Chapter 1.4 — Exercise 1.3 — Decimal Expansion

The Rule That Decides Everything

A rational number p/q, already reduced to its simplest form, has a terminating decimal exactly when the prime factorisation of q contains no prime other than 2 and 5.

If q's prime factorisation contains any other prime — 3, 7, 11, 13, and so on — the decimal is non-terminating and repeating instead. The reason traces back to how our number system itself is built: 10 = 2 × 5, so only denominators made purely of 2s and 5s can ever be converted exactly into a power of 10, which is the only kind of denominator a terminating decimal can have.

The Rule, Stated Both Ways

The rule above is really three separate guarantees working together, and it's worth being precise about each direction:

• If a rational number's decimal terminates, its simplest-form denominator is guaranteed to be of the form 2ᵐ × 5ⁿ.
• Conversely, if a rational number's simplest-form denominator is of the form 2ᵐ × 5ⁿ, its decimal is guaranteed to terminate.
• And if that denominator contains any other prime, the decimal is guaranteed to be non-terminating and repeating — not just "possibly" repeating.

Because all three directions hold without exception, checking the denominator's prime factors is never a shortcut that occasionally fails — it's a complete substitute for actually performing the division, every single time.

Five Fractions, by Long Division

The first question asks you to actually perform the division and classify the result:

Classifying Without Dividing at All

This is where the rule pays off — factorise the denominator instead of dividing, and you get the answer instantly. The one step that must never be skipped is simplifying the fraction first, since it is the reduced denominator that decides the outcome, not the original one.

• 13/3125 — 3125 = 5⁵, only the prime 5, so terminating.
• 11/12 — 12 = 2² × 3, and 3 is present, so non-terminating, repeating.
• 64/455 — 455 = 5 × 7 × 13, both 7 and 13 present, so non-terminating, repeating.
• 15/1700 — simplifies to 3/340, and 340 = 2² × 5 × 17; 17 is present, so non-terminating, repeating.
• 29/343 — 343 = 7³, only the prime 7, so non-terminating, repeating.
• 23/(2³×5²) — denominator is purely 2s and 5s, so terminating.
• 129/(2²×5⁷×7⁵) — 7 is present, so non-terminating, repeating.
• 9/15 — simplifies to 3/5, denominator 5¹, so terminating.
• 36/100 — simplifies to 9/25, denominator 5², so terminating.
• 77/210 — simplifies to 11/30, and 30 = 2 × 3 × 5; 3 is present, so non-terminating, repeating.

Notice how easy it is to get 9/15 wrong if you check the original denominator 15 = 3 × 5 — the factor of 3 there would wrongly suggest a repeating decimal. Simplify to 3/5 first, and the 3 in the numerator cancels the 3 in the denominator, leaving a clean 5.

Converting Without Division, the Other Direction

When a denominator is already known to be of the form 2ᵐ × 5ⁿ, there's a shortcut for writing out the decimal directly: multiply numerator and denominator by whatever is missing to turn the denominator into a clean power of 10.

• 13/25 — multiply by 4/4: 52/100 = 0.52
• 15/16 — 16 = 2⁴, multiply by 5⁴/5⁴ = 625/625: 9375/10000 = 0.9375
• 23/(2³×5²) — multiply by 5/5: 115/1000 = 0.115
• 7218/(3²×5²) — simplify 7218/9 = 802 first, giving 802/25; multiply by 4/4: 3208/100 = 32.08
• 143/110 — simplifies directly to 13/10 = 1.3

Whichever factor of 2 or 5 is missing from the denominator, that same factor has to be multiplied into both numerator and denominator — multiplying only one side changes the value of the fraction, which is the single most common slip when using this shortcut.

Working Backward: Decimal to Fraction

The final question reverses the whole process, starting from a decimal and asking what its denominator's prime factors reveal.

For a terminating decimal, the denominator falls straight out of the place value — three digits after the point means a denominator of 1000, seven digits means 10⁷, and so on, which is always some product of 2s and 5s, so the terminating outcome is guaranteed before you even simplify. For a repeating decimal, the standard trick is to subtract off the repeating part algebraically: 43.1212… minus 43 leaves 0.1212…, and multiplying by 99 (because a two-digit block repeats) clears the repetition entirely, leaving 12/99, which reduces to 4/33. The same subtract-and-multiply idea handles any repeating block, whatever its length.

Why a Repeating Decimal Actually Repeats

It's worth asking why 2/11 settles into 0.181818… forever instead of just producing an endless, unpatterned string of digits. Long division answers this directly. At every step of dividing 2 by 11, the remainder you get is always some whole number strictly less than 11 — the same 0 ≤ r < b condition from Euclid's Division Lemma at the start of this chapter. That means there are only 10 possible non-zero remainders the division can ever produce (1 through 10), so after at most 10 steps, some remainder has to repeat.

The moment a remainder repeats, the division is back in a state it has already been in before — so every digit and every remainder from that point on repeats too, forever, in exactly the same order. That's the entire mechanism behind a repeating decimal: not an accident, but a direct, guaranteed consequence of the fact that dividing by 11 only ever leaves finitely many possible remainders.

The Boundary This Chapter Is Building Toward

Every fraction in this exercise, terminating or not, is still a rational number — the decimal always eventually settles into a pattern, even if that pattern never stops. That is precisely the boundary Exercise 1.4 crosses next: irrational numbers have decimal expansions that neither terminate nor repeat, which is a genuinely different kind of behaviour from anything seen here. For the prime-factorisation tools this exercise leans on, see Exercise 1.2, or start from the beginning at the chapter introduction.