6. Converting octal numbers to decimal and binary numbers

Post date: 01-Jul-2014 04:54:55

Converting octal numbers to decimal and binary numbers

Converting octal numbers to decimal

The method of converting octal numbers to decimal numbers is similar as converting binary numbers to to decimal. However, place values increases by a factor of 8. Notify that octal numbers have the following range of digits 0,1,2,3,4,5,6,7 which count to 8. A number like 896 is not a valid octal numbers because it has 8 and 9 digits

To convert octal numbers to decimal, proceed as follows in the example:

Convert 752₈ to its base 10 equivalent.

Step1: write each number under its place value as shown below

Step2: multiply each number by its place value then add the results

Thus: (64x7) + (8x5) + (1x2)

448+40+2

= 490

Therefore, 752₈= 490₁₀

Convert octal fraction to decimal fraction

Problem 1: Convert ( 2 1 . 2 1 )₈= ( ? )₁₀

_{= 2 1 . 2 1}

↑ ↑

MSD LSD

= 2 x 8¹ + 1 x 8⁰ . 2 x 8^-1 + 1 x 8^-2

= 2 x 8 + 1 x 1 . 2 x ( 1 / 8 ) + 1 x ( 1 / 64 )

= 16 + 1 . ( 0. 2 5 ) + ( 0 . 0 1 5 6 2 5 )

= 17 + 0. 265625

= 17 . 265625

Therefore ( 2 1 . 2 1 )₈ = ( 1 7 . 2 6 5 6 2 5 )₁₀

Converting octal numbers to binary

Number system conversion table

Procedure

The highest digit in the octal number system is (7)₈, from the number system conversion table we can see that (111)₂ is the binary equivalent of the highest octal digit and three binary digits(bits) are required to represent the highest octal digit.
In the octal to binary conversion method, we will convert each digit from the given octal number into three bit binary number starting from the right most octal digit(LSD) to the left most digit(MSD) and at the end combine each three bit binary number to form the binary equivalent of given octal number.
The steps involved in the octal to binary conversion is explained with example below:

Example-1: Convert octal number (375)₈ to binary number (?)₂

Step 1: Write down each digit from the given octal number leaving some space between each octal digit as shown below

Octal Number -> 3 7 5

Step 2: If you have memorized the binary equivalents of all the octal digits this step becomes very easy, otherwise you can still refer the conversion table to find the binary equivalents of each octal digit.

Write down the three bit number below each octal digit as shown below:

Octal Number -> 3 7 5

↓ ↓ ↓

Binary Number -> 011 111 101

Hence, the binary equivalent of the given octal number is (11111101)₂

Convert octal fraction to decimal fraction

Problem 1: Convert ( 2 1 . 2 1 )₈= ( ? )₁₀

_{= 2 1 . 2 1}

↑ ↑

MSD LSD

= 2 x 8¹ + 1 x 8⁰ . 2 x 8^-1 + 1 x 8^-2

= 2 x 8 + 1 x 1 . 2 x ( 1 / 8 ) + 1 x ( 1 / 64 )

= 16 + 1 . ( 0. 2 5 ) + ( 0 . 0 1 5 6 2 5 )

= 17 + 0. 265625

= 17 . 265625

Therefore ( 2 1 . 2 1 )₈ = ( 1 7 . 2 6 5 6 2 5 )₁₀

Review questions

1. Convert ( 0.357 )₈= ( ? )₁₀

Answer: highlight on blank space belowto see answers

Therefore ( 0 . 3 5 7)₈ = ( 0 . 466796875 )₁₀

2. Convert ( 100.01 )₈= ( ? )₁₀

Answer: highlight on blank space belowto see answers

Therefore ( 1 0 0 . 0 1 )₈ = ( 64 . 0 1 5 6 2 5 )₁₀