June 2000 Different ways of looking at numbers There are all sorts of ways of writing numbers. We can use arithmetics with different bases, fractions, decimals, logarithms, powers, or simply words. Each is more convenient for one purpose or another and each will be familiar to anyone who has done some mathematics at school.
But, surprisingly, one of the most striking and powerful representations of numbers is completely ignored in the mathematics that is taught in schools and it rarely makes an appearance in university courses, unless you take a special option in number theory. Yet continued fractions are one of the most revealing representations of numbers. Numbers whose decimal expansions look unremarkable and featureless are revealed to have extraordinary symmetries and patterns embedded deep within them when unfolded into a continued fraction. Continued fractions also provide us with a way of constructing rational approximations to irrational numbers and discovering the most irrational numbers. Every number has a continued fraction expansion but if we restrict our ambition only a little, to the continued fraction expansions of 'almost every' number, then we shall find ourselves face to face with a simple chaotic process that nonetheless possesses unexpected statistical patterns. Modern mathematical manipulation programs like Mathematica have continued fraction expansions as built in operations and provide a simple tool for exploring the remarkable properties of these master keys to the secret life of numbers.
Concepts: Continued Fractions. A number is usually written down using the place-value system. There are other systems. One such system is Continued Fractions. Real numbers may be expressed as continued fractions, so may analytic functions.
The Nicest Way of Looking at Numbers Introducing continued fractions Consider the quadratic equation ( 1) Dividing by we can rewrite it as ( 2) Now substitute the expression for given by the right-hand side of this equation for in the denominator on the right-hand side: ( 3) We can continue this incestuous procedure indefinitely, to produce a never-ending staircase of fractions that is a type-setter’s nightmare: ( 4) This staircase is an example of a continued fraction. If we return to equation 1 then we can simply solve the quadratic equation to find the positive solution for that is given by the continued fraction expansion of equation 4; it is ( 5) Picking, we have generated the continued fraction expansion of the golden mean,: ( 6) This form inspires us to define a general continued fraction of a number as ( 7) where the are positive integers, called the partial quotients of the continued fraction expansion (cfe).
To avoid the cumbersome notation we write an expansion of the form equation 7 as ( 8) Continued fractions first appeared in the works of the Indian mathematician in the 6th century. He used them to solve linear equations. They re-emerged in Europe in the 15th and 16th centuries and attempted to define them in a general way.
Game Gta San Andreas Java 320x240 Jar. The term 'continued fraction' first appeared in 1653 in an edition of the book Arithmetica Infinitorum by the Oxford mathematician,. Their properties were also much studied by one of Wallis's English contemporaries,, who along with Wallis, was one of the founders of the Royal Society. Crack Para Registrar Bandicam. At about the same time, the famous Dutch mathematical physicist, made practical use of continued fractions in building scientific instruments. Share It For Iphone. Later, in the eighteenth and early nineteenth centuries, and explored many of their deep properties.
How long is a continued fraction? Continued fractions can be finite in length or infinite, as in our example above. Finite cfes are unique so long as we do not allow a quotient of in the final entry in the bracket (equation 8), so for example, we should write 1/2 as rather than as We can always eliminate a from the last entry by adding to the previous entry. If cfes are finite in length then they can be evaluated level by level (starting at the bottom) and will reduce always to a rational fraction; for example, the cfe. However, cfes can be infinite in length, as in equation 6 above. Infinite cfes produce representations of irrational numbers. If we make some different choices for the constant in equations 4 and 5 then we can generate some other interesting expansions for numbers which are solutions of the quadratic equation.