# NUMBER SYSTEM CONVERTING, Formulas and forms for Digital Systems Design. University of Sindh

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CONVERTING NUMBER SYSTEM BASE SYSTEM
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NUMBER SYSTEM CONVERSION

There are many methods or techniques which can be used to convert numbers from one base to another. We'll demonstrate here the following −

Decimal to Other Base System

Other Base System to Decimal

Other Base System to Non-Decimal

Shortcut method − Binary to Octal

Shortcut method − Octal to Binary

Shortcut method − Binary to Hexadecimal

Shortcut method − Hexadecimal to Binary

Decimal to Other Base System

Steps

Step 1 − Divide the decimal number to be converted by the value of the new base.

Step 2 − Get the remainder from Step 1 as the rightmost digit of new base number.

Step 3 − Divide the quotient of the previous divide by the new base.

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

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.

The last remainder thus obtained will be the Most Significant Digit of the new base number.

Example −

Decimal Number: 2910

Calculating Binary Equivalent −

Step Operation Result Remainder

Step 1 29 / 2 14 1

Step 2 14 / 2 7 0

Step 3 7 / 2 3 1

Step 4 3 / 2 1 1

leastsignificantdigit

totheleft

MSD

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Step 5 1 / 2 0 1

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit and the last remainder becomes the Most Significant Digit .

Decimal Number − 2910 = Binary Number − 111012.

Other Base System to Decimal System

Steps

Step 1 − Determine the column value of each digit .

Step 2 − Multiply the obtained column values by the digits in the corresponding columns.

Step 3 − Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Example

Binary Number − 111012

Calculating Decimal Equivalent −

Step Binary Number Decimal Number

Step 1 111012 ((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10

Step 2 111012 10

Step 3 111012 2910

Binary Number − 111012 = Decimal Number − 2910

Other Base System to Non-Decimal System

Steps

Step 1 − Convert the original number to a decimal number .

Step 2 − Convert the decimal number so obtained to the new base number.

Example

Octal Number − 258

Calculating Binary Equivalent −

LSD MSD

positional

thisdependsonthepositionofthedigitandthebaseofthenumbersystem

inStep1

16 + 8 + 4 + 0 + 1

base10

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Step 1 − Convert to Decimal

Step Octal Number Decimal Number

Step 1 258 ((2 × 81) + (5 × 80))10

Step 2 258 10

Step 3 258 2110

Octal Number − 258 = Decimal Number − 2110

Step 2 − Convert Decimal to Binary

Step Operation Result Remainder

Step 1 21 / 2 10 1

Step 2 10 / 2 5 0

Step 3 5 / 2 2 1

Step 4 2 / 2 1 0

Step 5 1 / 2 0 1

Decimal Number − 2110 = Binary Number − 101012

Octal Number − 258 = Binary Number − 101012

Shortcut method - Binary to Octal

Steps

Step 1 − Divide the binary digits into groups of three .

Step 2 − Convert each group of three binary digits to one octal digit.

Example

Binary Number − 101012

Calculating Octal Equivalent −

16 + 5

startingfromtheright

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Step Binary Number Octal Number

Step 1 101012 010 101

Step 2 101012 28 58

Step 3 101012 258

Binary Number − 101012 = Octal Number − 258

Shortcut method - Octal to Binary

Steps

Step 1 − Convert each octal digit to a 3 digit binary number .

Step 2 − Combine all the resulting binary groups into a single binary number.

Example

Octal Number − 258

Calculating Binary Equivalent −

Step Octal Number Binary Number

Step 1 258 210 510

Step 2 258 0102 1012

Step 3 258 0101012

Octal Number − 258 = Binary Number − 101012

Shortcut method - Binary to Hexadecimal

Steps

Step 1 − Divide the binary digits into groups of four .

Step 2 − Convert each group of four binary digits to one hexadecimal symbol.

Example

Binary Number − 101012

theoctaldigitsmaybetreatedasdecimalforthisconversion

of3digitseach

startingfromtheright

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Step 1 101012 0001 0101

Step 2 101012 110 510

Step 3 101012 1516

Binary Number − 101012 = Hexadecimal Number − 1516

Shortcut method - Hexadecimal to Binary

Steps

Step 1 − Convert each hexadecimal digit to a 4 digit binary number .

Step 2 − Combine all the resulting binary groups into a single binary number.

Example

Calculating Binary Equivalent −

Step 1 1516 110 510

Step 2 1516 00012 01012

Step 3 1516 000101012

Hexadecimal Number − 1516 = Binary Number − 101012