Number System Conversion and Boolean Logic : Class 11 NEB Notes and Numerical Solutions

Number system refers to the base, arrangements and positional value of digits in number system.

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INTRODUCTIONTO NUMBER SYSTEM

In every step of life, we definitely find the use of numbers. If you need to buy something, you will have to pay a certain amount of money for which you will have to count the money. Likewise, the shopkeeper will also count the goods to give you and same for your changes. Therefore, number system can simply be defined as a way to represent numbers.

TYPESOF NUMBER SYSTEM

1) Decimal Number System:

The decimal number system is the most commonly used number system in our daily life. This generally used number system is also known as the base 10 number system because it uses just the 10 symbols i.e. 0 to 9. It is also known as the denary number system because any numeric value there is, these system’s digits can easily represent them.The decimal system is specially used in the computer interface. The weight and position of the digit dictate the value represented by it.

In this system, each number consists of the digits that are located at different positions. The positions of the 1^st and the 2^nd digits towards the right side of the decimal point are ^-1 and ^-2. Similarly, the positions of the 1^st and 2^nd digits towards the left side of the decimal point are 0 and 1 respectively.

The value of the number is determined by adding the results out of the multiplication of the digits with the weight of their position. This method is called the expansion method. Under this method, the rightmost digit of the number is called the Least Significant Digit (LSD), as it has the lowest weight. Likewise, the leftmost digit of the number is called the Most Significant Digit (MSD), as it has the highest weight.

Examples:

(a) The weights and positions of each digit of the number 796 are as follows:

The above table indicates that:
The value of digit 7 = 7x10²=700
The value of digit 9 = 9x10¹=90
The value of digit 6 = 6x10⁰=6
The actual number can be formed by adding the values obtained by the digits as follows:
700+90+6=796
Here, the digit 7 in the number 796 is the most significant digit and 6 is the least significant digit.

(b) The weights and positions of each digit of the number 125.64 are as follows:

The above table indicates that:
The value of digit 1 = 1x10² = 100
The value of digit 2 = 2x10¹ = 20
The value of digit 5 = 5x10⁰= 5
The value of digit 6 = 6x10^-1= 0.6
The value of digit 4 = 4x10^-2= 0.4
The actual number can be formed by adding the values obtained by the digits as follows:
100+20+5+0.6+0.4=125.64

2) Binary Number System:

Binary Number System refers to the number system that uses only two symbols i.e. 0 and 1, that's 0why it is called a base 2 number system. It is also known as Binary Digit (BIT). This number system is especially used in the internal processing of computer system. When we count up from 0 in binary, symbols are much more frequently run out as this system only uses 0 and 1 and 2 do not exist. Therefore, we use 10 in this system because 10 is equal to 2 in decimal. The combination of binary numbers can be used to represent different quantities like 1001. In Binary, each digit’s positional value is twice the face value or place value of the digit of its right side. Each position’s weight is a power of 2.

According to the position and weight, the place value of the digits is as follows:

Example: Convert 101.11

The above table indicates that:
101.101 = 1x2²+0x2¹+1x2⁰+1x2^-1+1x2^-2
= 1x4+0+1x1+1/2+1/4
= 4+0+1+0.5+0.25
= 5.75

3) Octal Number System:

Octal Number System is the base 8 system. Like the decimal number system, this system is also used in the internal processing of computer system. It is the system that consists of eight digits i.e. {0, 1, 2, 3, 4, 5, 6, 7} which is used for the representation of long binary numbers short-handedly. In this system, each digit position represents a power of 8. The number 708 will not be valid in this system as 8 is not a valid digit.

According to the position and weight, the place value of the digits is as follows:

 

Example: Convert 128 to a decimal number.

128 = 1x8¹+2x8⁰
= 1x8+2x1
= 8+2
= 10

4) Hexadecimal NumberSystem:

Hexadecimal Number System is the number system that represents long binary numbers in shortcut method. It is a base 16 system as it consists of 16 digits i.e. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } where the alphabets represent the decimal numbers 10 to 15. This system is also used in the computer system, mainly in the memory management. As the name suggests, each digit's position represents a power of 16 in this system.

According to the position and weight, the place value of the digits is as follows:

Example: Convert 2A16 to a decimal number.

 

2A16 = 2x16¹+Ax16⁰
= 2x16+10x1
= 32+10
= 42

Number system conversion: binary, octal and hexadecimal

CONVERSION OF BINARY

It consists of :

• conversion from binary to decimalnumber system
• conversion from binary to octal number system
• conversion from binary to hexadecimal number system

 

Conversion of binary to decimal (base 2 to base 10):

The rules for conversion from binary to decimal are given below:

1. Multiply each bit by corresponding power of 2 (base).
2. Sum each product term to get a decimal equivalent number.

Note: A power of 2 is 0 for a left bit of binary point (or for a right most bit for the number that does not contain fractional part) and increase the power by one for each bit towards left and decrease power by one towards the right of binary point.

Example 1: convert (110011)₂ to decimal.

Solution:

(110011)₂ = 1×2⁵ + 1×2⁴ + 0×2³ + 0×2² + 1×2¹ + 1×2⁰

= 32 + 16 + 0 + 0 + 2 +1

= (51)₁₀

 

Example 2: convert (1011.101)₂ into decimal.

(1011.101)₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ + 1×2^-1 + 0×2^-2 + 1×2^-3

= 8 + 0 + 2 + 1 +0.5+ 0 +0.125

= 11 + 0.5 + 0.125

= (11.625)₁₀

Conversion of binary to octal (base 2 to base 8):

The rules for conversion from binary to decimal are given below:

1. Divide the given binary number into a group of three bits from binary point to left (or from right to left if no fractional part) and from binary point to the right. Append 0’s at leading or trailing or trailing position if necessary to make each group of 3 bits.
2. Substitute each group of three bits by octal equivalent from a table.
3. Collect octal digits to get an octal equivalent number.

 

Example: convert (10110101)₂ into octal.

Solution:

Dividing the given binary number (10110101)₂ into groups of 3 bits from right to left, and appending 0’s at leading position, we have,

Conversion of binary to hexadecimal (base 2 to base 16):

The rules for conversion from binary to hexadecimal are as given below:

1. Divide the given binary number into a group of four bits from binary point to left (or from right to left if no fractional part) and from binary point to the right. Append 0’s at leading or trailing position if necessary to make each group of 4 bits.
2. Substitute each group of four bits by hexadecimal equivalent symbol (letter or digit) from the table.
3. Collect Hexadecimal symbols to get Hexadecimal equivalent number

Example: convert (10101011101)₂ into hexadecimal

Solution:

Dividing the given binary number (10101011101)₂ into groups of 4 bits from binary point to left and binary point right and appending 0’s at leading or at trailing position to make each group of four bits, we have,

Conversion of Decimal

It consists of conversion from decimal to binary, octal and hexadecimal number system which are described below:

Conversion of decimal to binary (base 10 to base 2):

The rules for conversion from decimal to binary are as given below:

1. Divide the given number by 2 and note the remainder.
2. Repeatedly divide the quotient by two and note the remainder until quotient reduced to 0.
3. Collect the remainders, last obtained first and first obtained last to binary equivalent.

Example: convert (51)₁₀ into binary

Solution:

51÷2=25	remainder	= 1
25÷ 2=12	remainder	= 1
12÷ 2=6	remainder	= 0
6÷ 2=3	remainder	= 0
3÷ 2=1	remainder	= 1
1÷ 2=0	remainder	= 1

 

Therefore, (51)₁₀ = (110011)₂

 

Conversion of decimal to octal (base 10 to base 8):

The rules for conversion from decimal to binary are as given below:

1. Divide the given number by 8 and note the remainder.
2. Repeatedly divide the quotient by 8 and note the remainder until quotient reduced to 0.
3. Collect the remainders, last obtained first and first obtained last to get the octal equivalent.

 

Example: convert (177)₁₀ into octal

Solution:

177÷ 8 =22	remainder	= 1
22÷ 8 = 2	remainder	= 6
2÷ 8 = 0	remainder	= 2

 

Therefore, (177)₁₀ = (261)₈

 

Conversion of decimal to hexadecimal (base 10 to base 16):

Example: convert (77)₁₀ into hexadecimal

Solution:

77÷ 16=4	remainder	= 13 which means D
4÷ 16 =0	remainder	= 4

 

Therefore, (77)₁₀ = (4D)₁₆

 

Conversion of octal

It consists of conversion from octal to decimal, binary and hexadecimal number system which are described below:

Conversion of octal to decimal (base 8 to base 10):

Multiply the each octal digit by corresponding power of 8 and sum each product term to get decimal equivalent.

Example: convert (632)₈ to decimal.

Solution:

(632)₈ = 6×8² + 3×8¹ + 2×8⁰

= 384 + 24 + 2

= 410

Therefore, (632)₈ = (410)₁₀

Conversion from octal to binary (base 8 to base 2):

Substitute each octal digit by equivalent 3 bit binary from table and collect bits for each digit to get binary equivalent numbers.

Example: convert (741)₈ into binary.

Conversion from octal to hexadecimal (base 8 to base 16):

Actually, there is no direct method for converting from octal to hexadecimal so first, convert octal into binary or decimal and then convert binary or decimal to hexadecimal.

 

Conversion of hexadecimal

It consists of conversion from hexadecimal to decimal, binary and octal number system which are described below:

Conversion of hexadecimal to decimal (base 16 to base 10):

The rules for conversion from hexadecimal to decimal are as given below:

1. Multiply each digit by corresponding power of 16(base) as in decimal to binary.
2. Sum each product term to get decimal equivalent.

Example: convert (F4C)₁₆ into decimal.

Solution:

 

(F4C)₁₆ = F×16² + 4×16¹ + C×16⁰

= 15×256 + 4×16 + 12×1

= 3840 + 64 + 12

= 3916

Therefore, (F4C)₁₆ = (3916)₁₀

 

Conversion of hexadecimal to binary (base 16 to base 2):

Substitute each hexadecimal symbol by equivalent 4 bit binary from table and collect bits for each digit to get binary equivalent numbers.

Conversion of hexadecimal to octal (base 16 to base 8):

There is no direct method for converting from hexadecimal to octal, so first convert hexadecimal into binary or decimal and then convert binary or decimal to octal.

COMPLEMENT

In a computer system, subtraction is not performed directly as arithmetic subtraction. It is performed by the technique called complement. It is the process of repeated addition.

• There are two types of complement: r`s complement and (r-1)`s complement.

Where 'r' is the base of a number system.

In binary number system, there are two types of complement: 1`s complement and 2`s complement.

Similarly, decimal number system has 9`s and 10`s complement.

• 1`s Complement

1`s complement of a binary number is obtained by subtracting each bit by 1. We can get 1`s complement by simply replacing 1 by 0 and 0 by 1.

Example: 1`s complement of 1011 = 0100

Subtraction of binary numbers using 1`s complement

Steps are here as below:

1. Make the both numbers having the same number of bits.
2. Determine the 1`s complement of the number to be subtracted(subtrahend).
3. Add the 1`s complement to the given number from which we subtract (minuend).
4. If there exists` any additional bit (carry) in the result after addition, remove and add it to the result else (i.e. if there exists` no any carry)
5. Determine the 1`s complement of the result and prefix by a negative sign to get the final result.

Example: Subtract 1110000 from 1100000

• 2`s complement

The 2`s complement of a binary number is obtained by adding binary 1 to the 1`s complement of the number.

Subtraction using 2`s complement:

Steps are here as below:

1. Make the both numbers having the same number of bits.
2. Determine the 2`s complement of the number to be subtracted (subtrahend).
3. Add the 2`s complement to the given number from which we subtract (minuend).
4. If there exits no carry determine the 2`s complement of the result and prefix by a negative sign to get a final result.

Example:Subtract 1110000 from 1100000

• 9`s Complement and 10`s Complement

The 9`s complement of decimal number can be obtained by subtracting each digit of the number from 9.

For example, the 9`s complement of 3 is 6 (9-3=6), and 234 is 765 (999-234 =765).

The 10`s complement of decimal number can be obtained by adding 1 to the least significant digit of 9`s complement of that number. For example, 10`s complement of 3 is 7 (9-3=6+1=7), and 123 is 877.

Subtraction of decimal number using 9`s complement

Here are the steps are given below:

1. Make the both numbers having the same number of digits.
2. Determine the 9`s complement of the number from which we subtracted (subtrahend).
3. Add the 9`s complement to the given number from which we subtract (minuend).
4. If there exists` any additional digit (carry) in the result after addition, remove it and add it to the complement of the result and prefix by a negative sign to get the final result.

E.g. Subtract (123)₁₀ From (345)₁₀
9`s complement of 123= (999 -123) =876
Adding the 9`s complement with 345, i.e 345 + 876 = 1221

In the result, most significant digit 1 is the carryover. So add this carry over to remaining digits 221
i.e, 221 + 1 = 222
Hence, (222)₁₀ is the required result after subtracting (123)₁₀ from (345)_10.

Subtraction using 10`s complement:

Here are the steps are given below:

1. Make the both numbers having same numbers of digits.
2. Determine the 10`s complement of the number to be subtracted (subtrahend).
3. Add the 10`s complement to the given number from which we subtract (minuend).
4. If there exists`s any additional digit (carry) in the result after addition, remove it from the result and the remaining digits form the final result.
5. If there exists` no any carry then determine the 10`s complement of the result and prefix by the negative sign to get the final result.

Example: Subtract (123)₁₀ from (345)₁₀

10`s Complement of 123 = (999 - 123) = 876 + 1 = 877
Adding the 10`s complement with 345, i.e. 345 + 877 = 1222
In this result, most significant digit 1 is the carry over.So remove it to find the result.
Therefore, (222)₁₀ is the required result.

Binary Mathematics

1. Binary Addition

 

Rule for binary addition0+0=01+0=10+1=11+1=10 (0 with carry over 1)

Example: Binary addition101101+101111000100 sum

 

 

2. Binary Subtraction

Rule for binary subtraction 1-1=01-0=10-1=1 (with borrowing 1)0-0=0

Example: Binary addition101101 minuend-10111 subtrahend10110 difference

 

 

3. Binary Multiplication

Rule for binary multiplication1*1=11*0=00*1=00*0=0

Example: Binary multiplication1011 multiplicand *1011 multiplier10111011*0000** +1011***1111001 product

 

 

4. Binary Division

 

Rule for binary division1/1=11/0=not defined0/1=00/0=not defined

Example: Binary DivisionDivide 101011 by 110110) 101011 (111 quotient -1101001 -110111 -1101 remainder

 

Some Basic Terms Related with Number System

• MSB (Most Significant Bit)

The left most bit of a number is called MSB.

Example :

1010 = MBS

• LSB (Leas Significant Bit)

The right most bit of a number is called LSB.

Example:

1010 = LSB

 

BIT: Single binary number either 0 or 1

Nibble: Combination of 4 binary bits e.g. 1001

Byte: Combination of 8 binary bits e.g. 1001 0111

Logic Function and Boolean Algebra

Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations.

INTRODUCTION 

• Boolean algebra is used to design and simplify circuits of electronic devices.
• Each input and output are thought as a member of the set {0, 1}.
• The basic elements of circuits are called gates. Each type of gate implements Boolean operation.

Boolean Algebra

Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations. It makes possible to transform logical statements into mathematical symbols and to calculate the truth or falsity of related statements by using rules. It is named after George Boole, a 19th-century Mathematician and Philosopher, who was the first to try and to formalize what we call logic or reasoning.

This symbolic logic which was designed using variables and symbols is the language used to describe and manipulate logical statements and determine whether they are true or not. Through this, certain logical statements can be expressed in mathematical terms. For example, the symbols '•', '-' and '+' represent 'and', 'not'and 'or'respectively. Aside from its many applications in the design of computers, it serves as the foundation of information theory.

Boolean variable

A boolean variable is the variables which have only two states i.e. true/ false or right/ wrong or on/off or 0/1. As a computer is a binary system, it operates on an electronic signal which has only 2 possible states.

The signal that does not change its state with time is called constant signal and its value always remains the same i.e. either 1 or 0 whereas a variable signal continuously changes its state according to the time. At some point, the value of the variable signal may be 1 and at some another point, it might be 0.Therefore, these variables which consist of only two values i.e. 1 and 0 are Boolean variables or logic variables. These variables are denoted by English capital letters like A, B, X, Y, etc.

Boolean algebra as Switching algebra:

Boolean Algebra which is also known as the Switching algebra consists of two elements (0 and 1) has two binary operators called OR and AND and another unary operator called NOT. The unary operator performs on a single operand whereas a binary operator requires more than one operand.
Followings are their symbols:

 

Logic function (Boolean function)

Boolean function, commonly known as a logic function is an expression expressed algebraically with binary variables, logical operation symbols, parenthesis and equal sign. For a given value of the binary variables, the logic function can be either 0 or 1.

Example: Consider the logic function in algebraic expression:

F = X.Y.Z’+X.Y
Here, X, Y, Z are Boolean variables. The equation on the right-hand side above is known as an expression. Each occurrence of a variable or its complement in an expression is called literal.So, there are altogether three variables i.e. X, Y and Z and five literals; X, Y, Z’, X, and Y.

Logical circuit
By using transistors or Integrated Circuits (IC) or (LSI), the high and low-level voltage electrical signals called logical elements are produced. These logical elements are essential for the composition of a circuit for a specific operation called logical circuit.

 

Basic Logical/ Boolean Operation:

Introduction:

An operator is a special symbol that indicates the operation to be carried out between two operands. An operation is an action to be carried out upon operands. There are 3 basic Boolean Operations: AND, OR and NOT operations.

• AND operation:
  Known as logical multiplication, it is carried out by dot (.) operator or simply by AND. If the inputs are true, it generates true output. Otherwise, it generates false output. Its logical equation is written as C=A.B or C=A AND B. The truth table of AND operation is:

 

 

 

• OR operation
  Known as logical addition, it is carried out by plus (+) operator or simply by OR. If at least one input is true, it generates true output or else, it gives false output. The logical equation of OR operation is written as C=A+B or C=A OR B. The truth table of OR operation is given below:

 

 

 

 

• NOT Operation
  Also known as the logical compliment, it is carried out by prime (‘) operator or bar (‾). It generates the output opposite the input i.e. if the input is true, it generates false output and vice versa. It's logical equation can be written as C=A’. The truth table of OR operation is:

 

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