What is the decimal equivalent of the octal number (651.124)8?

Steps to Solve

1. Identify the position values of the digits The octal number consists of two parts: the whole number part (651) and the fractional part (0.124). Each digit's position in the octal system represents a power of 8.

For the whole number part:

• The rightmost digit (1) is at position 0: (8^0)
• The next digit (5) is at position 1: (8^1)
• The leftmost digit (6) is at position 2: (8^2)

For the fractional part:

• The rightmost digit (4) is at position -1: (8^{-1})
• The next digit (2) is at position -2: (8^{-2})
• The leftmost digit (1) is at position -3: (8^{-3})

2. Convert the whole number part to decimal Calculate each digit's decimal value and sum them up: $$ 6 \times 8^2 + 5 \times 8^1 + 1 \times 8^0 $$ Calculate:
   $$ 6 \times 64 + 5 \times 8 + 1 \times 1 = 384 + 40 + 1 = 425 $$

3. Convert the fractional part to decimal Calculate each digit's fractional value and sum them up: $$ 1 \times 8^{-1} + 2 \times 8^{-2} + 4 \times 8^{-3} $$ Calculate:
   $$ 1 \times \frac{1}{8} + 2 \times \frac{1}{64} + 4 \times \frac{1}{512} = \frac{1}{8} + \frac{2}{64} + \frac{4}{512} $$ Convert to common denominators:
   $$ \frac{64}{512} + \frac{16}{512} + \frac{4}{512} = \frac{84}{512} $$ Reduce the fraction:
   $$ \frac{84}{512} = \frac{21}{128} $$

4. Combine whole number and fractional parts Now, we add the whole number part and the fractional part: $$ 425 + \frac{21}{128} $$ This gives us the final decimal equivalent:
   $$ 425.1640625 $$