Definition
Decimal expansion
A decimal expansion is a number’s complete base-ten digit trail through smaller place values.
What it means
Read the digits as a trail moving through smaller place values. For 3/8, the trail reaches 0.375 and any later digits are zeros. For 2/11, the block 18 repeats forever. For π, the digits continue without settling into a repeating block.
Remember it
Look at the tail: zeros, a repeating block, or no repeating block.
Example
Dividing 7 by 20 gives 0.35. Its full expansion can be written 0.35000…, so 0.35 is a terminating form. Dividing 4 by 33 gives 0.121212…, where the block 12 repeats.
Watch for
A calculator display such as 0.333333 is only a rounded or cut-off prefix of the full decimal expansion 0.333… for 1/3.
Common mix-up
The ellipsis … says that digits continue; it does not by itself say they repeat. A visible pattern or repeat mark must show repetition.
Exact meaning
A decimal expansion is the complete base-ten digit representation of a number through tenths, hundredths, thousandths, and so on. It may have a terminating written form, eventually repeat a fixed block, or continue without eventually repeating.