Express the following octal numbers into their equivalent decimal numbers. (i) 145 (ii) 6760 (iii) 455 (iv) 10.75

Express the following octal numbers into their equivalent decimal numbers.

(i) 145

(ii) 6760

(iii) 455

(iv) 10.75

To convert numbers from octal to decimal, we use the positional value method. The base value of octal is 8.

Converting 145 from Octal to Decimal

1. Write down the positional values from right to left $8^0, 8^1, 8^2, ...$.

2. Multiply each digit by its positional value.

3. Sum the results.

$$ \begin{array}{cccc} 1 \cdot 8^2 & = & 1 \cdot 64 & = 64 \\ 4 \cdot 8^1 & = & 4 \cdot 8 & = 32 \\ 5 \cdot 8^0 & = & 5 \cdot 1 & = 5 \\ \end{array} $$

$$ 145_8 = 64 + 32 + 5 = 101 $$

Therefore, $(145)_{8} = (101)_{10}$

Converting 6760 from Octal to Decimal

1. Write down the positional values from right to left.

2. Multiply each digit by its positional value.

3. Sum the results.

$$ \begin{array}{cccc} 6 \cdot 8^3 & = & 6 \cdot 512 & = 3072 \\ 7 \cdot 8^2 & = & 7 \cdot 64 & = 448 \\ 6 \cdot 8^1 & = & 6 \cdot 8 & = 48 \\ 0 \cdot 8^0 & = & 0 \cdot 1 & = 0 \\ \end{array} $$

$$ 6760_8 = 3072 + 448 + 48 + 0 = 3568 $$

Therefore, $(6760)_{8} = (3568)_{10}$

Converting 455 from Octal to Decimal

1. Write down the positional values from right to left.

2. Multiply each digit by its positional value.

3. Sum the results.

$$ \begin{array}{cccc} 4 \cdot 8^2 & = & 4 \cdot 64 & = 256 \\ 5 \cdot 8^1 & = & 5 \cdot 8 & = 40 \\ 5 \cdot 8^0 & = & 5 \cdot 1 & = 5 \\ \end{array} $$

$$ 455_8 = 256 + 40 + 5 = 301 $$

Therefore, $(455)_{8} = (301)_{10}$

Converting 10.75 from Octal to Decimal

1. Write down the positional values from right to left for the integer part, and from left to right (negative exponents) for the fractional part.

2. Multiply each digit by its positional value.

3. Sum the results.

$$ \begin{array}{cccc} 1 \cdot 8^1 & = & 1 \cdot 8 & = 8 \\ 0 \cdot 8^0 & = & 0 \cdot 1 & = 0 \\ 7 \cdot 8^{-1} & = & 7 \cdot 0.125 & = 0.875 \\ 5 \cdot 8^{-2} & = & 5 \cdot 0.015625 & = 0.078125 \\ \end{array} $$

$$ 10.75_8 = 8 + 0 + 0.875 + 0.078125 = 8.953125 $$

Therefore, $(10.75)_{8} = (8.953125)_{10}$