Questions: Convert the number from octal to binary form. 513eight

Solution Steps

To convert a number from octal to binary, we can first convert the octal number to a decimal number and then convert the decimal number to binary. This involves two main steps:

1. Convert the octal number to decimal by multiplying each digit by 8 raised to the power of its position index (starting from 0 from the right).
2. Convert the resulting decimal number to binary by repeatedly dividing the number by 2 and recording the remainders.

To convert the octal number \( 513_8 \) to decimal, we use the formula:

\[ 513_8 = 5 \cdot 8^2 + 1 \cdot 8^1 + 3 \cdot 8^0 \]

Calculating each term:

\[ 5 \cdot 8^2 = 5 \cdot 64 = 320 \] \[ 1 \cdot 8^1 = 1 \cdot 8 = 8 \] \[ 3 \cdot 8^0 = 3 \cdot 1 = 3 \]

Adding these values together gives:

\[ 320 + 8 + 3 = 331 \]

Thus, \( 513_8 = 331_{10} \).

Next, we convert the decimal number \( 331_{10} \) to binary. We repeatedly divide by 2 and record the remainders:

\[ 331 \div 2 = 165 \quad \text{remainder } 1 \] \[ 165 \div 2 = 82 \quad \text{remainder } 1 \] \[ 82 \div 2 = 41 \quad \text{remainder } 0 \] \[ 41 \div 2 = 20 \quad \text{remainder } 1 \] \[ 20 \div 2 = 10 \quad \text{remainder } 0 \] \[ 10 \div 2 = 5 \quad \text{remainder } 0 \] \[ 5 \div 2 = 2 \quad \text{remainder } 1 \] \[ 2 \div 2 = 1 \quad \text{remainder } 0 \] \[ 1 \div 2 = 0 \quad \text{remainder } 1 \]

Reading the remainders from bottom to top, we find:

\[ 331_{10} = 101001011_2 \]

Final Answer