Middle school maths, year 2 · unit 1
Rational Numbers and Recurring Decimals
Divide 1 by 3 and you get 0.333…, with 3s for ever. A number without an end sounds strange, but it is about as well-behaved as numbers get: the same digits simply repeat. We start by pulling the tape out of an adding machine.
Divide, and it ends or it repeats
A board to adjustA fraction is a division: 1/4 is 1 ÷ 4. Drag the gold handle on the tape coming out of the machine to the right to pull out the decimal.
The denominator alone tells you
A board to arrangeYou can tell whether the decimal ends without dividing. Drag each fraction card onto the right tray.
Line up the tails and they cancel
A board to adjustHow do you turn a decimal that never ends back into a fraction? Drag the top tape to the left until everything after its decimal point matches the row below.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- A decimal that stops is a terminating decimal; one that goes on for ever is an infinite decimal. An infinite decimal in which the same run of digits repeats is a recurring decimal, and the run is its repeating block. 0.1818… is written short with dots over the block.
- A fraction written as a decimal either terminates or recurs, because there are only so many possible remainders and sooner or later one of them comes back.
- If the denominator in lowest terms has no prime factors other than 2 and 5, the decimal terminates; any other prime factor and it recurs. Always cancel down first.
- Call the recurring decimal x, multiply by 10 or 100 according to the length of the block, subtract, and the tail vanishes. For 0.333…, 10x − x = 3, so x = 1/3.
- Terminating and recurring decimals can all be written as fractions: they are rational numbers.