Higher Algebra Hall & Knight Recurring Continued Fractions (Chapter 27) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 27 ‘Recurring Continued Fractions’ is a reliable set of answers to the exercise questions in this chapter. These solutions are a perfect study guide for competitive exams like JEE and NEET along with Class 12 Maths. You will get to solve some numerical examples about how a periodic continued fraction is equivalent to quadratic surds. In this chapter of Higher Algebra by Hall and Knight, You will also learn the technique to convert a quadratic surd into a continued fraction. Other topics of significant weightage covered in this chapter are the recurring quotients, partial quotient at the end of a period, equal partial quotients, and the penultimate convergents of the periods.

Hall & Knight Algebra Mathematics book with solutions for Recurring Continued Fractions have 2 exercises with 49 questions in total. These exercises consist of a diverse variety of questions ranging from formula-based problems to word problems. The word problems will require a mathematical interpretation for conversion into an algebraic expression which can be solved using the formulae and rules explained in this chapter. Solving these questions will give you an idea of the level and pattern of questions asked in important exams such as NEET, JEE, etc. It is advised that you solve the questions of Higher Algebra by Hall & Knight ‘Recurring Continued Fractions’ using Instasolv as a source of reference.

Instasolv is the platform where you will get answers to all the queries that will arise while solving the exercise questions. Our team of math experts will assist you in building the right aptitude for approaching a problem of recurring continued fractions. Each solution complies with the latest guidelines for competitive exams like JEE, therefore, you may use these answers for guidance at any time. This set of solutions for Higher Algebra By H.S. Hall and S.R. Knight book is also apt for a quick revision of the concepts in the question-answer format right before your exams.

Important Topics Covered under Hall and Knight Higher Algebra Solutions Chapter 27: Recurring Continued Fractions


A given quadratic surd can be converted and expressed in the form of an infinite continued fraction, the quotients of which are recurring in nature. This has been explained in the chapter with the help of numerical examples.

Periodic Continued Fraction

All the periodic continued fractions are equal to one of the roots of a quadratic equation in which the coefficients are rational numbers.

Let x and y be the continued fraction and the periodic part respectively, such that

m, n, …u, v are positive integers

The equation is given as,

, which gives the value of y (assumed as the periodic part) as its roots which are real and bear opposite signs.

Substituting the value of y in x and rationalizing the denominator, we get x in the form.

Quadratic Surd

We follow the following rules and formula to convert a given quadratic surd in the form of a quadratic fraction,

If we assume N to be a positive integer which is not a perfect square, a1 is the greatest integer in 

Now if b1 is the greatest integer in, the


so on.

Therefore,can be expressed in the form of an infinite continued fraction.

This fraction will have periodic quotients in the form of.

,… as the first, second, … quotients.

Also, the numbers a1, r1, b1,… are positive integers along with the quantities a2, a3,… and r2,r3,… being positive integers.

Complete and Partial Quotients

The first complete quotient and the subsequent quotients can be proved to be recurring in nature.

Similarly, the partial quotients also recur and each cycle has 2a12 number of partial quotients at most.

Size of Period

It can be proved that a period begins with the second partial quotient and ends with the partial quotient which is two times the quotient of the first partial quotient. Mathematically,  and

Penultimate Convergents of the Recurring Periods

Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 27: Recurring Continued Fractions

  1. In exercise 27.a, you will be required to find the sixth convergent after conversion of given surds into continued fractions, limits of the error, express roots of a given expression as a continued fraction. 
  2. Other questions covered in this exercise will require you to evaluate the difference between the given infinite continued fractions.
  3. In exercise 27.b, the questions are such that they require conversion of surds into continued fractions and find the fourth or fifth convergent to each. 
  4. This exercise will also require you to prove some important results that you may use in the topics ahead.

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  5. The solutions are a perfect guide for JEE and NEET like competitive exams and will help you revise all the essential formulae of the chapter.