Prepared By

Asst. Lect. Mohammed Salim

Department of IT

1 LFU 2014

Contents

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Binary Addition, Subtraction, Multiplication,

Division

1's Complement of Binary Number.

2's Complement of Binary Number.

Signed Numbers & Binary Coded Decimal (BCD)

Gray Code

Excess-3 Code

Notes

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What is Binary Arithmetic?

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Binary arithmetic is essential part of all the digital

computers and many other digital system such as

mobile phones .

Binary numbers are NUMBERS: That means you

can add, subtract, multiply, and divide.

2 + 2 = 4 , and

in Binary: 10 + 10 = 100

Binary Addition Rules

Rules:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10= 0 with 1 to carry

1 + 1 + 1 =11 = 1 with 1 to carry

11_

11 0011

+ 1011

11 1110

carried bits

Addition Example & Practice

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Ex: 1110 + 1010 = 11000

Ex: 1001 + 111= 10000

Ex: 1111 0000 + 1111 = 1111 1111

Now take a few minutes to try these two:

111 1000 + 1111

1000 1100 + 1100 0110

Subtraction

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The rules for binary subtraction are:Binary

1002

- 0012

-------

0112

0-1 = 1; with borrow of 1

from next column

0 -1 (borrow) - 0 = 1, with

borrow of 1

1 - 1 (borrow) - 0 = 0.

Answer = 0112.

Subtraction Example & Practice

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Now take a few minutes to try these two:

100001 – 11111= ?

11100 - 1111 = ?

Multiplication

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Division

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Ex: 11010/101 = 101

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1's Complement of Binary Number As the binary system has base radix = 2. So the two types of

complements for the binary system are 2's complement and 1's

complement.

Complements are used in the digital computers in order to simplify

the subtraction operation and for the logical manipulations, and also

to represent negative numbers.

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. Example of 1's Complement is:

Note: All complements should deal with a specific byte or bit size. For example the number above

is 5bits long.

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1's Complement Problems

2's Complement of Binary Number

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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

Example of 2's Complement is as follows.

The problems of multiple representations of 0 and the need for the end-around

carry are solved by a system called two's complement. In two's complement,

negative numbers are represented by the bit pattern which is one greater (in an

unsigned sense) than the ones' complement of the positive value. In two's-

complement, there is only one zero, represented as 00000000.

2's Complement & Signed Numbers

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Signed Binary Numbers

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Binary Number

Representation

Binary Coded Decimal (BCD)

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Binary Coded Decimal (BCD):

In computing and electronic systems, binary-coded decimal (BCD) is

a class of binary encodings of decimal numbers where each decimal

digit is represented by a fixed number of bits, usually four or eight,

Gray Code

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The reflected binary code, which invented by Frank

Gray, is a binary numeral system where two

successive values differ in only one bit.

The Gray code was originally designed

to prevent false output from

electromechanical switches. Today, Gray

codes are widely used to facilitate error

correction in digital communications such

as digital terrestrial television and some

cable TV systems.

Binary to Gray code conversion

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It is easy to convert binary code to gray, by following these steps:

(1) The M.S.B. of the gray code will be exactly equal to the first bit of

the given binary number.

(2) Now the second bit of the code will be exclusive-or of the first

and second bit of the given binary number, i.e if both the bits are

same the result will be 0 and if they are different the result will be 1.

(3)The third bit of gray code will be equal to the exclusive-or of the

second and third bit of the given binary number. Thus

the Binary to gray code conversion goes on. One example given

below can make your idea clear on this type of conversion.

Let (01001) be the given binary number

Thus the equivalent gray code is 01101.

Gray to Binary code conversion

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It is also very easy process. Just follow these steps:

(1) The M.S.B of the binary number will be equal to the M.S.B of the

given gray code.

(2) Now if the second gray bit is 0 the second binary bit will be same

as the previous or the first bit. If the gray bit is 1 the second binary bit

will alter. If it was 1 it will be 0 and if it was 0 it will be 1.

(3) This step is continued for all the bits to do Gray code to binary

conversion.

One example given below will make your idea clear.

Let the gray code be 01101

The binary code = (01001)

More Examples on Gray Code Conversion

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Excess 3 Code

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It is a complementary BCD code and numeral system. It is basically a binary code which is made by adding 3 to the equivalent decimal of a binary number and again converting it into binary number.

Examples:

Excess-3 of 12 is 0001 0010 + 0011 0011 = 0100 0101 (45) .

Excess-3 of 6 is 0110(6) + 0011(3)= 1001(9) .