Exercise · ZW-CONTINUED-FRACTION-009 · ★★☆ Intermediate
Write the continued fraction of \( 27/11 \) using the Euclidean algorithm.
Choose one option.
Show answer
- B. [2; 2, 5]
Explanation
We have \( 27 = 2 \cdot 11 + 5 \), \( 11 = 2 \cdot 5 + 1 \) and \( 5 = 5 \cdot 1 \), so the expansion is [2; 2, 5]. The option [2; 5, 2] reverses the order of the quotients, and [3; 1, 4] uses a wrong initial quotient.
Review the concept: Continued fraction
A continued fraction is an expression of the form a₀ + 1/(a₁ + 1/(a₂ + ...)), where a₀ is an integer and the remaining terms aᵢ are positive integers.
\[x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}\]
More practice problems: continued fraction
- ZW-CONTINUED-FRACTION-001 What is the term a_0 of the continued fraction of 415/93 ?
- ZW-CONTINUED-FRACTION-002 What is the value of the rational [2; 3] ?
- ZW-CONTINUED-FRACTION-003 How many terms does the continued fraction of 415/93 have according to the Euclidean algorithm?
- ZW-CONTINUED-FRACTION-004 Which rational number does the continued fraction [1; 1] represent?
- ZW-CONTINUED-FRACTION-005 What is the term a_1 of the continued fraction of 415/93 ?
- ZW-CONTINUED-FRACTION-006 Compute the convergent p_1/q_1 of the continued fraction [4; 2, 6, 7] .
- ZW-CONTINUED-FRACTION-007 Compute the convergent p_2/q_2 of the continued fraction [4; 2, 6, 7] .
- ZW-CONTINUED-FRACTION-008 What is the value of [0; 1, 2] ?