The octal number (651.124)8 is equivalent to ______

Understand the Problem

The question is asking for the decimal equivalent of the given octal number (651.124), meaning we need to convert this octal number into the decimal number system.

Answer

$425.1640625$

Steps to Solve

Identify the integer and fractional parts

The octal number consists of two parts: the integer part (651) and the fractional part (124). We will convert each part separately.

Convert the integer part

To convert the integer part from octal to decimal, we use the formula: $$ \text{Decimal} = d_n \cdot 8^n + d_{n-1} \cdot 8^{n-1} + \ldots + d_1 \cdot 8^1 + d_0 \cdot 8^0 $$ For 651:

$6 \cdot 8^2 = 6 \cdot 64 = 384$
$5 \cdot 8^1 = 5 \cdot 8 = 40$
$1 \cdot 8^0 = 1 \cdot 1 = 1$

Adding these values: $$ 384 + 40 + 1 = 425 $$

Convert the fractional part

To convert the fractional part from octal to decimal, we use: $$ \text{Decimal} = d_{-1} \cdot 8^{-1} + d_{-2} \cdot 8^{-2} + \ldots $$ For 124:

$1 \cdot 8^{-1} = 1 \cdot \frac{1}{8} = 0.125$
$2 \cdot 8^{-2} = 2 \cdot \frac{1}{64} = 0.03125$
$4 \cdot 8^{-3} = 4 \cdot \frac{1}{512} = 0.0078125$

Adding these values: $$ 0.125 + 0.03125 + 0.0078125 = 0.1640625 $$

Combine both parts

Now, we add the integer and fractional results: $$ 425 + 0.1640625 = 425.1640625 $$

The decimal equivalent of the octal number 651.124 is $425.1640625$.

More Information

The octal number system uses base 8, in which the digits range from 0 to 7. This method of conversion highlights the differences between numeral systems. Each digit is multiplied by the base raised to the power of its position.

Tips

Neglecting the position values: Remember to raise the base (8) to the power of the position; failing to do this will yield incorrect results.
Confusing integer and fractional conversions: Ensure you treat the integer and fractional parts separately and correctly apply their respective formulas.

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