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Binary arithmetic operations with Bits and Bytes

Binary arithmetic operations in Computer is a digital machine which can solve arithmetical as well as logical problems. Being a digital in nature it only understands machine language which is made up of ‘zeroes’ and ‘ones’. The only data that a computer can understand is on and off.

BINARY CODE

The memory chips inside the computer are divided into thousands of tiny compartments called bits.

Each bit has an electronic switch or gate.

On represents the gate is open and letting electricity go through. The computer reads on or open switches as a number 1. Closed gates are off because the electricity is blocked and cannot get through. The computer reads off bits as 0.

It is by grouping these bits together to form a series of 1/0 commands, that data is formed.Eight bits are grouped together to form a byte. In this group of eight, there are 256 possible combinations of 1/0. The grouping of 1/0 within a byte is called Binary code.A binary code is hence a combination of 0s’ and 1s’ grouped together as a combination representation of instructions to the computer.

When any user types the letter, say, A on the keyboard, electrical signals are sent from the keyboard to the CPU. The CPU turns the signals into binary code. Then, the computer reads the code and sends it on to the monitor to display the letter A.

Binary arithmetic operations in computer are made up of electrical devices, so it couldn’t use paper to keep track of data. Instead, it uses voltages. A one is represented by a high voltage, while a zero is represented by a low voltage.

DIGITAL AND ANALOG OPERATION

Computers fall broadly into two classes: analog and digital. In analog computers, a number is represented in the machine by a physical quantity as measured by some system of units. Physical quantity can be anything like voltage, current etc. In digital computers, representation of a number is carried out by an algorithm termed as binary number system. This algorithm transforms the number into a sequence of 1’s and 0’s.

There are two basic types of data which are stored and processed by computers, namely characters and numbers.Characters include letters and special symbols. The other types of data are decimal numbers such as 1234,485 etc. The characters and numbers fed to a computer, and the output from the computer, must be in a form which is usable by people For this purpose, natural language symbols and decimal digits are appropriate. These constitute the external data representation.

The representation of data inside a computer must match the technology used by the computer to store and process data . The data inside the computer are represented in the form of Binary strings i.e. 0 and 1. These constitute the internal representation of data.

DATA REPRESENTATION

Data within the computer is represented in the following types:

Binary Number System
Decimal Number System
Octal Number System
Hexadecimal Number System

Binary Number System

Under this system the data is represented in the form of 0s’ and 1s’. This system has only two numbers and hence the base of this system is 2. It is the only system being used by computers for manipulation.

REASONS FOR BINARY NUMBER SYSTEM AT THIS LEVEL

Being electronic machine the binary system is the most ideal system for manipulation and understanding.
It provides a shorthand notation of using only two numbers 0 and 1.
It can do any calculation which a decimal number would do.
Examples of Binary Number System are 110₂, 0110₂, 11002 etc.

Decimal Number System

The Decimal Number System is the system which is composed of numbers from 0 to 9. It has ten digits and the base of this system is also 10.

Examples of Decimal Number System are 123₁₀ , 456₁₀ , 789₁₀ etc.

Octal Number System

The Octal number System has digits starting from 0 to 7. The base of this system is 8.

Examples of Octal number System are 123₈ , 236₈ , 564₈ etc.

Hexadecimal Number System

Under the Hexadecimal number System , there are 0 to 9 digits and alphabets from A to F. A represents 10 , B represents 11, C represents 12, D represents 13, E represents 14 and F represents 15. The base of this system is 16.

Examples of Hexadecimal Number System are A4₁₆, 1AB₁₆, 934₁₆ etc.

BINARY ADDITION

With only two numerals, 1 (one) and 0 (zero) , counting in binary is pretty simple. Just keep in mind the following:

· 0+0=1

· 0+1=1

· 1+0=1

· 1+1=10

· 1+1+1=11

With that in mind we can count by starting at 0. Add 1 to get the next number, namely 1. Add 1 again to get 10. Then:

To calculate the next number:

Add the first (rightmost) digits to get 10.

Write the low digit below the line and carry the 1 just as you would when adding decimal numbers.

Next add the high digit of 11 to the 1 you carried…………..

……….to get 10, and write the 10 below the line just as you would when adding decimal numbers.

Since, 1+1=0 carry 1, 1+1=0 carry 1 and finally, 1+1+1=01+1=11 (as the answer)

We would count in binary as follows:

BINARY SUBTRACTION AND MULTIPLICATION

Finding Complement of a Binary Number:

Complement of any binary number is calculated by replacing all 1’s by 0’s and all 0’s and all 0’s by 1’s.

E.g. the complement of 1101 is 0010.

To subtract any binary number from another binary number, the complement of the number which has to be subtracted is calculated, (subtrahend), then the number so obtained is added to the original number from which the number was to be subtracted. If the result so obtained has a carryover (1), then it is again added to the result obtained and is taken away from the last result.

It there is no carry over then the result so obtained is re complemented and the number so obtained is the answer and is shown along with a negative sign.

e.g. Subtract 1011 from 1101.

Step 1. Calculate the complement of 1011 - 0100

Step 2. Add the number 1101 to the complement so obtained = 0100

Step 3. 1101+0100

1101

- 0100

10001 (1 is the carry over)

So it is again added to the result,

10001

- 1

0010

(the carry over is left with the original result so obtained is not considered after it is added).

Therefore the answer will be 0010.

BINARY MULTIPLICATION

The multiplication of binary numbers is same as decimal numbers.

The following rules are applied to find the product of any binary numbers.

0x0=0

1x0=0

0x1=0

1x1=1

e.g. The product of two binary numbers 1010 and 101 =

1010

X101

1010

0000x

1010xx

110010 (answer)

CONVERSION FROM ONE NUMBER SYSTEM TO ANOTHER

Numbers expressed in decimal are much more meaningful to us than the values expressed in any other number system because we all are using decimal number system in our day to day life. Thus, we are often required to convert numbers in other number systems to decimal .

Positional value of a digit in a positional number system can be found in such way-

If the position from right is n, base is b and the face value is f then positional value is f x (b)^n-1

e.g. positional value of 6 in different number systems are as follows:

(1642)₁₀ = 6 × 10^3-1= 6 × 10² = 600

(1642)₈ = 6 × 8^3-1= 6 × 8² = 384

(1642)₁₆ = 6 × 16^3-1= 6 × 16² = 1536

CONVERTING TO DECIMAL FROM ANOTHER BASE

The following steps are used to convert to a base 10 value from any other number system:

1. Determine the positional value of the number system.

2. Multiply the obtained value by the digits in the corresponding position.

3. Sum the products calculated in step 2.

CONVERSION FROM BINARY TO DECIMAL

Example 1.

Convert 11001 to decimal number.

Solution:

First Method:

Step 1. Determine the positional value.

Step 2.

Multiply column value by corresponding column digits.

16×1 = 16

8×1 = 8

4×0 = 0

2×0 = 0

1×1 = 1

Step 3.

Sum the products .

16+8+0+0+1 = 25

(11001)₂ = (25)₁₀

Second Method:

Example 2.

Convert (10101)₂ into decimal number.

Solution:

Example 3.

Conversion from octal to decimal.

Solution:

Example 4.

Convert (2057)₈into decimal number.

Solution:

Example 5:

Conversion from hexadecimal to decimal .

Solution:

CONVERSION FROM DECIMAL TO A NEW BASE (DIVISION REMAINDER TECHNIQUE)

The following steps are used to convert a number from base 10 to a new base:

1. Divide the decimal number by the value of the new base.

2. Record the remainder from step 1 as the rightmost digit least significant digit (LSD) of the new base number.

3. Divide the quotient of the previous division by the new base.

4. Record the remainder from step 3 as the next digit of the new base number.

Repeat steps 3 and 4 until the quotient becomes less than the new base. The last remainder will be the most significant digit (MSD) of the new base number.