Question

change the following octal to binary :
1•) $( 546)_{8}$
2•)$( 737)_{8}$
( need all steps )

Answer 1

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$\mathbb{\underline{\green{Type\: Of\: Problem :}}}$

Conversion of Octal to Binary No System

$\mathbb{\underline{\green{Approach\: To\: Problem :}}}$

¶¶¶ $\textbf{Method -1}$
$\mathcal{A}$. Convert the original number to a decimal number (base 10).

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Steps:

[1]. Find the positional value for each digit
(It depends on the digit's postion & Base of No. system i.e., 8)

[2]. Multiply the obtained positional values by the digits in the corresponding position.

[3]. Add the products calculated in above step.

The total gives the decimal equivalent of the Given Octal No.

$\mathcal{B}$. Convert the decimal number so obtained to the binary number.

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Steps:

[1]. Divide the decimal number to be converted by the 2.

[2]. Note the remainder from Step 1 as the rightmost digit (least significant bit) of the binary No.

[3]. Divide the quotient of the previous by 2.

[4]. Note the remainder from Step 3 as the next digit (to the left) of the binary No.

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 Bit (MSB) of the binary number.

The binary result is obtained by placing all the remainders in order with the least significant bit (LSB) being at the top and the most significant bit (MSB) being at the bottom

^^ Shortcut:

• If the decimal number being divided is even then the remainder will be equal to 0.
•If the decimal number is odd then the remainder will be 1.

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¶¶¶ $\textbf{Method -2}$

Steps :

1. Express the each of octal digit in binary equivalent of 3-bit (by considering it as decimal)

2. Combine all the obtained 3-bit digits (or binary groups) into a single binary No.
This is the equivalent binary No of given octal No.

^^ 3 digits are preferred because the base of octal = 8 = $2^{3}$

^ Here 2 refers to any value either 1 or 0.

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$\mathcal{\underline{\red{SOLUTION:}}}$

Given:

$\mathbb{Question- 1}$

$(546)_{8} = (?)_{2}$

¶¶ Method -1:
• Convert to Decimal

$(546)_{8} = ((5 x 8^{2}) + (4×8^{1}) + (6×8^{0}))$

$(546)_{8} = ( 320+ 32 + 6) = (358)_{10}$

• Convert to binary
2 | 358
2 | 179 - 0 (L.S.B)
2 | 89 - 1
2 | 44 - 1
2 | 22 - 0
2 | 11 - 0
2 | 5 - 1
2 | 2 - 1
2 | 1 - 0
.• | 0 - 1 (M.S.B)

$(358)_{10} = (101100110)_{2}$

•°• $\underline{\underline{(546)_{8} = (358)_{10} = (101100110)_{2}}}$

¶¶ Method - 2:
$(546)_{8} =?$

5 = 101
4 = 010
6 = 110

•°• $\underline{\underline{(546)_{8} =(101010110)_{2}}}$

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$\mathbb{Question- 2}$
$(737)_{8} = (?)_{2}$

¶¶ Method -1:
• Convert to Decimal

$(737)_{8} = ((7 x 8^{2}) + (3×8^{1}) + (7×8^{0}))$

$(737)_{8} = (448 + 24 + 7) = (479)_{10}$

• Convert to binary
2 | 479
2 | 239 - 1 (L.S.B)
2 | 119 - 1
2 | 59 - 1
2 | 29 - 1
2 | 14 - 1
2 | 7 - 0
2 | 3 - 1
2 | 1 - 1
.• | 0 - 1 (M.S.B)

$(479)_{10} = (111011111)_{2}$

•°• $\underline{\underline{(737)_{8} = (479)_{10} = (111011111)_{2}}}$

¶¶ Method - 2:
$(737)_{8} =?$

7 = 111
3 = 011
7 = 111

•°• $\underline{\underline{(737)_{8} =(111011111)_{2}}}$

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