system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a binary number which can be represented perfectly, where it cannot be represented perfectly in a decimal form?

There are certain fractions which cannot be represented using a finite amount of digits. For example, 1/3 = 0.33333..... We can never truly represent 1/3 as a decimal with a finite number of digits.

Similarly, there are certain numbers which cannot be represented perfectly in binary - 1/3 is one of these numbers. However, 0.1 is also one of these numbers. You can never correctly represent 1 tenth in binary with a finite number of digits, even though you CAN in base 10.

Are there any numbers which can be represented in binary, but cannot be represented in denary? If not, why not? I've tried to think of an example but can't, and it's a toughie to google too.

My guess is that there are no such numbers, because all binary numbers can be made up by adding together the fractions 1/(2^n) depending on the place of the current digit.
gpt
If a number can be represented with finite digit in base b, it has to be of the form k/b^n for some integer k and n. Right? Because if it has n digits after the point, you multiply it by b^n (which equals shifting all digits left by n) and you get an integer, which is k.

So you want a number of the form a/2^(n), but not of the form b/10^(m).

But you just do

Your number = a/2^n = (5^n a)/(5^n 2^(n)) = b/10^n

After defining b = 5^(n) * a. 

So all number with a finite expansion in base 2 also have a finite expansion in base 10.

This in general holds if the first base divides the second.