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Real Numbers - Decimal Expansions of Rational Numbers

Grade 10CBSE

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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A rational number x=pqx = \frac{p}{q} (where pp and qq are co-prime) has a terminating decimal expansion if the prime factorization of the denominator qq is of the form 2n5m2^n 5^m, where nn and mm are non-negative integers. Visually, imagine the long division process ending at a remainder of zero after a finite number of steps.

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If the prime factorization of the denominator qq of a rational number pq\frac{p}{q} (in simplest form) contains prime factors other than 22 or 55, then the decimal expansion is non-terminating repeating (recurring). In a division table, you would see the remainders starting to repeat in a specific cycle, causing a bar to be placed over the repeating digits.

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The condition of being co-prime is essential. Before analyzing the denominator qq, you must simplify the fraction pq\frac{p}{q} to its lowest terms by canceling out common factors between the numerator and denominator.

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The number of decimal places after which the decimal expansion of pq\frac{p}{q} terminates is equal to the maximum value of nn and mm in the factorization q=2n5mq = 2^n 5^m. For example, if the denominator is 23Γ—512^3 \times 5^1, the expansion will terminate after max⁑(3,1)=3\max(3, 1) = 3 decimal places.

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To convert a rational number with a terminating expansion into a decimal without performing long division, multiply both the numerator and denominator by suitable powers of 22 or 55 to make the denominator a power of 1010 (i.e., 10k=(2Γ—5)k10^k = (2 \times 5)^k).

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Irrational numbers, unlike rational numbers, have decimal expansions that are non-terminating and non-repeating. Visually, these numbers cannot be represented as a repeating pattern or a finite sequence of digits.

πŸ“Formulae

Rational number form: x=pq,q≠0x = \frac{p}{q}, q \neq 0

Condition for termination: q=2nΓ—5mΒ whereΒ n,m∈{0,1,2,...}q = 2^n \times 5^m \text{ where } n, m \in \{0, 1, 2, ...\}

Number of decimal places: max(n,m)\text{max}(n, m)

Conversion property: p2n5m=pΓ—2aΓ—5b10k\frac{p}{2^n 5^m} = \frac{p \times 2^a \times 5^b}{10^k}

πŸ’‘Examples

Problem 1:

Without performing long division, determine if 133125\frac{13}{3125} has a terminating or non-terminating repeating decimal expansion. If it terminates, find its decimal expansion.

Solution:

Step 1: Write the denominator in prime factored form. 3125=5Γ—5Γ—5Γ—5Γ—5=553125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 Step 2: Check the form 2n5m2^n 5^m. Here, q=55q = 5^5, which can be written as 20Γ—552^0 \times 5^5. Since the prime factors are only 55, the expansion is terminating. Step 3: To find the decimal expansion, make the powers of 22 and 55 equal in the denominator: 1355=13Γ—2555Γ—25=13Γ—32(5Γ—2)5=416105\frac{13}{5^5} = \frac{13 \times 2^5}{5^5 \times 2^5} = \frac{13 \times 32}{(5 \times 2)^5} = \frac{416}{10^5} Step 4: Convert to decimal: 416100000=0.00416\frac{416}{100000} = 0.00416

Explanation:

We identify the prime factors of the denominator. Since it only contains 55, it follows the 2n5m2^n 5^m rule. We then multiply the numerator and denominator by 252^5 to create a denominator of 10510^5, allowing for an easy decimal shift.

Problem 2:

State whether 77210\frac{77}{210} will have a terminating or non-terminating repeating decimal expansion.

Solution:

Step 1: Simplify the fraction to its lowest terms (co-prime). 77210=7Γ—117Γ—30=1130\frac{77}{210} = \frac{7 \times 11}{7 \times 30} = \frac{11}{30} Step 2: Factorize the denominator of the simplified fraction. 30=2Γ—3Γ—530 = 2 \times 3 \times 5 Step 3: Check the factors. The prime factorization of the denominator contains the factor 33, which is not 22 or 55. Step 4: Conclusion: Since the denominator qq contains a prime factor other than 22 or 55, the decimal expansion is non-terminating repeating.

Explanation:

It is crucial to simplify the fraction first. If we had factored 210210 immediately, we might have seen the 77 and thought it was non-terminating, but 77 was actually a common factor with the numerator. However, even after simplification, the factor 33 remains in the denominator, confirming it is non-terminating repeating.