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Binary arithmetic is essential part of all digital computers and many other digital system.

1. Binary Addition: It is a key for binary subtraction, multiplication, division. The four rules of the binary addition are.

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 with a carry bit 1.

2. Binary Subtraction:

Subtraction and Borrow, these two words will be used very frequently for the binary subtraction.There are four rules of the binary substration. The four rules of the binary Subtraction are

0 – 0 = 0

0 – 1 = 0 with a borrow 1

1 – 0 = 1

1 – 1 = 0

The radix or base is the number of unique digits, including zero that a positional numeral system uses to represent numbers.

It is a system of naming or representing numbers, as the decimal system or the binary system. The number system is broadly classified into two types. Namely, positional and non-positional number systems.

1. Non-positional number system:

A non-positional number system does not rely on the position of the number, but rather symbols to dictate the value of a number. Roman number is a non-positional number system. Non-positional number system cannot be used to do arithmetic calculations effectively.

2. Positional number system:

It is a system of writing numbers in which the position of a digit affects its value. The Decimal number system and Binary number system are examples of a positional number system. The positional number system can be used to do arithmetic calculations effectively.

1100₍₂₎ + 101₍₂₎ = 10001₍₂₎

Binary equivalent of 2567(10) is 101000000111₍₂₎

Octal equivalent of 2567₍₁₀₎ is 5007₍₈₎ 2567₍₁₀₎ binary in broken into set of three bits is 101 000 000 111 5₍₈₎ 0₍₈₎ 0₍₈₎ 7₍₈₎

Hexadecimal equivalent of 2567₍₁₀₎ is A07₍₁₆₎ 2567₍₁₀₎ binary is divided into set of four bits is 1010 0000 0111 A₍₁₆₎ 0₍₁₆₎ 7₍₁₆₎

EBCDIC (Extended Binary Coded Decimal Interchange Code is a binary code for alphabetic and numeric characters that IBM developed for its larger operating systems. In an EBCDIC file, each alphabetic or numeric character is represented by an 8-bit binary number. 256 possible characters (letters of the alphabet, numerals, and special characters) are defined.

ASCII 7 bit and ASCII 8 bit are the two types.

Anon-positional number system does not rely on the position of the number, but rather symbols to dictate the value of a number.

F = 15, A = 10 C = 12 E = 14

61440 + 2560 + 192 + 14 = 64206₍₁₀₎

Step 1 : 357₍₈₎ = 011 101 111₍₂₎

Step 2: 11101111… group of 4 binary digits from LSB 1110₍₂₎ 1111₍₂₎

8 + 4 + 2 = 14₍₁₀₎ = E₍₁₆₎ 8 + 4 + 2+ 1 = 15₍₁₀₎ =F₍₁₆₎

= EF₍₁₆₎

011 101 111₍₂₎

1536 + 128 + 40 + 7 = 1711(10)

8 + 4+1 = 13 in hexadecimal is D 8 + 2 = 10 in hexadecimal is A

Convert 101110(2) to octal.

                    4 + 1       4 + 2

Decimal number system and Binary number system.

Binary Coded Decimal.

Weighted binary codes are those binary codes which obey the positional weight principle. Each position of the number represents a specific weight. Several systems of the codes are used to express the decimal digits 0 through 9. In these codes, each decimal digit is represented by a group of four bits.

32 + 8 + 4 +1 =45

The term “binary” means “two”. Thus, the binary number system is a system of numbers based on two possible digits – 0 and 1. Each binary digit, or “bit”, is a single 0 or 1, which directly corresponds to a single “switch” in a circuit. Binary codes are used in computers because they are used to represent information in the digital world.

A positional number system that has 8 digits which are 0,1,2,3,4,5,6 and 7 is called octal number system. It is also called a base 8.

The weight of the LSB of an octal number system is 0.

The least weight of the digit in a number is called Least Significant Bit.

A positional number system that has 16 digits which are 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, and F is called hexadecimal number system which is also called as base 16.

The weight of the MSB of the hexadecimal number system is 15.

The highest weight of the digit in a number is called Most Significant Bit.

Binary digit.

• Decimal number system
• Binary number system.
• Octal number system.
• Hexadecimal number system.

The radix or base is the number of unique digits, including zero that a positional numeral system uses to represent numbers.

It is a set of symbols for representing characters. For example, most computers use ASCII codes to represent characters.

Computer systems use electronic circuits which exist in only one of two states i.e., on a state or off state. The binary number system is a method of representing numbers that count by using combinations of only two numerals: zero (0) and one (1).

The decimal number 259 is represented as

2 x 10² + 5 x 101 + 9 x 100

= (2 x 100) + (5 x 10) +(9 x 1)

= 200 + 50 + 9 = 259

The non-positional number system consists of different symbols and each of these symbols represents a value irrespective of its position. In positional number system, each digit gets its value by its position in the digits.

Add an extra digit at the front of the binary number, to indicate whether the number is positive or negative. This digit is called a sign bit.

A negative number is represented in binary number system by adding an extra digit 1 at the front of binary number.

Non-positional number system cannot be used to do arithmetic calculations effectively.

It is a system of writing numbers in which the position of a digit affects its value.

A positional number system that has only 2 digits which are 0 and 1, is called a binary number system it is also called as base 2.

The positional number system that has 10 digits which are 0,1,2,3,4,5,6,7,8 and 9 is called decimal number system. Decimal is also called base 10 because it has 10 digits.

In this type of binary code, the positional weights are not assigned. The examples of non weighted codes are Excess-3 code and Gray code.

Representing a signed number with 1’s complement is done by changing all the bits that are 1 to 0 and all the bits that are 0 to 1.

For example, 1’s complement of 1010100 is 0101011.

Representing a signed number with 2’s complement is done by adding 1 to the 1’s complement representation of the number.

Extended Binary Coded Decimal Interchange Code.

American Standard Code of Information Interchange.

Representing a signed number with 1’s complement is done by changing all the bits that are 1 to 0 and all the bits that are 0 to 1.

28(10) = 11100(2)

25(10) = 01(2)

1. ASCII code: 

ASCII (American Standard Code for Information Interchange) is the most widely used coding system to represent data. ASCII is used on many personal computers and minicomputers. ASCII is a 7-bit code that permits 27 = 128 distinct characters. The 128 different combinations that can be represented in 7 bits are plenty to allow for all the letters, numbers and special symbols. An eight-bit was added. This allowed an additional 128 characters to be represented. The extra 128 combinations are used for symbols such as Ç ü è ©, ®, Æ, etc.

2. EBCDIC code:

(Extended Binary Coded Decimal Interchange Code) Pronounced “ebsuh-****.” The binary code for text as well as communications and printer control from IBM. This data code originated with the System/360 and is still used in IBM mainframes and most IBM midrange computers. It is an 8-bit code

(256 combinations) that stores one alphanumeric character or two decimal digits in a byte.

Complements are used in the digital computers in order to simplify the subtraction operation and for logical manipulations.

The two types of complements are 1’s complement and 2’s complement.

1. 1’s complement: 

The 1 ’s complement of a number is found by changing all 1’s to 0’s and all 0’s to 1’s. This is called as taking complement or 1’s complement. For example, 10101 is 01010 in 1’s complement.

2. 2’s complement:

The 2’s complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1 ’s complement of the number. 2’s complement = 1’s complement + 1

For example, 10101₍₂₎ 1’s complement is 01010 and 01010 + 1 = 01011 is 2’s complement.