Convert the number (210.25)10 to base 8.

To convert the decimal number (210.25) in base 10 to base 8, we need to handle the integer part (210) and the fractional part (0.25) separately. Let's start with the integer part.

Converting the Integer Part (210) to Base 8

1. Divide the number by 8 and record the remainder.
2. Use the quotient obtained from step 1 and divide by 8 again, recording the remainder.
3. Repeat this process until the quotient is 0.
4. The base 8 number is the remainders read in reverse order (from last to first).

Let's apply these steps to the integer part 210:

Division by 8	Quotient	Remainder
210 ÷ 8	26	2
26 ÷ 8	3	2
3 ÷ 8	0	3

Reading the remainders in reverse order, we get the base 8 representation of the integer part: 322.

Converting the Fractional Part (0.25) to Base 8

1. Multiply the fractional part by 8.
2. The whole number part of the result is the next digit in the base 8 representation.
3. Take the new fractional part and multiply by 8 again.
4. Repeat this process until the fractional part is 0 or until you have enough digits for your desired precision.
5. The base 8 number is the sequence of whole number parts obtained.

Let's apply these steps to the fractional part 0.25:

Multiplication by 8	Whole Number Part	Fractional Part
0.25 × 8	2	0.00

Since the fractional part becomes 0 after the first multiplication, we stop here.

The base 8 representation of the fractional part is .2.

Combining Both Parts

Now, we combine the integer and fractional parts to get the full number in base 8:

(210.25) in base 10 = (322.2) in base 8

So, the decimal number 210.25 converted to base 8 is 322.2.