Number · Fractions & decimals
Chapter 1 · 3
The idea
Recurring decimals and fractions
Why the ×10ⁿ-and-subtract trick turns any recurring decimal into a fraction, how to choose the right multiplier, and which fractions terminate (and why the denominator's primes decide it).
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Number · Fractions & decimals
Recurring decimals and fractions
Why the ×10ⁿ-and-subtract trick turns any recurring decimal into a fraction, how to choose the right multiplier, and which fractions terminate (and why the denominator's primes decide it).
Why it works
Shift, subtract, and the tail cancels
A recurring decimal like never ends — so you can't "just write it over a power of ten". The trick: shift it so the tails line up, then subtract — the infinite tail cancels itself.The two tails are identical — both are forever — so the subtraction wipes them out exactly, leaving ordinary whole numbers. This is why is exactly , not "about" it (and why it is NOT — the tail matters).
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The rest of the explanation, plus 2 worked examples you step through move by move.
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