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Age of Empires 3. The Age of Empires 3 Gold Edition is a . The Age of Empires 3 Gold Edition is a modification of .. The Age of Empires 3 Gold Edition does not ship with this game in a full game disc, but it will be a .
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The Age of Empires 3 Gold Edition, free and safe download. The Age of Empires 3 Gold Edition latest version: The Age of Empires III Gold Edition is a modification of . 4.03 The Age of Empires Gold Edition is a modification of .Q:

Binary splitting versus binary expansion

Suppose that we have a binary sequence,
$10101010101010\ldots$
which is a binary expansion of a rational number. How can I construct a binary sequence representing the same value?

A:

The simplest way to write the decimal representation of $1/3$ as a sequence of $1$ and $0$ is
$$
1/3=1/3\cdot10^0+1/3\cdot10^{ -1}+1/3\cdot10^{ -2}+\ldots.
$$
You can then convert this sequence into binary by a simple look up table.

A:

To get binary sequences for all rational numbers, you can use periodicity. Let $a,b$ be positive integers such that the sequence
$$
x_n = 1+\frac{a}{2^{n+1}}+\frac{b}{2^{n+1}}
$$
has period $2^{n+1}$ (that is, the sequence repeats after at least one complete period). Then $x_{2^k}$ has binary expansion
$$
1\,0\,1\,0\,1\,0\,1\,0\,1\,0\,1\,0\,1\,0\,1\,0\,1
50b96ab0b6

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