Question

Could someone please show me how to convert 37.223_10 to hexadecimal/base 16?

(1) It's the fractional part that confuses me.
(2) I get that the whole part is 25.
(3) The answer seems to be: 25.3916872b020c49ba5e353f7ced916872b020c49ba5e353f7ced916872b02..._16

Any help in solving the fractional part would be greatly appreciated!

Answer 1

What do fractional parts mean in base 10? 0.223 means:
\[2 \cdot 10^{-1} + 2 \cdot 10^{-2} + 3 \cdot 10^{-3}\]
The same is true for fractional parts in base 16. 0.2ae5 has the value of:
\[2 \cdot 16^{-1} + 10 \cdot 16^{-2} + 14 \cdot 16^{-3} + 5 \cdot 16^{-4}\]
So in base 10 that fraction is 0.1675567...

Now, how can you convert a fraction to another base? Very similiar to how you convert integer numbers. If you recal my answers to your other question about base conversion, I gave you two methods to convert numbers.

Take for example 0.3 (base 10). What is this number in base 16?
Using the first method, you need to start with the question of how many times 16^-1 fits into 0.3. The answer is 4.79999999999... times. The first digit of the fractional part is 4, and we continue our calculations with 0.3 - 4 * 16^-1 and see how many times 16^-2 fits into that. And so on.

The other method is a bit simpler. You multiply 0.3 by 16, which gives 4.79999... as a result. So the first digit of the fraction is 4. Now multiply 0.79999.... by 16 to get the next digit and so on.