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Higher Algebra Hall & Knight Continued Fractions (Chapter 25) Solutions

Hall & Knight Higher Algebra Solutions for Chapter 25 ‘Continue Fractions’ is created to help you attain thorough knowledge in algebra by solving the exercise questions of this chapter. In this chapter of Higher Algebra by Hall and Knight, You will learn to convert a given fraction into a continued fraction and also, you will be introduced to convergents. You will learn about the law of formation of the successive convergents, convergents gradually approximating continued fractions and convergents being nearer to the continued fraction as compared to the fractions with relatively smaller denominators.

Hall & Knight Algebra Mathematics book with solutions for Continued Fractions includes two exercises covering a total of 31 questions. These questions range from the beginner to the advanced level. Solving the questions will help you develop an understanding of continued fractions which in turn will develop a strong framework for you to understand the topics of Physics and Maths. The questions in this chapter will also help you develop conceptual clarity for IIT JEE and NEET exams.

At Instasolv, we have provided you with the solutions of the Hall and Knight Higher Algebra book with extreme care. We get our set of answers cross-checked twice by our team of experts. Such a standard procedure has helped us in maintaining 100% accuracy in our solutions. The answers at Instasolv are highly reliable, as we write precise steps with adequate reasoning. You will get acquainted with these interesting topics of algebra effortlessly with an equally interesting and explanatory set of solutions by our team of maths experts.

Important Topics for Hall and Knight Higher Algebra Solutions Chapter 25: Continued Fractions

Introduction

A continued fraction is the term given to an algebraic expression of the form,

a+bc+de+…

The variables a, b, c, d, e…, can be of any type but for now, we will assume a simpler version as a1+a2a3+a4a5+…=a1+1a2+1a2+1a3where a1, a2, a3, a4, a5and so on, will be positive integers. 

If the quotients a1, a2, a3, a4, a5go on infinitely, then they are known as infinite continued fractions while if they are finite, they are termed as terminating fractions.

Conversion of any Fraction into a Continued Fraction

If we consider m/n as the fraction, then while dividing m by n, let the quotient be a1 and the remainder be p, therefore

mn=a1+pn=a1+1np;

Now, in the next step, divide n by p such that the quotient is a2 and the remainder is q, then

np=a2+qp=a2+1pq;

And thus, we can go on with the next step by dividing p by q with the quotient as a3 and remainder r.

If we reach a stage when the division is exact and procedure of the above steps terminates, then it is said to be a terminating continued fraction.

Convergents

In the above procedure, the quotients obtained after every division, in the above case, a1, a2, a3,…, are known as the first convergent, second convergent, third convergent, and so on, respectively.

The convergents are less than the continued fractions and greater than the same alternatively. 

Law of Formation of the Successive Convergents

  • If a continued fraction is denoted as 

a1+1a2+1a2+1a3+, then the first 3 convergents will be given as a11, a1a2+1a2,a3a1a2+1+a1a3a2+1

Similarly, for the (n+1)th convergent, pn+1=an+1(anpn-1+pn-2)+pn-1an+1(anqn-1+qn-2)+qn-1

Where, pn=anpn-1+pn-2 and qn=anqn-1+qn-2

  • If k is the complete quotient at any stage and an the partial quotient in a continued fraction denoted by x

Then, x=kpn-1+pn-2kqn-1+qn-2

  • If pnqn is the nth convergent, then

pnqn-1pn-1qn=-12

  • A fraction that has the denominator less than the denominator of the convergent is less near to the given continued fraction than the latter.

Important Results

  1. Every convergent is relatively closer to the continued fraction than any of its preceding convergents.
  2. The convergents that are in the odd order always increase but remain lesser than the continued fraction.
  3. The convergents that are in the even-order always decrease but remain greater than the continued fraction.

Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 25: Continued Fractions

  • There are 2 exercises in Higher Algebra by Hall & Knight Continued Fractions, namely 25.a and 25.b covering all the topics in this chapter thoroughly in 18 and 13 questions respectively.
  • In exercise 25.a, you will get to solve questions based on terminating and infinite continued fractions. You will also get to solve questions based on the first, second, etc. convergents and the laws of formation of convergents.
  • Exercise 25.b is based on the theories that convergents gradually approximating continued fractions and convergents being nearer to the continued fractions than fractions with smaller denominators.

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