Fundamental- Unit 3

8/11/2019 Fundamental- Unit 3

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Fundamentals of Information Technology Unit 3

Sikkim Manipal UniversityDDE Page No. 52

Unit 3 Number Systems

Structure

3.0 Introduction

Objectives

3.1 Decimal Number System

3.2 Binary Number System

Binary Addition and Subtraction

Binary Multiplication and Division

3.3 Conversion from Decimal Numbers to Binary

Converting Fractions Decimal Value to Binary

3.4 Negative Numbers

3.5 Representing Negative Numbers Using Complements

Complements in Binary Number System

3.6 Gates

OR Gate

AND Gate

NOT Gate

3.7 Summary

3.8 Terminal Questions

3.9 Answers to Terminal Questions

3.0 Introduction

Any quantity is measured in some system. The quantity measured is

represented in some numbers. There are different number systems. In each

number system, different symbols are used to represent the numbers. The

different number systems are Decimal, Octal, Binary etc.


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Objectives:

After studying this unit you will be able to explain:

Decimal Number System

Binary Number System

Binary Addition and Subtraction

Binary Multiplication And Division

Conversion From Decimal Numbers To Binary

Negative Numbers

Representing Negative Numbers Using Complements

Complements In Binary Number System

Gates: OR, AND, NOT

3.1 Decimal Number System

In this ten symbols are used to represent the numbers and hence it is called

Decimal number system. The ten symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

These are called Arabic numerals.

It is necessary to learn only the 10 basic numerals and the positional

notational system in order to measure any desired quantity or to count any

required figure. After memorizing the addition and multiplication tables and

learning a few simple rules, we can perform all arithmetic operations.

Example-1: Let us represent the quantity 127. The actual meaning of the

number 127 can be seen more clearly if we notice that it is said as one

hundred and twenty seven Basically, the number is a contraction of 1 x 100

+ 2 x 10 + 7. The important point is that the value of each digit is determined

by its position. For example, the 3 in 300 has a different value than the 3 in30. We show this verbally by saying three hundred and thirty different

verbal representations have been invented for numbers from 10 to 20

(eleven, twelve, . .), but from 20 upward we break only at powers of 10


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(hundreds, thousands, lakhs, crores etc.). Written numbers are always

contracted, however, and only the basic 10 numerals are used, regardless

of the size of the integer written.

The base, or radix, of a number system is defined as the number of different

digits which can occur in each position in the number system. The decimal

number system has a base, or radix, of 10. Thus the system has 10 different

digits (0, 1, 2, 3, 4 , 9), any one of which may be used in each position in

a number.

Example-1:

In the decimal system, there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9

These digits can represent any value, for example: 754.

The value is formed by the sum of each digit, multiplied by the base(in this

case it is 10because there are 10 digits in decimal system) in power of digit

position (counting from zero):

Position of each digit is very important! For example, if you place "7" to the

end: 547 it will be another value:


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Important note:Any number in power of zero is 1; even zero in power of

zero is 1:

3.2 Binary Number System

In this number system two symbols are used to represent the numbers and

hence it is called binary number system. The two symbols are 0 and 1. The

same type of positional notation is used in the binary number system as in

the decimal system.

Although the same positional notation system is used, the instead of powers

of 10 as used in decimal number system the binary system uses powers of

2. As was previously explained, the number 127 actually means 1 102+ 2

101+ 7 100. In the binary system, the same number (127) is represented

as 1111111, meaning 1 26+ 1 25+ l 24+123+ l 22+ l 21+ l 20.

3.2.1 Binary Addition and Subtraction

In the same manner as decimal addition is performed, Binary addition is

also carried out.

The table for binary addition is as follows:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 and a carry-over of 1

Carry-overs are performed in the same manner as in decimal arithmetic.

Since 1 is the largest digit in the binary system, any sum greater than 1

requires that a digit be carried over. For example, 010 plus 010 binary


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requires the addition of the two 1s in the second position to the left, with a

carry-over. Since 1 + 1 = 0 plus a carry-over of 1, the sum of 010 and 010 is

100.

Here are three more examples of binary addition:

DECIMAL BINARY DECIMAL BINARY DECIMAL BINARY

6 110 15 1111 31/4 11.01

7 111 20 10100 53/4 101.11

13 1101 35 100011 9 1001.00

It is necessary to establish a procedure for subtracting a larger digit from a

smaller digit in subtraction. The only case in which this occurs with binary

numbers is when 1 is subtracted from 0. The remainder is 1, but it is

necessary to borrow 1 from the next column to the left.

This is the binary subtraction table.

0 0 = 0

1 0 = 1

1 1 = 0

0 1 = 1 with a borrow of 1

A few examples will make the procedure for binary subtraction clear:

DECIMAL BINARY DECIMAL BINARY DECIMAL BINARY

9 1001 16 10000 61/4 110.01

5 101 3 11 41/2 100.1

4 100 13 1101 13/4 1.11

3.2.2 Binary Multiplication and Division

Binary Multiplication:

The table for binary multiplication is also given below:0 x 0 = 0

1 x 0 = 0

0 x 1 = 0

1 x 1 = 1


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The following examples of binary multiplication show the simplicity of each

operation. It is only necessary to copy the multiplicand if the digit in the

multiplier is 1 and to copy all 0s if the digit in the multiplier is a 0.

The following two examples of binary multiplication illustrate the simplicity of

each operation. If the digit in the multiplier is 1 then copy multiplicand and

copy all 0s if the digit in the multiplier is a 0.

DECIMAL BINARY DECIMAL BINARY

09 1001 1.25 1.01

x 10 x 1010 x 2.5 x 10.1

90 0000 625 101

1001 250 1010

0000 3.125 11.001

1001

1011010

Binary Division:

Binary division is, like any number system is very simple.

0 1 = 0

1 1 = 1

Note:Division by zero is not done in any number system.

Following are the examples of division:

Example 1:

DECIMAL BINARY

5 101

5) 20 101) 11001

101

101

101


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Example 2:

DECIMAL BINARY

2.416... 10.011010101...

12)29.0000 1100)11101.00

24 1100

50 10100

48 1100

20 10000

12 1100

80 1000072 1100

8 ...

To convert the quotient obtained in the second example from binary to

decimal, we would proceed as follows:

10.011010101 = 1 x 21 = 2.0

0 x 20 = 0.0

0 x 21 = 0.0

1 x 22 = 0.25

1 x 23 = 0.125

0 x 24 = 0.0

1 x 25 = 0.03125

0 x 26 = 0.0

1 x 27 = 0.0078125

0 x 28 = 0.00

1 x 29 = 0.001953125

2.416015625

Therefore, 10.011010101 binary equals approximately 2.416 decimal.


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3.3 Conversion from Decimal Numbers to Binary

Many methods are available for converting a decimal number to a binary

number.

In the first method simply subtract all powers of 2, which can be subtracted

from the decimal number until nothing remains. The highest power of 2 is

subtracted first, then the second highest, etc.


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A good way to organize this conversion is to list the divisions in table

form as below:

Reading from bottom to top, the final answer is 10112. Remember that thefirst division gives us the least significant digit of our answer, and the final

division gives us the most significant digit of our answer. Also, the result of

the final division is always 0.

3.3.1 Converting Fractions Decimal Value to Binary

When converting a fractional decimal value to binary, we need to use a

slightly different approach. Instead of dividing by 2, we repeatedly multiply

the decimal fraction by 2. If the result is greater than or equal to 1, we add a

1 to our answer. If the result is less than 1, we add a 0 to our answer.


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3.4 Negative Numbers

For writing negative numbers a standard convention is adopted and it

consists of placing a sign symbol before a number that is negative. For

example, negative 27 is written as -27. If 27 is to be added to + 45, we

write

+ 45 + (27) = 18

But if a negative number is subtracted from a positive number, for example

the above expression can be written as +45(27) = + 45 + 27 = 72 (- andbecomes +).

In binary machines each of the binary digit is represented by a switch which

can be used to represent two values but one at a time either ON or OFF.


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As an example, given a set of six switches, any number from 000000 to

111111 may be represented by the switches if we define a switch with its

contacts closed as representing a 1 and a switch with open contacts as

representing a 0. If we desire to increase the total range of numbers that we

can represent so that it will include the negative numbers from 000000 to

111111, another bit (or switch) will be required. We then treat this bit as a

sign bit and place it before the magnitude of the number to be represented.

The convention adopted generally is that when the sign bit is a 0, the

number represented is positive, and when the sign bit is a 1, the number is

negative. Let us consider the previous example where we have used six

switches to represent the quantity and now to specify the sign of the quantity

whether positive or negative one more switch will have to be used. When

the contacts of this seventh switch are open, the number will be a positive

number equal to the magnitude of the number stored in the other six

switches; and if the switch for the sign bit is closed, the number represented

by the seven switches will be a negative number with a magnitude

determined by other six switches.

Let us consider the following example when seven switches are used.

37 = 1100101

The seventh switch represents the negative sign, which is 1.

+37 = 0100101

The seventh switch represents the positive sign, which is 0.

3.5 Representing Negative Numbers Using Complements

The negative numbers can be represented in complement form so that a

machine can be made to add and subtract, using only circuitry for adding.

3.5.1 Complements in Binary Number System

There are two types of complements in this and they are 2s complement

and 1s complement.


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The 2s complement of a binary number is formed by simply subtracting

each bit of the number from the radix minus one and adding a 1 to the least

significant bit. Since the radix in the binary number system is 2, each bit of

the binary number is subtracted from 1. The application of this rule is

actually very simple; every 1 in the number is changed to a 0 and every 0 to

a 1. Then a 1 is added to the least significant bit of the number formed. The

2s complement of 10110 is formed by the following steps:

Step 1:Change 1 bit to 0 and vice versa so 01001

Step 2:Add 1 to the Least significant bit so 01001 +

1

01010

So the 2s complement of 10010 is 01010.

Similarly the 2s complement of 10010 is 01110. Subtraction using the 2s

complement system involves forming the 2s complement of the subtrahend

and then adding this complement to the minuend. For instance,

Example 1:

11001 11001

10100 = + 01100

00101 1 00101

Carry is dropped

Example 2

10110 10110

01110 = + 10010

01000 1 01000

Carry is dropped


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Subtraction using the 1s complement system is also straightforward. The

1s complement of a binary number is formed by changing each 1 in the

number to a 0 and each 0 in the number to a 1. For instance, the 1s

complement of 11101 is 00010, and the 1s complement of 00011 is 11100.

When subtraction is performed in the 1s complement system, any end-

around carry is added to the least significant bit. For instance,

Example 1:

11001 11001

10100 = + 01011

00101 00100

1 Carry generated is added

00101

Example 2

10110 10110

01110 = + 10001

01000 00111

1 Carry generated is added

01000

Note:Observe the difference in 2s and 1s complement subtraction for the

same quantity.

3.6 Gates

A gate is an electronic circuit which operates on one or more input signal to

produce an output signal. There are different gates like OR, AND, NOT etc.


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3.6.1 OR Gate:

The following table gives the combinations of input and the output for each

of the combinations. This table is also called as truth table of OR gate.

Inputs

X Y

Output

Z

0 0 0

1 0 1

0 1 1

1 1 1

In the above table X and Y are the Inputs and Z is the Output. As shown in

the table when both the Inputs are 0, then the Output is 0. If any one of the

Input or both the Inputs are 1, then the Output is 1.

Logical addition table:

0 + 0 0

0 + 1 1

1 + 0 1

1 + 1 1

ORgate is used to realize the logical addition operation.

3.6.2 AND Gate:

The following table gives the combinations of input and the output for each

of the combinations. This table is also called truth table AND gate.


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Inputs

X Y

Output

Z0 0 0

1 0 0

0 1 0

1 1 1

In the above table, X and Y are the Inputs and Z is the Output. As shown in

the above table, the Output is 1 only when both the Inputs are 1 and in all

other cases the output is 0.

Logical multiplication table:

0 . 0 0

0 . 1 0

1 . 0 0

1 . 1 1

AND gate is used to realize the logical multiplication operation

3.6.3 NOT Gate

Singular or unary operations define an operation on a single variable. The

familiar example of unary operation is -, so we can write -7, -9 or Z, that

means we are to take the negative of these values. The operation

complementation means inversion of a quantity and this operation is defined

by the following table.

InputX

OutputZ

0 1

1 0


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The above table gives the output for each input. This table is also called

truth table NOT gate. The operation complementation or inversion of a

quantity can be realized by the help of NOT gate.

3.7 Summary

In Binary number system two symbols are used to represent the numbers

and hence it is called binary number system.

The two symbols are 0 and 1.

The same type of positional notation is used in the binary number system as

in the decimal system. In the same manner as decimal arithmetic is

performed, Binary addition, subtraction, multiplication and division is also

carried out. The negative numbers are represented in complement form so

that a machine can be made to add and subtract, using only circuitry for

adding. There are two types of complements in this and they are 2s

complement and 1s complement.

A gate is an electronic circuit which operates on one or more input signal to

produce an output signal. There are different gates like OR, AND, NOT etc.

Self Assessment Questions

1. Symbols are used to represent the numbers and it is called as

__________

2. For example, negative 27 is written as -27. If 27 is to be added to 45,

we denote as _____________

3. In decimal system, there are 10 digits. (True/False)

4. Carry-over is performed in the same manner as in decimal arithmetic.

(True/False)


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Answers to Self Assessment Questions

1. Decimal number system

2. + 45 + (27) = 18

3. True

4. True

3.8 Terminal Questions

1. What is Decimal number system?

2. How do you convert Fractional Decimal value to Binary Numbers?

3. Convert the following from Binary to Decimal:

a) 11111

b) 10110

c) 11001

d) 00101

4) Perform the following Arithmetic operations in Binary

a) 11011 + 11110

b) 11.01 + 101.11c) 11101100

d) 10010101

e) 1011 * 110

f) 111* 101

g) 1111 111

h) 111 11

5) Perform the following subtraction using 1s and 2s complement system

a) 1110111110b) 11001 10011

6) Write the truth table of OR, AND, NOT gates.


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3.9 Answers to Terminal Questions

1. Refer section 3.1

2. Refer section 3.3.1




6. Refer section 3.6.1, 3.6.2 & 3.6.3

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Fundamental- Unit 3