What is a binary expansion of a real number? Related to this question I asked.
I want to know what is exactly meant by a binary expansion of a number, real or natural.
Can someone show me via example, how would you binary expand a real number like 1.575? And can someone show me how 535 would be broken up?
My next question is why does this work? Why do binary expansions work? Why can't I get the same thing out of a power of 3 or 4 or 5 for that matter?
Why does powers of 2 work?
 A: For any base $b\geq 2$ (of course, $b\in\mathbb N$) and any real number $x$, you can write the number in base $b$ as
$$x = a_na_{n-1}\dots a_0.b_1b_2b_3\dots$$
where this means that $$x=a_0 + a_1\cdot b + a_2\cdot b^2 + \cdots + a_nb^n + b_1b^{-1} + b_2b^{-2} + \cdots$$ and all numbers $a_i$ and $b_i$ are elements of $\{0,1,\dots, b-1\}$
this can be done no matter what number $b$ you choose as the base. For example, in base $10$, the number $16$ is written as $16$ because $16=1\cdot 10 + 6\cdot 10^{0}$. In base $2$, the number is written as $10000$, because $$16=0\cdot 2^0 + 0\cdot 2^1 + 0\cdot 2^2 + 0\cdot 2^3 + 1\cdot 2^4.$$

This is also true for numbers that are not integers, for example, in base $10$, you write $\frac14$ as $0.25$ because $$\frac14 = 2\cdot 10^{-1} + 5\cdot 10^{-2}$$ while in base $2$, it's written as $0.01$ because $$\frac14 = 0\cdot 2^{-1} + 1\cdot 2^{-2}.$$
However, for non-integers, the expansions can sometimes be (more like usually is:) infinite. For example, the number $\frac13$ is written as $0.3333333\dots$ in base $10$, but in base $3$, it's simply $0.1$.

Naturally, for bases higher than $10$, we don't have enough symbols to write all the numbers, but we can make more up. For example, the base $16$ system uses the letters $a,b,c,d,e,f$ to represent the numbers $10,11,12,13,14,15$, meaning that $31$ in base $16$ is written as $$1f$$ because $$31 =1\cdot 16 + f\cdot 16^0.$$
Similarly, in that base, $$\frac{14}{16} = 0.e$$
