Content Connections Continued Fractions (Rationales) and Radicals Objective To explore applications of continued fractions and radicals. Background In this project we are going to extend your knowledge of rationales and radicals into some neat things: Continued fractions (which are rationales) and continued radicals, in this case continued square roots. Below is an example of a continued fraction, and what happens if you take it to infinity. 2+ Now, to work these out we need to know one shortcut. We know the reciprocal of a number called "n" is equal to So that means that _ = the reciprocal of "n" must be true for every number "n", including if "n" is a FRACTION! So therefore: since _ is the reciprocal of ; Now, you try these three: (A) (B) (C) So let us work out the first continued fraction above to a few terms:Content Content Connections Connections Proof of diagram # 2 BA 1+- 1+ 59 1+ 35 Prove that the golden ratio, " "equals the continued fraction below: 2+ 75 If we multiply each side by 4 we get: 1+- 1+- 59 1 2 4X , 07 7 = 2.80952380952 ... 1+1+ ...... Let us call the left hand side of this continued fraction "x", so we now have: Obviously we will need many more terms to get close to the value of n. X = 1 + Now here is a neat, continued fraction, in fact it is the simplest continued fraction |1+ . because it just uses 1's. See diagram 2 below: 1+1+.... Starting with the right hand side of the first denominator, we notice that from there on is the same as the whole continued fraction on the right hand side. Therefore it must also equal * ", so we can rewrite the right hand side as : 1+ - 1 + 1 + . 1+ x Now multiply both sides by (1 + x), or what we call, "Cross-multiply", we get: x(1 + x) = 1 Which gives us 0.618033988749895... Or x + x2 = 1, Or x? + x -1 = 0, If we take this to infinity, and take its reciprocal, we get the value of the golden ratio, . and solving for "x" using the quadratic formula we get x = 0.6180339887 which is one form of the golden ratio: so therefore that continued fraction, which is the P = 7 - = 1.618033988749895.. simplest of all continued fractions just equals the golden ratio, " 0.618033988749895... Q. E. D. On the next page is a really neat proof for how we know this.O Content Content Connections Connections Here is a continued radical, this one by the great mathematician from India, Srinivasa Ramanujan (whose life was shown in the movie, "The Man Who Knew Infinity" with Dev Above we got the right-hand side of the golden ratio, let's see what we get here with the Patel playing the lead role): following proof 3= 1+2 143/1+ 414 .. Proof of diagram # 3 Prove that the golden ratio, "" equals the continued radical below: Exercise: (D) Work this out up to the number 4, in the above continued radical. D = 1 + 1 + V1 +... Let us call the left hand side of this continued radical "x", so we now have: x= 1+ 1+ 1 + V1+ ..... Starting with the right-hand side of the first square root, we notice that from there on is the same as the whole continued radical on the right hand side. Therefore it must also equal " x ". so we can rewrite the right hand side as : x = V1+ x Now, squaring both sides, we get: x? = V1+x Or: x? = 1+x Or x2 - x - 1 =0, and solving for "x" using the quadratic formula we get x = 1.6180339887 which is the other form of the golden ratio: so therefore that continued radical, which is the Now let's take the simplest of all continued radicals where we just use 1's: simplest of all continued radicals just equals the golden ratio, " We'll call this diagram 3 Q. E. D. 0 = 1 + 1 + 1 + v 1+ ..... Again, the golden ratio appears. There are two values that are usually assigned to the golden ratio. If you flip over any fraction (which is a ratio) you get another ratio between the two numbers, so the ratios 7:8 or 8:7 as fractions become or Now if I take the reciprocal of the golden ratio, I get the other form of the golden ratio: it is still golden!! 1.618033988749895 = 0.618033988749895