Convert the number (102.10) to base 2, 8, 16.

To convert the decimal number 102.10 to base 2 (binary), base 8 (octal), and base 16 (hexadecimal), we need to handle the integer part (102) and the fractional part (0.10) separately.

Conversion to Base 2 (Binary)

Integer Part: 102

1. Divide the number by 2 and write down the remainder.
2. Use the quotient as the new number to be divided by 2.
3. Repeat the process until the quotient is 0.
4. The binary number is the remainders read in reverse order.

Let's convert the integer part 102 to binary:

Step | Quotient | Remainder | Binary
-------------------------------------
  1  |   51     |     0     |   0
  2  |   25     |     1     |   10
  3  |   12     |     1     |   110
  4  |    6     |     0     |   0110
  5  |    3     |     0     |   00110
  6  |    1     |     1     |   100110
  7  |    0     |     1     |  1100110

So, 102 in decimal is 1100110 in binary.

Fractional Part: 0.10

1. Multiply the fractional part by 2.
2. The whole number part of the result is the next binary digit (0 or 1).
3. Use the new fractional part (result - whole number part) for the next multiplication.
4. Repeat the process until the fractional part is 0 or until you reach the desired precision.

Let's convert the fractional part 0.10 to binary:

Step | Fraction * 2 | Whole Number | Fractional Part | Binary
--------------------------------------------------------------
  1  |    0.20      |       0      |      0.20      |   .0
  2  |    0.40      |       0      |      0.40      |   .00
  3  |    0.80      |       0      |      0.80      |   .000
  4  |    1.60      |       1      |      0.60      |   .0001
  5  |    1.20      |       1      |      0.20      |   .00011
  ... (and so on, until desired precision or repeating pattern)

The fractional part will not resolve to a finite binary number in this case because 0.10 in decimal is a repeating fraction in binary. For the sake of this example, let's stop at 4 digits after the decimal point, which gives us 0.0001 in binary.

So, 0.10 in decimal is approximately 0.0001 in binary.

Combining both parts, the number 102.10 in decimal is approximately 1100110.0001 in binary.

Conversion to Base 8 (Octal)

Integer Part: 102

The process is similar to binary conversion, but instead of dividing by 2, we divide by 8.

Step | Quotient | Remainder | Octal
-------------------------------------
  1  |   12     |     6     |   6
  2  |    1     |     4     |   46
  3  |    0     |     1     |  146

So, 102 in decimal is 146 in octal.

Fractional Part: 0.10

The process is similar to binary conversion, but instead of multiplying by 2, we multiply by 8.

Step | Fraction * 8 | Whole Number | Fractional Part | Octal
--------------------------------------------------------------
  1  |    0.80      |       0      |      0.80      |   .0
  2  |    6.40      |       6      |      0.40      |   .06
  3  |    3.20      |       3      |      0.20      |   .063
  4  |    1.60      |       1      |      0.60      |   .0631
  ... (and so on, until desired precision or repeating pattern)

Again, for the sake of this example, let's stop at 3 digits after the decimal point, which gives us 0.063 in octal.

So, 0.10 in decimal is approximately 0.063 in octal.

Combining both parts, the number 102.10 in decimal is approximately 146.063 in octal.

Conversion to Base 16 (Hexadecimal)

Integer Part: 102

The process is similar to binary and octal conversion, but instead of dividing by 2 or 8, we divide by 16.

Step | Quotient | Remainder | Hexadecimal
-------------------------------------------
  1  |    6     |     6     |   6
  2  |    0     |     6     |  66

So, 102 in decimal is 66 in hexadecimal.

Fractional Part: 0.10

The process is similar to binary and octal conversion, but instead of multiplying by 2 or 8, we multiply by 16.

Step | Fraction * 16 | Whole Number | Fractional Part | Hexadecimal
--------------------------------------------------------------------
  1  |     1.60      |       1      |      0.60      |   .1
  2  |     9.60      |       9      |      0.60      |   .19
  ... (and so on, until desired precision or repeating pattern)

Again, for the sake of this example, let's stop at 2 digits after the decimal point, which gives us 0.19 in hexadecimal.

So, 0.10 in decimal is approximately 0.19 in hexadecimal.

Combining both parts, the number 102.10 in decimal is approximately 66.19 in hexadecimal.

In summary, the decimal number 102.10 converts to the following in binary, octal, and hexadecimal:

Base	Conversion Result
Binary (2)	1100110.0001 (approx.)
Octal (8)	146.063 (approx.)
Hexadecimal (16)	66.19 (approx.)

Please note that the fractional parts are approximations due to the repeating nature of the decimal fraction in other bases.