Dot notation marks the repeating block
One dot over a single repeating digit (); a dot over the first and last digit of a repeating block marks everything in between (, ). Only the dotted digits repeat — any digits before them do not.
Each decimal place is 10× smaller
Reading left to right after the point: tenths , hundredths , thousandths , So in the is tenths, the is hundredths and the is thousandths. Place value is what makes ordering and rounding work.
Terminates only if denominator is 2s and 5s
Put the fraction in its lowest terms first. If the denominator's only prime factors are and/or the decimal terminates (e.g. , since ). Any other prime factor (3, 7, 11, ) forces it to recur (e.g. , since ).
Order decimals 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 () — a longer string of digits does NOT mean a bigger number ().
Drawn from real examiner reports.
Rounding too early (too few dp)
Cutting a decimal short part-way through a calculation (premature rounding) loses accuracy and the final accuracy mark; the value should be kept to plenty of decimal places (or in the calculator memory) until the very end.
A commonly reported examiner note noted that "premature rounding cost some students the final accuracy mark" and advised keeping a good number of decimal places.
A recurring 9 equals a round value
For an upper bound students wrote recurring decimals such as or instead of the correct terminating bounds and ; examiners also note that terminating decimals like or were not accepted. The exact bound is the terminating value, not a string of 9s.
A commonly reported examiner note records some students gave $28.4\dot{9}$ / $17.\dot{4}\dot{9}$ for bounds and examiners stressed that $28.499$ / $17.499$ were not accepted.
Wrong powers of 10: tails don't cancel
When a recurring decimal is turned into a fraction, students multiply by powers of at random (e.g. and when one digit recurs) so the repeating tails do not line up and cannot be subtracted away.
A commonly reported examiner note records that where the recurring decimal gave $\tfrac{756}{990}$, zero-scoring students "used incorrect combinations to subtract" instead of matching $x$ with $100x$ (or $10x$ with $1000x$).
Reading which digits actually recur
In only the repeats () — the does not. Misreading the block (treating the as recurring, or reading a two-digit block as one) leads to the wrong multipliers and the wrong fraction.
Forgetting to simplify before judging
To decide if a fraction terminates, cancel to lowest terms first. looks like it has a factor of , but and , so it terminates (). Judging from the un-simplified denominator gives the wrong verdict.
Pad with trailing zeros to compare
Give every decimal the same number of places () so you compare like with like digit by digit — this kills the "more digits = bigger" error and makes the ordering obvious.
Set up x and the scaled equation first
In a "show that" proof, define as the decimal and write the scaled equation (e.g. ) whose recurring tail matches before subtracting. Skipping this setup loses the method marks even if the fraction is right.
Carry the exact fraction, not a decimal
When a later part needs an exact value, use the fraction form (e.g. ) rather than a rounded decimal, so the final area or length stays exact instead of drifting to something like .
Every column after the decimal point is ten times smaller than the one to its left:
| Digit | |||||
|---|---|---|---|---|---|
| Place | units | tenths | hundredths | thousandths | |
| Worth |
So . Getting the place value right is what makes ordering and rounding work.
In the number 3.504, what is the value of the digit 4?
Write these decimals in order, smallest first: .
