Decimal expansion of rational numbers

Step 1: Let (x=0.12333\ldots). Step 2: Then (100x=12.333\ldots) and (1000x=123.333\ldots). Subtracting gives (900x=111), so (x=\frac{111}{900}=\frac{37}{300}). Step 3: Separate the non-repeating and repeating parts before multiplying.

Step 1: The numerator (5) cancels one factor of (5) from (5^6). Step 2: The reduced denominator becomes (2^4\cdot 5^5). The larger exponent is (5), so the decimal terminates after (5) places. Step 3: Factors (2) or (5) in the numerator can reduce the decimal length.

Step 1: (0.0\overline{6}=0.0666\ldots). Step 2: This equals (\frac{1}{15}), since (\frac{1}{15}=0.0666\ldots). Hence (q=15). Step 3: The initial zero shows that the recurring part starts after a delay.

Step 1: In (0.125000\ldots), only zeros occur after a point. Step 2: So it is a terminating decimal and equals (\frac{125}{1000}=\frac{1}{8}), which is rational. Step 3: Trailing zeros do not make a decimal non-terminating.

Step 1: The denominator contains (2), (5), and (13). Step 2: For a terminating decimal, (13) must not remain in the reduced denominator. So (a) must contain the factor (13). Step 3: Powers of (2) and (5) may remain, but other prime factors must cancel.

Step 1: A reduced denominator must contain only (2) and (5). Step 2: (80=2^4\cdot 5), so it is possible. (48), (84), and (98) contain primes like (3) or (7). Step 3: If lowest form is given, check the prime factors of the denominator directly.

Step 1: (\frac{99}{9900}) reduces to (\frac{1}{100}) because (9900\div 99=100). Step 2: The reduced denominator is (100=2^2\cdot 5^2), so the decimal terminates. Step 3: With large numbers, check reduction carefully by division.

Step 1: (\frac{27}{990}=\frac{3}{110}). Step 2: (110=2\cdot 5\cdot 11), so (11) remains in the denominator. Hence the decimal is non-terminating recurring. Step 3: If a reduced denominator has a prime other than (2) or (5), it will not terminate.

Step 1: The reduced denominator has only powers of (2) and (5). Step 2: The number of decimal places equals the larger exponent. Since (r>s), the larger exponent is (r). Step 3: Remember (\max(r,s)) for decimal places.

Step 1: (0.375=\frac{375}{1000}). Step 2: Dividing numerator and denominator by (125) gives (\frac{3}{8}). So the denominator is (8). Step 3: Always reduce after converting a decimal to a fraction.

Step 1: (0.\overline{25}) has a repeating block, so it is rational. Step 2: It is not terminating because the decimal does not end. (\sqrt{2}) and (\pi) are irrational. Step 3: Rational numbers can be terminating or non-terminating recurring.

Step 1: (56=2^3\cdot 7). Step 2: For a terminating decimal, (7) must not remain in the reduced denominator. Therefore (m) must contain (7). Step 3: Cancel all denominator primes other than (2) and (5).

Step 1: (3125=5^5). Step 2: The denominator has power (0) of (2) and power (5) of (5). So the decimal terminates after (5) places. Step 3: Remembering (3125=5^5) helps in quick factorisation.

Step 1: In (x=0.21818\ldots), the non-repeating part is (2) and the repeating part is (18). Step 2: First use (10x=2.1818\ldots), then (1000x=218.1818\ldots) so the recurring parts align. Step 3: Choose powers of (10) based on the lengths of the non-repeating and repeating parts.

Step 1: The denominator is (2^6\cdot 5^3), and the fraction is in lowest form because (37) does not cancel. Step 2: The larger exponent is (6), so the decimal terminates after (6) places. Step 3: Do not add the exponents for decimal places.

Step 1: Rational numbers have either terminating or non-terminating recurring decimals. Step 2: (\sqrt{5}) is irrational, so its decimal is non-terminating non-recurring. Step 3: To identify non-terminating non-recurring decimals, look for irrational numbers.

Step 1: (3^0=1), so there is actually no factor (3) in the denominator. Step 2: The denominator is (2^2\cdot 5^3), containing only (2) and (5). Hence the decimal terminates. Step 3: Do not get confused by a zero exponent.

Step 1: The repeating block (142857) has (6) digits. Step 2: For a purely recurring decimal, before reducing, the denominator has the same number of (9)'s. So it is (999999). Step 3: The number of repeating digits tells the number of (9)'s.

Step 1: (16=2^4), so (2^4) cancels from the denominator. Step 2: The reduced denominator is (2^3\cdot 5^4). The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Include powers hidden in the numerator during cancellation.

Step 1: The block (36) repeats in the decimal. Step 2: A recurring decimal is always rational, but it is not terminating. So it is rational and non-terminating recurring. Step 3: When a repeating block appears, identify the number as rational.