimal Conversion

Decimal value of any binary number can be found by adding weights of all bits that are 1 and discarding the weights of all bits that are 0

Solve this..

(a) 10102

(b) 101112

Answer : ?

Answer : ?

(c) 10101102 Answer : ?

Convert the following binary numbers to decimal

Decimal to Binary Conversion

For whole number conversion, use the repeated division-by-2 process and record the remainder

For fractional number conversion, use repeated multiplication by 2 until the fractional product is 0

or until the desired number of decimal places is

reached

2 5 = 12 + 1 2

1 2 = 6 + 0 2

6 = 3 + 0 2

3 = 1 + 1 2

1 = 0 + 1 2 MSB LSB 2510 = 1 1 0 0 1 2

Remainder


Whole number


Carry . 0 1 0 1 0.3125 x 2 = 0.625 0 0.625 x 2 = 1.25 1 0.25 x 2 = 0.50 0 0.5 x 2 = 1.00 1

The Answer: 1 1 0 0 1.0 1 0 1

MSB LSB

Fractional number

Solve this..

(a) 3910

(b) 5810

Answer : ?

Answer : ?

(c) 0.37510 Answer : ?

Convert the following decimal numbers to binary

Binary Arithmetics

Binary Addition

Binary Subtraction

Binary Multiplication

Binary Division

Binary Addition

Four basic rules for adding binary digits (bits) are:

0 + 0 = 0 (Sum of 0 with a carry of 0)



1 + 1 = 1 0 (Sum of 0 with a carry of 1)

Examples Perform the following binary additions:

(a) 100 + 10

1 0 0

1 0 +

1 1 0 (Answer)

(b) 111 + 11

1 1 1

1 1 +

1 0 1 0 (Answer)

Solve this..

(a) 11 + 01

(b) 111 + 110

Answer : ?

Answer : ?

(c) 1001 + 101:

Answer : ?

Perform the following binary additions:

Binary Arithmetics

Binary Addition

Binary Subtraction


Binary Division

Binary Subtraction

Four basic rules for subtracting binary digits (bits) are:

0 - 0 = 0

1 - 1 = 0

1 - 0 = 1

1 0 - 1 = 1 (0 1 with a borrow of 1)

Examples Perform the following binary subtractions:

(a) 101 011

1 0 1

0 1 1 -

0 1 0 (Answer)

(b) 110 101

1 1 0

1 0 1 -

0 0 1 (Answer)

Solve this..

(a) 101 100

(b) 1110 - 11

Answer : ?

Answer : ?

(c) 1100 - 1001:

Answer : ?

Perform the following binary subtractions

Binary Arithmetics

Binary Addition

Binary Subtraction


Binary Division


Four basic rules for muliplying binary digits (bits) are:

0 x 0 = 0

0 x 1 = 0

1 x 0 = 0

1 x 1 = 1

Examples

Multiply 111 and 101:

1 1 1

1 0 1 x

1 1 1

0 0 0

1 1 1

1 0 0 0 1 1 (Answer)

Solve this..

(b) 110 x 111:

(c) 1101 x 1010:

Answer : ?

Answer : ?

(a) 11 x 11:

Answer : ?

Binary Arithmetics

Binary Addition

Binary Subtraction


Binary Division

Binary Division

Division in binary follows the same procedure as division in

decimal

Example:

Perform the binary divisions of 110 11

1 1 0 1 1

1 0

1 1

0 0 0

(Answer)

Solve this..

(a) 100 10

Answer : ?

(b) 1100 100:

Answer : ?

Divide the binary numbers as indicated:

1s Complement

Changing all the 1s to 0s and all the 0s to 1s

Example:

1 1 0 1 0 0 1 0 1 Binary number

0 0 1 0 1 1 0 1 0 1s complement

2s Complements

Find the 1s complements of the numbers

1 1 1 0 1 0 1 0 1 Binary number

0 0 0 1 0 1 0 1 0 1s complement

Step 1:

Step 2: Add 1 to the 1s complements

0 0 0 1 0 1 0 1 0 1s complement

+ 1 Add 1

0 0 0 1 0 1 0 1 1 2s complement

Solve this..

(a) 00010110

Answer : ?

(b) 10010001

Answer : ?

Determine the 2s complement of each binary number:

Signed Numbers

Left most is the sign bit 0 is for positive, 1 is for negative

Sign & magnitude

00011001 = +25

sign bit magnitude bits

Sign-Magnitude Numbers

Sign bit

0 = positive

1 = negative

31 bits for magnitude

0110010.. 00101110010101

The left-most is the sign bit and the remaining bits are the magnitude bits

Signed Numbers (Cont.) 1s complement

The negative number is the 1s complement of the corresponding positive number

Example

+25 is 00011001 So, -25 is 11100110

2s complement

The positive number same as sign magnitude and 1s complement

The negative number is the 2s complement of the corresponding positive number.

Example

+25 is 00011001 So, -25 is 11100111

Solve this..

Express +19 and -19 (as an 8-bit number) in

i. sign magnitude

ii. 1s complement

iii. 2s complement

Answer : ?

Answer : ?

Answer : ?

Ch02a Numbering Sytems

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