5.7.1 Patterns in Real Numbers

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• Each number is half of the previous number. We can find the next number by multiplying the current number by $\frac{1}{2}$ or by 0.5, or by dividing by 2.
• The non-zero digits of the decimals are powers of 5 ($5^1$ , $5^2$ , $5^3$, etc.), and each time there is an additional decimal place.
• 0.5 can be written as $\frac{5}{10}$, 0.25 as $\frac{25}{100}$, and 0.125 as $\frac{125}{1,000}$. The numerators are powers of 5, and the denominators are powers of 10, so the next two numbers are $\frac{5^4}{{10}^4}$, which is $\frac{65}{10,000}$ or 0.0625, and $\frac{5^5}{{10}^5}$, which is $\frac{3,125}{100,000}$ or 0.03125.
• The denominators in the fractions are powers of 2.