The octal number system provides a convenient way of converting large binary numbers into more compact and smaller groups. The binary number system is a base 2 number system since it only uses the digits 0 and 1. Octal is a base 8 number system since it uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.
Binary and octal numbers often used in computing applications, so it’s fairly common to need to convert from one to the other. The base 8 system is often used in legacy computing applications because a single octal digit can represent three binary bits, which are cleanly divisible in 6, 12, 24, and 36-bit computer systems.
Converting a number from Binary to Octal Number System
There are two ways to convert binary to octal, which are explained below.
1- The direct method of binary to octal conversion
2- Convert binary to decimal and then decimal to octal
1- Direct Method
To convert the numbers from octal to binary, proceed with the steps given below:
◆ Step 1 − The first step is to break the binary number into groups of three digits, starting from the right to the left. The reason for this is that a group of three bases 2 numbers, or 2³ is equal to 8, which is evenly divisible into the base 8 number system.
◆ Step 2 − At this point, each group of three binary digits can be converted to an octal digit using the following table.
Examples :
Example 1 - Convert the binary number 10010110 into an octal number.
First make groups of three digits from the right most to the left and note that you can add 1 or 2 zeroes before the leftmost digit or after the rightmost digit to make complete group of 3 bits
10 010 110 ⟶ 010 010 110
So, (10010110)₂ = (226)₈
Example 2 - Convert the binary number (1010011.11101)₂ into an octal number.
First make groups of three digits from the right most to the left and note that you can add 1 or 2 zeroes before the leftmost digit or after the rightmost digit to make complete group of 3 bits.
1 010 011. 111 01 ⟶ 001 010 011. 111 010
So,(1010011.11101)₂ = (123.72)₈
2- Convert binary to decimal and then decimal to octal
In this method, First, you need to convert a binary into another base system (e.g., into decimal, or hexadecimal). Then you need to convert it to an octal number. Since number numbers are a type of positional number system. That means the weight of the positions from right to left are 8⁰, 8¹, 8², 8³, and so on for the integer part, and the weights of the positions from left to right are 8-¹, 8-², 8-³and so on. for the fractional part.
Examples :
Example 1 - Convert the binary number 10010110 into an octal number.
First, convert this into a decimal number
= (10010110)2
= 0x2⁰+1x2¹+1x2²+0x2³+1x2⁴+0x2⁵+0x2⁶+1x2⁷
= 0+2+4+0+16+0+0+128
= (150)₁₀
Then, convert it into an octal number
= (150)₁₀
= 2x8²+2x8¹+6x8⁰
= (226)₈ which is the answer.
Example 2 -
First, convert the binary number to a decimal number.
(1011101)₂ = (1 x 2⁰) + (0 x 2¹) + (1 x 2²)+ (1 x 2³) + (1 x 2⁴) + (0 x 2⁵) + ( 1 x 2⁶)
= 64 + 0 + 16 + 8 + 4 + 0 + 1
= 93
(1011101)₂ = (93)₁₀
The next step is to convert the decimal number to an octal number by dividing 93 by 8.
93 divided by 8 will give 5 as the remainder and 11 as the quotient
11 divided by 8 will give 3 as the remainder and 1 as the quotient
1 divided by 8 will give 1 as the remainder and 0 as the quotient
Collect the remainder in reverse order we get 1 3 5
So, The binary number (1011101)₂ = (135)₈
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