Decimals — place value, ordering, terminating & recurring

Key learnings

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  Each decimal place is ten times smaller than the one before

  Reading left to right after the point: tenths $\tfrac{1}{10}$, hundredths $\tfrac{1}{100}$, thousandths $\tfrac{1}{1000}$, $\ldots$ So in $3.504$ the $5$ is $5$ tenths, the $0$ is $0$ hundredths and the $4$ is $4$ thousandths. Place value is what makes ordering and rounding work.

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  Order decimals by lining up the point and comparing place by place

  Compare the whole-number part first; if equal, compare tenths, then hundredths, and so on. Pad with trailing zeros so every number has the same number of decimal places ($0.4 = 0.40 = 0.400$) — a longer string of digits does NOT mean a bigger number ($0.4 > 0.39$).

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  A fraction terminates only if its denominator is built from 2s and 5s

  Put the fraction in its lowest terms first. If the denominator's only prime factors are $2$ and/or $5$ the decimal terminates (e.g. $\tfrac{7}{40}$, since $40 = 2^3 \times 5$). Any other prime factor (3, 7, 11, $\ldots$) forces it to recur (e.g. $\tfrac{1}{6}$, since $6 = 2 \times 3$).

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  Recurring-decimal dot notation marks the repeating block

  One dot over a single repeating digit ($0.\dot{3} = 0.333\ldots$); a dot over the first and last digit of a repeating block marks everything in between ($0.\dot{1}\dot{7} = 0.171717\ldots$, $0.\dot{2}3\dot{1} = 0.231231\ldots$). Only the dotted digits repeat — any digits before them do not.