Step 1 Number the first 25 lines on your paper, (1,2,3…)

Step 1

Number the first 25 lines on your paper, (1,2,3…)

Step 2

Write any two whole numbers on the first two lines

Step 3

Add the two numbers and write the sum on the third line

Step 4

Add the last two numbers and write the sum on the next line

Continue this process

(add the last two, write the sum) until you have 25 numbers on your

list).

Select any number among the last five

on your list and divide it by the

number above it

Remember I do not know your original two numbers

or any of the 25 numbers on your sheet of paper

So I can’t know which of the last 5 numbers you have chosen to divide by the number above it

Now I need to concentrate on the number presently shown

on your calculator.If I close my eyes and think about your

number, I will be able to prove to you that I know what your number will be.

If you select any number between the last five (#21 to #25) and divides it by the number above it, the answer will always be1.618033989…, which just happens to

be the Golden mean! (provided, of course, you have done all the addition

correctly in steps 3-5 above)

It’s an incredible bit of mathematical trivia. Begin with any two whole numbers, make

a Fibonacci-type addition list, take the ratio of two consecutive entries, and the ratio approaches the Golden Mean!

The further out we go, the more accurate it becomes.

That’s why we need 25 numbers to obtain sufficient

accuracy.The proof requires familiarity with the

Fibonacci Sequence, pages of algebra, and a knowledge of limits, all of which go far beyond the scope of explanation.

If you divide one of your last five numbers by the next number (instead of the previous number), the result is the same decimal without the leading 1.

A B

Euclid of Alexandria (325 – 265 BC.)

In Book VI of the Elements, Euclid defined the "extreme and mean ratios" on a line segment. He wished to find the point (P) on line segment AB such that, the small segment is to the large segment as the large segment is to the whole segment.

P

In other words how far along the line is P such that:

ABPB

PBAP

We use a different approach to Euclid and use algebra to help us find this ratio, however the method is essentially the same.

1 Let AP be of unit length and PB = . Then we require such that

11

12

01 2 ...... 61803312

51

Solving this quadratic and taking the positive root. We get the irrational number shown.

is the Greek letter Phi.

Fibonacci Sequence

1716

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

3 3

4 6 4

55 10 10

6156 15 20

7 72121 35 35

88 285628 56 70

9 36 84 126 126 84 36 9

10 45 210 252 210 120 45120 10

11 55 330 462 462 330 165165 55 11

12 66 495 792 924 792 495220 220 66 12

13 78 286 715 1287 1716 1287 715 286 78 13

Add the numbers shown along each of the shallow diagonals to find another well known sequence of numbers.

1 2 3 8 13

21 55 89

1 5

34 144 233 377

The sequence first appears as a recreational maths problem about the growth in population of rabbits in book 3 of his famous work, Liber – abaci (the book of the calculator).

Fibonacci travelled extensively throughout the Middle East and elsewhere. He strongly recommended that Europeans adopt the Indo-Arabic system of numerals including the use of a symbol for zero “zephirum”

The Fibonacci Sequence

Leonardo of Pisa 1180 - 1250