Decimal expansion of rational numbers

Step 1: (8=2^3). Step 2: The denominator has only the prime factor (2), so the decimal expansion terminates. Step 3: In exams, first check the prime factors of the simplified denominator.

Step 1: (6=2\times3). Step 2: Since the denominator also has (3), the decimal will not terminate but will repeat. Step 3: Any prime factor other than (2) or (5) prevents termination.

Step 1: For terminating decimals, look at the denominator in lowest form. Step 2: If the denominator is of the form (2^m5^n), the decimal terminates. Step 3: Always reduce the fraction before applying the rule.

Step 1: (\frac{15}{40}=\frac{3}{8}). Step 2: Since (8=2^3), the denominator has only (2), so the decimal terminates. Step 3: Do not judge from the original denominator; reduce first.

Step 1: (16=2^4). Step 2: To make the denominator a power of (10), multiply by (5^4), so it can terminate within four decimal places. Step 3: Focus on the larger exponent of (2) and (5).

Step 1: (45=3^2\times5). Step 2: The factor (3) stops termination, but the number is rational, so the decimal repeats. Step 3: A non-terminating decimal of a rational number is recurring.

Step 1: (40=2^3\times5). Step 2: It contains only (2) and (5), so (\frac{9}{40}) gives a terminating decimal. Step 3: Quickly factor the denominators in options.

Step 1: (15=3\times5). Step 2: The factor (3) makes the decimal non-terminating, and since the number is rational, it is recurring. Step 3: Be alert when a factor other than (2) or (5) appears.

Step 1: (0.75=\frac{75}{100}). Step 2: Reducing gives (\frac{75}{100}=\frac{3}{4}). Step 3: Write terminating decimals with denominators like (10,100,1000) first.

Step 1: The exponent of (2) is (3) and the exponent of (5) is (2). Step 2: The larger exponent is (3), so the decimal terminates within three places. Step 3: Choose the larger exponent, not the smaller one.

Step 1: (125=5^3). Step 2: Multiplying by (2^3) can make the denominator (10^3), so the decimal terminates in three places. Step 3: For (5^3), think of three decimal places.

Step 1: The fraction is in lowest form and the denominator is (7). Step 2: Since the denominator has a factor other than (2) or (5), the decimal is non-terminating recurring. Step 3: (\frac{22}{7}) is rational, so its non-terminating decimal must repeat.

Step 1: A terminating decimal can be converted into a fraction with denominator (10), (100), (1000), and so on. Step 2: Therefore, it can be written as (\frac{p}{q}), so it is rational. Step 3: Do not mistake terminating decimals for irrational numbers.

Step 1: The decimal expansion of a rational number is either terminating or non-terminating recurring. Step 2: (0.10100100010000\ldots) has no fixed repeating block, so it is not rational. Step 3: Learn to distinguish recurring from non-recurring decimals.

Step 1: (455=5\times7\times13), and the fraction is in lowest form. Step 2: The denominator contains (7) and (13), so the decimal does not terminate, but it repeats. Step 3: A denominator with factors other than (2) and (5) gives a recurring decimal.

Step 1: (\frac{35}{50}=\frac{7}{10}). Step 2: Since (10=2\times5), the decimal terminates. Step 3: Simplification is essential because the denominator changes after reducing.

Step 1: (\frac{77}{210}=\frac{11}{30}). Step 2: Since (30=2\times3\times5), the factor (3) remains, so the decimal is recurring. Step 3: Always check the denominator after reducing.

Step 1: (\frac{6}{15}=\frac{2}{5}). Step 2: The reduced denominator is (5), so the decimal terminates. Step 3: If a factor like (3) cancels during reduction, the decimal may terminate.

Step 1: For a terminating decimal, the denominator must be made only from (2) and (5). Step 2: So its form is (2^m5^n). Step 3: (m) or (n) may be zero, so only (2) or only (5) is also allowed.

Step 1: (40=2^3\times5). Step 2: The larger exponent is (3), so the decimal terminates in three places. Step 3: Indeed, (\frac{1}{40}=0.025), which has three decimal places.

Step 1: Multiply (20) by (5) to make (100). Step 2: (\frac{3}{20}=\frac{15}{100}=0.15). Step 3: Converting the denominator to (10), (100), or (1000) is a quick method.

Step 1: Multiply (25) by (4) to make (100). Step 2: (\frac{7}{25}=\frac{28}{100}=0.28). Step 3: When converting to decimal, multiply numerator and denominator by the same number.

Step 1: Dividing (1) by (3) gives the digit (3) repeatedly. Step 2: Hence, (\frac{1}{3}=0.\overline{3}). Step 3: The bar means that the digit repeats continuously.

Step 1: (\frac{1}{11}=0.\overline{09}). Step 2: Multiplying by (2) gives (\frac{2}{11}=0.\overline{18}). Step 3: Put the complete repeating block under the bar.

Step 1: A decimal with a fixed repeating block is called a recurring decimal. Step 2: Every recurring decimal can be written as a fraction, so it is rational. Step 3: A fixed repeat is a strong sign of rationality.