ࡱ> gifW 0bjbjVV <<-,8)fQ)S)S)S)S)S)S)$\+-Zw)9^w))%%%Q)%Q)%%(M(?z9 B(=))0)!(,X.!X.M(X.M(%w)w)%)X. : Module 6 Chapters 10 and 11 Continued Fractions and Fibonacci numbers. Continued Fractions A continued fraction is a way to represent numbers that are improper fractions or, in some cases, transcendental numbers. A continued fraction takes a whole LOT of room on a page so we quickly move to an alternate representation. For example:  EMBED Equation.DSMT4  Improper fraction (fully reduced); continued fraction; decimal; mixed number Each of these representations has its uses! The continued fraction is called finite and simple because it ends and each numerator is 1. There are 2 fraction bars in the continued fraction of the number. Now, proper fractions do not have a continued fraction representation only mixed number and improper fractions. 0.25 =  EMBED Equation.DSMT4  no chance for a continued fraction here! Note that we use the Euclidean Algorithm as a way to get the CF for an improper fraction!  EMBED Equation.DSMT4  There is a second way to get CFs  EMBED Equation.DSMT4  Both ways are similar. You really do need a fully reduced fraction to do this work (numerator, denominator) = 1. Another example:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Simple finite continued fraction! Another one:  EMBED Equation.DSMT4  Four fraction bars! Simple finite continued fraction. Popper 11, Question 1 Every fraction can be represented as a continued fraction. A True B False Continued fractions and Diophantine equations Heres a surprising tie in between continued fractions and Diophantine equations! Heres an example that is not in the book its the same process, though. Step 1 Get the continued fraction Step 2 Chop off the last fraction Step 3 Reconstitute a new improper fraction Step 4 Subtract the new fraction from the original one Step 5 Multiply and subtract the numerators getting an equation Step 5 Multiply both sides of the equation by the common denominator Take  EMBED Equation.DSMT4  Notice that I always use a prime number in my improper fractions its a guaranteed way to make sure the fraction is fully reduced! The CF work - Step 1:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Step 2: chop of the :  EMBED Equation.DSMT4  Step 3 Reconstitute the new improper fraction:  EMBED Equation.DSMT4  Step 4 Subtract the new fraction from the original one  EMBED Equation.DSMT4  Step 5 Multiply and subtract the numerators getting an equation  EMBED Equation.DSMT4  Step 5 Multiply both sides of the equation by the common denominator  EMBED Equation.DSMT4  Which is:  EMBED Equation.DSMT4  Now look at it hard:  EMBED Equation.DSMT4  Theres a Diophantine equation! With the initial solution showing! Popper 11, Question 2 Diophantine Equations have continued fractions as solutions. A True B False Now for infinite repeating continued fractions All irrationals of the form  EMBED Equation.DSMT4  have infinite repeating continued fractions. Well definitely need new notation for this! Example  EMBED Equation.DSMT4  is a nonrepeating, nonterminating decimalno place to put a repeat superscripted bar  EMBED Equation.DSMT4  How cool is that? A nice compact and REPEATING representation? The square root of 10 has period 1. Example  EMBED Equation.DSMT4  Period 2 Other common notation:  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4  period 6 Popper 11, Question 3  EMBED Equation.DSMT4  has period A 3 B 4 Well explore only one theorem on how to find these. There has been a LOT of work done and lots is known. Theorem on page 127 If an irrational number has the form  EMBED Equation.DSMT4  where a is a natural number, then its continued fraction is  EMBED Equation.DSMT4  Lets illustrate this theorem! a = 1  EMBED Equation.DSMT4  Will this work for  EMBED Equation.DSMT4  a = 3  EMBED Equation.DSMT4  a = 4  EMBED Equation.DSMT4  nope! Lets check the theorem: 5 =  EMBED Equation.DSMT4  From a calculator:  EMBED Equation.DSMT4  From the theorem:  EMBED Equation.DSMT4  2 + = 2.25  EMBED Equation.DSMT4  = 2.2352941 2 + 6 fraction bars = 2.2360648 Take a limit to inifintysure! Popper 11, Question 3  EMBED Equation.DSMT4  A True B False Now lets take a minute and look at continued fractions from a set perspective Reals  Algebraic: Rationals (finite simple cf) and some irrationals (infinite repeating cf) Transcendental: Some have cf but theyre definitely not simple! Example for a transcendental number:  EMBED Equation.DSMT4  The nice thing is that you can see the pattern. The problem is that its not a repeating pattern so theres nowhere for a repeat bar. This is an infinite complicated continued fraction! Example e, the Euler number e is irrational and approximately 3.7, non-terminating and non-repeating It shows up in lots of natural phenomena and in continuously compounded interest in man-made phenomena. It has an infinite complicated continued fraction with a visible pattern:  EMBED Equation.DSMT4  Again, no place for a repeat bar But for the first time theres a pattern to see with some of these irrational numbers! Popper 11, Question 5 Only algebraic numbers have continued fraction representations. A True B False Homework hint: Google Continued Fraction for homework help! End video 1 Chapter 11: Fibonacci Numbers Pages 133 144 Fibonacci was an Italian mathematician. He discovered a sequence of natural numbers created by addition that shows up frequently in natural phenomena. If you google the topic, youll get millions of hits. There is even a Fibonacci Society! The sequence:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Why 3? Well if you try it with n = 1, you get  EMBED Equation.DSMT4  and thats not ok. Side note: on the final youll have a calculator, a list of primes, and a list of Fibonacci numbers on links. If you dont see the links, get some help from the proctors! Now lets look at n = 10  EMBED Equation.DSMT4  so we really need a list of these to do things! Popper 12, Question 1 Given  EMBED Equation.DSMT4  what is  EMBED Equation.DSMT4 ?  EMBED Equation.DSMT4  Now lets work with our formula a little bit:  EMBED Equation.DSMT4  The second equation is used almost as much as the first equation so be comfy with either, please. Popper 12, Question 2  EMBED Equation.DSMT4  A 322 B  EMBED Equation.DSMT4  C  EMBED Equation.DSMT4  D  EMBED Equation.DSMT4  E 12 Lets take a really nice continued fraction and see where it leads us:  EMBED Equation.DSMT4  it doesnt get simpler than this one  EMBED Equation.DSMT4  Does this one converge to a number? 1 1/1  EMBED Equation.DSMT4  1 1 + 1 2/1  EMBED Equation.DSMT4  2  EMBED Equation.DSMT4  3/2  EMBED Equation.DSMT4  1.5  EMBED Equation.DSMT4  5/3  EMBED Equation.DSMT4  1.6 1.625 1.618 non-terminating and non-repeating decimal  EMBED Equation.DSMT4  Which, after some algebra turns out to be  EMBED Equation.DSMT4  Lets look at that algebra (page 133)  EMBED Equation.DSMT4  If I drop below the first fraction bar, notice that Ive got x again  EMBED Equation.DSMT4  incredible but true Now, multiply both sides by x which we know is NOT zero! And then subtract to get a quadratic = 0  EMBED Equation.DSMT4  Now let s use the quadratic formula with A = 1, B = "1, and C = "1.  EMBED Equation.DSMT4  And we ignore the negative answer because its a negative number and not the result of adding positives. Now, in the homework you need to look up PHI, this number. Its used often in art! This is an amazing result because the algebra agrees with our work on the continued fraction. And the trick to getting the algebra to work was so clever! Can you see how wed do this to any repeating continued fraction? Popper 12, Question 3  EMBED Equation.DSMT4  A True B False Properties of Fibonacci Numbers Theorem:  EMBED Equation.DSMT4 . The big capital sigma means add from the bottom index to the top. So this says, the sum of n Fibonacci numbers is  EMBED Equation.DSMT4 . Lets illustrate this theorem. Ill go up to n = 8.  EMBED Equation.DSMT4  Adding the first 8 Fibonacci numbers: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 54 Checking the list  EMBED Equation.DSMT4  Perfect! It works. Another illustration:  EMBED Equation.DSMT4  On the left, well, I added them but didnt want to type them! Now theres a nifty illustration of WHY this works on page 136. Its a process called telescoping. And it has lots and lots of cancelling in it. Its actually fun to check out. Popper 12, Question 4  EMBED Equation.DSMT4  A True B False Another property and theorem:  EMBED Equation.DSMT4  Lets illustrate this using n = 5:  EMBED Equation.DSMT4  And perhaps most astonishing of all: The Fibonacci numbers can be found in Pascals Triangle on diagonals! Page 137 The rising diagonals 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 And on page 138, we have the Fibonacci numbers and Areas  And from this we find another theorem to illustrate:  EMBED Equation.DSMT4  Doesnt that look awful? Lets do it with n = 6 weve got the first 6 Fibonacci numbers in the area above. The height is 8 ( EMBED Equation.DSMT4 ) and the length is 13 ( EMBED Equation.DSMT4 ). Yep!  Homework hint: Well go back to geometry and learn about the geometric mean If we have b is the geometric mean of a and c for 3 natural numbers we mean the result of this proportionality:  EMBED Equation.DSMT4  we only want the positive number Now we switch subjects for the very last time this semester at the bottom of page 139. Lets take a line of length 1 and split it into 2 carefully chosen pieces: x and 1 x in a way that the ancient Greeks thought was particularly pleasing:  Now we want  EMBED Equation.DSMT4  We only want the positive number so we ignore the minus! Notice how close this is to Phi! Now take the reciprocal of both sides:  EMBED Equation.DSMT4  In the book are the steps that show you that for the reciprocal of x, we do have Phi! The ancient Greeks called this way of breaking up a line: The Golden Section. 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