Convert 20.45 in base eight to to base ten

To convert \( 20.45_8 \) (base 8) to base 10, we separate the integer and fractional parts and handle each individually:

### Integer part:
The integer part is 20 in base 8. We convert it to base 10 by expanding it as:

\[
20_8 = 2 \times 8^1 + 0 \times 8^0 = 2 \times 8 = 16_{10}
\]

### Fractional part:
The fractional part is 0.45 in base 8. We convert it by multiplying each digit by \( 8^{-n} \) where \( n \) is the position of the digit after the decimal point:

\[
0.45_8 = 4 \times 8^{-1} + 5 \times 8^{-2}
\]

\[
= 4 \times \frac{1}{8} + 5 \times \frac{1}{64}
\]

\[
= 0.5 + 0.078125 = 0.578125_{10}
\]

### Final conversion:
Now, we combine the integer and fractional parts:

\[
20.45_8 = 16 + 0.578125 = 16.578125_{10}
\]

So, \( 20.45_8 = 16.578125_{10} \).

Would you like more details, or do you have any questions?

Here are some related questions you might consider:
1. How do you convert a base 10 decimal to base 8?
2. What is the difference between converting integer and fractional parts between bases?
3. How do you convert numbers in other bases like base 2 or base 16?
4. Can irrational numbers be expressed exactly in other bases?
5. What happens to repeating decimals when converted between different bases?

**Tip:** When working with fractional parts in different bases, always pay attention to how digits are affected by negative powers of the base.

Learn how to convert the number 20.45 from base 8 to base 10. This detailed solution explains the process of converting both integer and fractional parts, using key mathematical formulas and concepts. Suitable for students in Grades 9-11.

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